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<!-- -->
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<!-- Main Page Body -->
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<p>I would like to thank my supervisor, Professor Johannes Knolle and
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co-supervisor Professor Derek Lee for guidance and support during this
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long process.</p>
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<p>Dan Hdidouan for being an example of how to weather the stress of a
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PhD with grace and kindness.</p>
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<p>I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process.</p>
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<p>Dan Hdidouan for being an example of how to weather the stress of a PhD with grace and kindness.</p>
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<p>Arnaud for help and guidance…</p>
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<p>Carolyn, Juraci, Ievgeniia and Loli for their patience and
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support.</p>
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<p>Carolyn, Juraci, Ievgeniia and Loli for their patience and support.</p>
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<p>Nina del Ser</p>
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<p>Brian Tam for his endless energy on our many many calls while we
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served as joint Postgraduate reps for the department.</p>
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<p>All the students in CMTH, Halvard, Tom, Chris, Krishnan, David,
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Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM
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in Munich, Alex and Rohit.</p>
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<p>Gino, Peru and Willian for their collaboration on the Amorphous
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Kitaev Model.</p>
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<p>Brian Tam for his endless energy on our many many calls while we served as joint Postgraduate reps for the department.</p>
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<p>All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM in Munich, Alex and Rohit.</p>
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<p>Gino, Peru and Willian for their collaboration on the Amorphous Kitaev Model.</p>
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<p>Mr Jeffries who encouraged me to pursue physics</p>
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<p>All the gang from Munich, Toni, Mine, Mike, Claudi.</p>
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<p>Dan Simpson, the poet in residence at Imperial and one of my
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favourite collaborators during my time at Imperial.</p>
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<p>Lou Khalfaoui for keeping me sane during the lockdown of March 2022.
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Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design
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expertise and patience.</p>
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<p>Dan Simpson, the poet in residence at Imperial and one of my favourite collaborators during my time at Imperial.</p>
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<p>Lou Khalfaoui for keeping me sane during the lockdown of March 2022. Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design expertise and patience.</p>
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<p>All the I-Stemm team, Katerina, Jeremey, John, ….</p>
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<p>And finally, I’d like the thank the staff of the Camberwell Public
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Library where the majority of this thesis was written.</p>
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<p>We thank Angus MacKinnon for helpful discussions, Sophie Nadel for
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input when preparing the figures.</p>
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<p>And finally, I’d like the thank the staff of the Camberwell Public Library where the majority of this thesis was written.</p>
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<p>We thank Angus MacKinnon for helpful discussions, Sophie Nadel for input when preparing the figures.</p>
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</main>
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@ -27,14 +27,12 @@ image:
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<br>
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<nav aria-label="Table of Contents" class="page-table-of-contents">
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<ul>
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<li><a href="#the-falikov-kimball-model"
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id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
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<li><a href="#chap:2-background" id="toc-chap:2-background">2 Background</a></li>
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<li><a href="#the-falikov-kimball-model" id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
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<ul>
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<li><a href="#the-model" id="toc-the-model">The Model</a></li>
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<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
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||||
Diagrams</a></li>
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||||
<li><a href="#long-ranged-ising-model"
|
||||
id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
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||||
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase Diagrams</a></li>
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||||
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
|
||||
</ul></li>
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||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
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@ -49,14 +47,12 @@ id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
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<!-- Table of Contents -->
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<!-- <nav id="TOC" role="doc-toc">
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||||
<ul>
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||||
<li><a href="#the-falikov-kimball-model"
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||||
id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
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||||
<li><a href="#chap:2-background" id="toc-chap:2-background">2 Background</a></li>
|
||||
<li><a href="#the-falikov-kimball-model" id="toc-the-falikov-kimball-model">The Falikov Kimball Model</a>
|
||||
<ul>
|
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<li><a href="#the-model" id="toc-the-model">The Model</a></li>
|
||||
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase
|
||||
Diagrams</a></li>
|
||||
<li><a href="#long-ranged-ising-model"
|
||||
id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
|
||||
<li><a href="#phase-diagrams" id="toc-phase-diagrams">Phase Diagrams</a></li>
|
||||
<li><a href="#long-ranged-ising-model" id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
|
||||
</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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||||
</ul>
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@ -68,688 +64,190 @@ id="toc-long-ranged-ising-model">Long Ranged Ising model</a></li>
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<p>2 Background</p>
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<hr />
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</div>
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<section id="chap:2-background" class="level1">
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<h1>2 Background</h1>
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</section>
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<section id="the-falikov-kimball-model" class="level1">
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<h1>The Falikov Kimball Model</h1>
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<section id="the-model" class="level2">
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<h2>The Model</h2>
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<p>The Falikov-Kimball (FK) model is one of the simplest models of the
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correlated electron problem. It captures the essence of the interaction
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between itinerant and localized electrons. It was originally introduced
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to explain the metal-insulator transition in f-electron systems but in
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its long history it has been interpreted variously as a model of
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electrons and ions, binary alloys or of crystal formation <span
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||||
class="citation"
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||||
data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a
|
||||
href="#ref-hubbardj.ElectronCorrelationsNarrow1963"
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role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006"
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role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span
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class="math inline">\(d_i\)</span> and light fermions <span
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class="math inline">\(c_i\)</span> and with chemical potential fixed at
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half-filling, the model reads</p>
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<p>The Falikov-Kimball (FK) model is one of the simplest models of the correlated electron problem. It captures the essence of the interaction between itinerant and localized electrons. It was originally introduced to explain the metal-insulator transition in f-electron systems but in its long history it has been interpreted variously as a model of electrons and ions, binary alloys or of crystal formation <span class="citation" data-cites="hubbardj.ElectronCorrelationsNarrow1963 falicovSimpleModelSemiconductorMetal1969 gruberFalicovKimballModelReview1996 gruberFalicovKimballModel2006"> [<a href="#ref-hubbardj.ElectronCorrelationsNarrow1963" role="doc-biblioref">1</a>–<a href="#ref-gruberFalicovKimballModel2006" role="doc-biblioref">4</a>]</span>. In terms of immobile fermions <span class="math inline">\(d_i\)</span> and light fermions <span class="math inline">\(c_i\)</span> and with chemical potential fixed at half-filling, the model reads</p>
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<p><span class="math display">\[\begin{aligned}
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H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} -
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\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle
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i,j\rangle} c^\dagger_{i}c_{j}.\\
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H_{\mathrm{FK}} = & \;U \sum_{i} (d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
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\end{aligned}\]</span></p>
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<p>Here we will only discuss the hypercubic lattices, i.e the chain, the
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square lattice, the cubic lattice and so on. The connection to the
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Hubbard model is that we have relabel the up and down spin electron
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states and removed the hopping term for one species, the equivalent of
|
||||
taking the limit of infinite mass ratio <span class="citation"
|
||||
data-cites="devriesSimplifiedHubbardModel1993"> [<a
|
||||
href="#ref-devriesSimplifiedHubbardModel1993"
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||||
role="doc-biblioref">5</a>]</span>.</p>
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<p>Like other exactly solvable models <span class="citation"
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||||
data-cites="smithDisorderFreeLocalization2017"> [<a
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||||
href="#ref-smithDisorderFreeLocalization2017"
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role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model
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||||
possesses extensively many conserved degrees of freedom <span
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||||
class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. The Hilbert
|
||||
space therefore breaks up into a set of sectors in which these operators
|
||||
take a definite value. Crucially, this reduces the interaction term
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<span class="math inline">\((d^\dagger_{i}d_{i} -
|
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\tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> from being
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quartic in fermion operators to quadratic. This is what makes the FK
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model exactly solvable, in contrast to the Hubbard model.</p>
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<p>Due to Pauli exclusion, maximum filling occurs when each lattice site
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is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle
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= 2\)</span>. Here we will focus on the half filled case <span
|
||||
class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground
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||||
state phenomenology as the model is doped away from the half-filled
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state can be rich <span class="citation"
|
||||
data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a
|
||||
href="#ref-jedrzejewskiFalicovKimballModels2001"
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role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990"
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role="doc-biblioref">8</a>]</span> but from this point we will only
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||||
consider the half-filled point.</p>
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<p>At half-filling and on bipartite lattices, FK the model is
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||||
particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with
|
||||
the particle hole operator <span
|
||||
class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a
|
||||
consequence the energy spectrum is symmetric about <span
|
||||
class="math inline">\(E = 0\)</span> and this is the Fermi energy. The
|
||||
particle hole operator corresponds to the substitution <span
|
||||
class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i,
|
||||
d^\dagger_i \rightarrow d_i\)</span> where <span
|
||||
class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
|
||||
<span class="math inline">\(-1\)</span> for the even sublattice <span
|
||||
class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
|
||||
href="#ref-gruberFalicovKimballModel2005"
|
||||
role="doc-biblioref">9</a>]</span>. The absence of a hopping term for
|
||||
the heavy electrons means they do not need the factor of <span
|
||||
class="math inline">\(\epsilon_i\)</span>. See appendix <a
|
||||
href="../6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html#particle-hole-symmetry">A.1</a>
|
||||
for a full derivation of the PH symmetry.</p>
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<div id="fig:simple_DOS" class="fignos">
|
||||
<p>Here we will only discuss the hypercubic lattices, i.e the chain, the square lattice, the cubic lattice and so on. The connection to the Hubbard model is that we have relabel the up and down spin electron states and removed the hopping term for one species, the equivalent of taking the limit of infinite mass ratio <span class="citation" data-cites="devriesSimplifiedHubbardModel1993"> [<a href="#ref-devriesSimplifiedHubbardModel1993" role="doc-biblioref">5</a>]</span>.</p>
|
||||
<p>Like other exactly solvable models <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">6</a>]</span> and the Kitaev Model, the FK model possesses extensively many conserved degrees of freedom <span class="math inline">\([d^\dagger_{i}d_{i}, H] = 0\)</span>. The Hilbert space therefore breaks up into a set of sectors in which these operators take a definite value. Crucially, this reduces the interaction term <span class="math inline">\((d^\dagger_{i}d_{i} - \tfrac{1}{2})\;(c^\dagger_{i}c_{i} - \tfrac{1}{2})\)</span> from being quartic in fermion operators to quadratic. This is what makes the FK model exactly solvable, in contrast to the Hubbard model.</p>
|
||||
<p>Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, <span class="math inline">\(\langle n_c + n_d \rangle = 2\)</span>. Here we will focus on the half filled case <span class="math inline">\(\langle n_c + n_d \rangle = 1\)</span>. The ground state phenomenology as the model is doped away from the half-filled state can be rich <span class="citation" data-cites="jedrzejewskiFalicovKimballModels2001 gruberGroundStatesSpinless1990"> [<a href="#ref-jedrzejewskiFalicovKimballModels2001" role="doc-biblioref">7</a>,<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">8</a>]</span> but from this point we will only consider the half-filled point.</p>
|
||||
<p>At half-filling and on bipartite lattices, FK the model is particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with the particle hole operator <span class="math inline">\(\mathcal{P}H\mathcal{P}^{-1} = -H\)</span>. As a consequence the energy spectrum is symmetric about <span class="math inline">\(E = 0\)</span> and this is the Fermi energy. The particle hole operator corresponds to the substitution <span class="math inline">\(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\)</span> where <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the even sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">9</a>]</span>. The absence of a hopping term for the heavy electrons means they do not need the factor of <span class="math inline">\(\epsilon_i\)</span>. See appendix <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">A.1</a> for a full derivation of the PH symmetry.</p>
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||||
<figure>
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||||
<img src="/assets/thesis/background_chapter/simple_DOS.svg"
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||||
data-short-caption="Cubic Lattice dispersion with disorder"
|
||||
style="width:100.0%"
|
||||
alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
|
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<figcaption aria-hidden="true"><span>Figure 1:</span> The dispersion
|
||||
(upper row) and density of states (lower row) obtained from a cubic
|
||||
lattice model <span class="math inline">\(H = \sum_{i} V_i
|
||||
c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle}
|
||||
c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With not external
|
||||
potential. (b) With a static charge density wave background <span
|
||||
class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density
|
||||
wave background with 2% binary disorder.</figcaption>
|
||||
<img src="/assets/thesis/background_chapter/simple_DOS.svg" id="fig:simple_DOS" data-short-caption="Cubic Lattice dispersion with disorder" style="width:100.0%" alt="Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j} in one dimension. (a) With not external potential. (b) With a static charge density wave background V_i = (-1)^i (c) A static charge density wave background with 2% binary disorder." />
|
||||
<figcaption aria-hidden="true">Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model <span class="math inline">\(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\)</span> in one dimension. (a) With not external potential. (b) With a static charge density wave background <span class="math inline">\(V_i = (-1)^i\)</span> (c) A static charge density wave background with 2% binary disorder.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>We will later add a long range interaction between the localised
|
||||
electrons at which point we will replace the immobile fermions with a
|
||||
classical Ising field <span class="math inline">\(S_i = 1 -
|
||||
2d^\dagger_id_i = \pm\tfrac{1}{2}\)</span> which I will refer to as the
|
||||
spins.</p>
|
||||
<p>We will later add a long range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field <span class="math inline">\(S_i = 1 - 2d^\dagger_id_i = \pm\tfrac{1}{2}\)</span> which I will refer to as the spins.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
|
||||
\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
|
||||
\end{aligned}\]</span></p>
|
||||
<p>The FK model can be solved exactly with dynamic mean field theory in
|
||||
the infinite dimensional <span class="citation"
|
||||
data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a
|
||||
href="#ref-antipovCriticalExponentsStrongly2014"
|
||||
role="doc-biblioref">10</a>–<a
|
||||
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
|
||||
role="doc-biblioref">13</a>]</span>.</p>
|
||||
<p>The FK model can be solved exactly with dynamic mean field theory in the infinite dimensional <span class="citation" data-cites="antipovCriticalExponentsStrongly2014 ribicNonlocalCorrelationsSpectral2016 freericksExactDynamicalMeanfield2003 herrmannNonequilibriumDynamicalCluster2016"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>–<a href="#ref-herrmannNonequilibriumDynamicalCluster2016" role="doc-biblioref">13</a>]</span>.</p>
|
||||
</section>
|
||||
<section id="phase-diagrams" class="level2">
|
||||
<h2>Phase Diagrams</h2>
|
||||
<div id="fig:fk_phase_diagram" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg"
|
||||
data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams"
|
||||
style="width:100.0%"
|
||||
alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> Schematic Phase
|
||||
diagram of the Falikov-Kimball model in dimensions greater than two. At
|
||||
low temperature the classical fermions (spins) settle into an ordered
|
||||
charge density wave state (antiferromagnetic state). The schematic
|
||||
diagram for the Hubbard model is the same. Reproduced from <span
|
||||
class="citation"
|
||||
data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a
|
||||
href="#ref-antipovCriticalExponentsStrongly2014"
|
||||
role="doc-biblioref">10</a>,<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">14</a>]</span></figcaption>
|
||||
<img src="/assets/thesis/background_chapter/fk_phase_diagram.svg" id="fig:fk_phase_diagram" data-short-caption="Fermi-Hubbard and Falikov-Kimball Temperatue-Interaction Phase Diagrams" style="width:100.0%" alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
|
||||
<figcaption aria-hidden="true">Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a href="#ref-antipovCriticalExponentsStrongly2014" role="doc-biblioref">10</a>,<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">14</a>]</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>In dimensions greater than one, the FK model exhibits a phase
|
||||
transition at some <span class="math inline">\(U\)</span> dependent
|
||||
critical temperature <span class="math inline">\(T_c(U)\)</span> to a
|
||||
low temperature ordered phase <span class="citation"
|
||||
data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||||
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||||
role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons
|
||||
this corresponds to them occupying only one of the two sublattices A and
|
||||
B this is known as a charge density wave (CDW) phase. In terms of spins
|
||||
this is an AFM phase.</p>
|
||||
<p>In the disordered region above <span
|
||||
class="math inline">\(T_c(U)\)</span> there are two insulating phases.
|
||||
For weak interactions <span class="math inline">\(U << t\)</span>,
|
||||
thermal fluctuations in the spins act as an effective disorder potential
|
||||
for the fermions, causing them to localise and giving rise to an
|
||||
Anderson insulating state <span class="citation"
|
||||
data-cites="andersonAbsenceDiffusionCertain1958"> [<a
|
||||
href="#ref-andersonAbsenceDiffusionCertain1958"
|
||||
role="doc-biblioref">16</a>]</span> which we will discuss more in
|
||||
section <a
|
||||
href="../2_Background/2.3_Disorder.html#bg-disorder-and-localisation">2.3</a>.
|
||||
For strong interactions <span class="math inline">\(U >>
|
||||
t\)</span>, the spins are not ordered but nevertheless their interaction
|
||||
with the electrons opens a gap, leading a Mott insulator analogous to
|
||||
that of the Hubbard model <span class="citation"
|
||||
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
|
||||
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
|
||||
role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>By contrast, in the one dimensional FK model there is no
|
||||
finite-temperature phase transition (FTPT) to an ordered CDW phase <span
|
||||
class="citation" data-cites="liebAbsenceMottTransition1968"> [<a
|
||||
href="#ref-liebAbsenceMottTransition1968"
|
||||
role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial
|
||||
for the physics of both localisation and FTPTs. In one dimension,
|
||||
disorder generally dominates: even the weakest disorder exponentially
|
||||
localises <em>all</em> single particle eigenstates. Only longer-range
|
||||
correlations of the disorder potential can potentially induce
|
||||
localisation-delocalisation transitions in one dimension <span
|
||||
class="citation"
|
||||
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
|
||||
href="#ref-aubryAnalyticityBreakingAnderson1980"
|
||||
role="doc-biblioref">19</a>–<a
|
||||
href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
|
||||
role="doc-biblioref">21</a>]</span>. Thermodynamically, short-range
|
||||
interactions cannot overcome thermal defects in one dimension which
|
||||
prevents ordered phases at non-zero temperature <span class="citation"
|
||||
data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
|
||||
href="#ref-goldshteinPurePointSpectrum1977"
|
||||
role="doc-biblioref">22</a>–<a
|
||||
href="#ref-kramerLocalizationTheoryExperiment1993"
|
||||
role="doc-biblioref">24</a>]</span>.</p>
|
||||
<p>However, the absence of an FTPT in the short ranged FK chain is far
|
||||
from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY)
|
||||
interaction mediated by the fermions <span class="citation"
|
||||
data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
|
||||
href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">25</a>–<a
|
||||
href="#ref-yosidaMagneticPropertiesCuMn1957"
|
||||
role="doc-biblioref">28</a>]</span> decays as <span
|
||||
class="math inline">\(r^{-1}\)</span> in one dimension <span
|
||||
class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a
|
||||
href="#ref-rusinCalculationRKKYRange2017"
|
||||
role="doc-biblioref">29</a>]</span>. This could in principle induce the
|
||||
necessary long-range interactions for the classical Ising background to
|
||||
order at low temperatures <span class="citation"
|
||||
data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">30</a>,<a
|
||||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||||
role="doc-biblioref">31</a>]</span>. However, Kennedy and Lieb
|
||||
established rigorously that at half-filling a CDW phase only exists at
|
||||
<span class="math inline">\(T = 0\)</span> for the one dimensional FK
|
||||
model <span class="citation"
|
||||
data-cites="kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">32</a>]</span>.</p>
|
||||
<p>Based on this primacy of dimensionality, we will go digging into the
|
||||
one dimensional case. In chapter <a
|
||||
href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#fk-model">3</a>
|
||||
we will construct a generalised one-dimensional FK model with long-range
|
||||
interactions which induces the otherwise forbidden CDW phase at non-zero
|
||||
temperature. To do this we will draw on theory of the Long Range Ising
|
||||
Model which is the subject of the next section.</p>
|
||||
<p>In dimensions greater than one, the FK model exhibits a phase transition at some <span class="math inline">\(U\)</span> dependent critical temperature <span class="math inline">\(T_c(U)\)</span> to a low temperature ordered phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">15</a>]</span>. In terms of the immobile electrons this corresponds to them occupying only one of the two sublattices A and B this is known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.</p>
|
||||
<p>In the disordered region above <span class="math inline">\(T_c(U)\)</span> there are two insulating phases. For weak interactions <span class="math inline">\(U << t\)</span>, thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating state <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">16</a>]</span> which we will discuss more in section <a href="../2_Background/2.4_Disorder.html#bg-disorder-and-localisation">2.3</a>. For strong interactions <span class="math inline">\(U >> t\)</span>, the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading a Mott insulator analogous to that of the Hubbard model <span class="citation" data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a href="#ref-brandtThermodynamicsCorrelationFunctions1989" role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>By contrast, in the one dimensional FK model there is no finite-temperature phase transition (FTPT) to an ordered CDW phase <span class="citation" data-cites="liebAbsenceMottTransition1968"> [<a href="#ref-liebAbsenceMottTransition1968" role="doc-biblioref">18</a>]</span>. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In one dimension, disorder generally dominates: even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in one dimension <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">19</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">21</a>]</span>. Thermodynamically, short-range interactions cannot overcome thermal defects in one dimension which prevents ordered phases at non-zero temperature <span class="citation" data-cites="goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">22</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">24</a>]</span>.</p>
|
||||
<p>However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">25</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">28</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in one dimension <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">29</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>,<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">31</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the one dimensional FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">32</a>]</span>.</p>
|
||||
<p>Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3</a> we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.</p>
|
||||
</section>
|
||||
<section id="long-ranged-ising-model" class="level2">
|
||||
<h2>Long Ranged Ising model</h2>
|
||||
<p>The suppression of phase transitions is a common phenomena in one
|
||||
dimensional systems and the Ising model serves as a great illustration
|
||||
of this. In terms of classical spins <span class="math inline">\(S_i =
|
||||
\pm \frac{1}{2}\)</span> the standard Ising model reads</p>
|
||||
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij
|
||||
\rangle} S_i S_j\]</span></p>
|
||||
<p>Like the FK model, the Ising model shows an FTPT to an ordered state
|
||||
only in two dimensions and above. This can be understood via Peierls’
|
||||
argument <span class="citation"
|
||||
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||||
role="doc-biblioref">31</a>,<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">32</a>]</span> to be a consequence of the low
|
||||
energy penalty for domain walls in one dimensional systems.</p>
|
||||
<p>Following Peierls’ argument, consider the difference in free energy
|
||||
<span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span>
|
||||
between an ordered state and a state with single domain wall in a
|
||||
discrete order parameter. If this value is negative it implies that the
|
||||
ordered state is unstable with respect to domain wall defects, and they
|
||||
will thus proliferate, destroying the ordered phase. If we consider the
|
||||
scaling of the two terms with system size <span
|
||||
class="math inline">\(L\)</span> we see that short range interactions
|
||||
produce a constant energy penalty <span class="math inline">\(\Delta
|
||||
E\)</span> for a domain wall. In contrast, the number of such single
|
||||
domain wall states scales linearly with system size so the entropy is
|
||||
<span class="math inline">\(\propto \ln L\)</span>. Thus the entropic
|
||||
contribution dominates (eventually) in the thermodynamic limit and no
|
||||
finite temperature order is possible. In two dimensions and above, the
|
||||
energy penalty of a domain wall scales like <span
|
||||
class="math inline">\(L^{d-1}\)</span> which is why they can support
|
||||
ordered phases. This argument does not quite apply to the FK model
|
||||
because of the aforementioned RKKY interaction. Instead this argument
|
||||
will give us insight into how to recover an ordered phase in the one
|
||||
dimensional FK model.</p>
|
||||
<p>In contrast the long range Ising (LRI) model <span
|
||||
class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one
|
||||
dimension.</p>
|
||||
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
|
||||
S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
|
||||
<p>Renormalisation group analyses show that the LRI model has an ordered
|
||||
phase in 1D for <span class="math inline">\(1 < \alpha <
|
||||
2\)</span> <span class="citation"
|
||||
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
|
||||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
|
||||
role="doc-biblioref">33</a>]</span>. Peierls’ argument can be
|
||||
extended <span class="citation"
|
||||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">30</a>]</span> to long range interactions to
|
||||
provide intuition for why this is the case. Again considering the energy
|
||||
difference between the ordered state <span
|
||||
class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span>
|
||||
and a domain wall state <span
|
||||
class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>.
|
||||
In the case of the LRI model, careful counting shows that this energy
|
||||
penalty is <span class="math display">\[\Delta E \propto
|
||||
\sum_{n=1}^{\infty} n J(n)\]</span></p>
|
||||
<p>because each interaction between spins separated across the domain by
|
||||
a bond length <span class="math inline">\(n\)</span> can be drawn
|
||||
between <span class="math inline">\(n\)</span> equivalent pairs of
|
||||
sites. The behaviour then depends crucially on the sum scales with
|
||||
system size. Ruelle proved rigorously for a very general class of 1D
|
||||
systems, that if <span class="math inline">\(\Delta E\)</span> or its
|
||||
many-body generalisation converges to a constant in the thermodynamic
|
||||
limit then the free energy is analytic <span class="citation"
|
||||
data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a
|
||||
href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
|
||||
role="doc-biblioref">34</a>]</span>. This rules out a finite order phase
|
||||
transition, though not one of the Kosterlitz-Thouless type. Dyson also
|
||||
proves this though with a slightly different condition on <span
|
||||
class="math inline">\(J(n)\)</span> <span class="citation"
|
||||
data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a
|
||||
href="#ref-dysonExistencePhasetransitionOnedimensional1969"
|
||||
role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
|
||||
there are a few cases to consider:</p>
|
||||
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite
|
||||
range interactions, the Ising model is exactly solvable and mean field
|
||||
theory is exact <span class="citation"
|
||||
data-cites="lipkinValidityManybodyApproximation1965"> [<a
|
||||
href="#ref-lipkinValidityManybodyApproximation1965"
|
||||
role="doc-biblioref">35</a>]</span>. This limit is the same as the
|
||||
infinite dimensional limit.</p>
|
||||
<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very
|
||||
slowly decaying interactions. <span class="math inline">\(\Delta
|
||||
E\)</span> does not converge as a function of system size so the
|
||||
Hamiltonian is non-extensive, a topic not without some considerable
|
||||
controversy <span class="citation"
|
||||
data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a
|
||||
href="#ref-grossNonextensiveHamiltonianSystems2002"
|
||||
role="doc-biblioref">36</a>–<a
|
||||
href="#ref-wangCommentNonextensiveHamiltonian2003"
|
||||
role="doc-biblioref">38</a>]</span> that we will not consider further
|
||||
here.</p>
|
||||
<p>For <span class="math inline">\(1 < \alpha < 2\)</span>, we get
|
||||
a phase transition to an ordered state at a finite temperature, this is
|
||||
what we want!</p>
|
||||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of
|
||||
domain walls diverges logarithmically, and this turns out to be a
|
||||
Kostelitz-Thouless transition <span class="citation"
|
||||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">30</a>]</span>.</p>
|
||||
<p>Finally, for <span class="math inline">\(2 < \alpha\)</span> we
|
||||
have very quickly decaying interactions and domain walls again have a
|
||||
finite energy penalty, hence Peirels’ argument holds and there is no
|
||||
phase transition.</p>
|
||||
<p>One final complexity is that for <span
|
||||
class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span>
|
||||
renormalisation group methods show that the critical point has
|
||||
non-universal critical exponents that depend on <span
|
||||
class="math inline">\(\alpha\)</span> <span class="citation"
|
||||
data-cites="fisherCriticalExponentsLongRange1972"> [<a
|
||||
href="#ref-fisherCriticalExponentsLongRange1972"
|
||||
role="doc-biblioref">39</a>]</span>. To avoid this potential confounding
|
||||
factors we will park ourselves at <span class="math inline">\(\alpha =
|
||||
1.25\)</span> when we apply these ideas to the FK model.</p>
|
||||
<p>Were we to extend this to arbitrary dimension <span
|
||||
class="math inline">\(d\)</span> we would find that thermodynamics
|
||||
properties generally both <span class="math inline">\(d\)</span> and
|
||||
<span class="math inline">\(\alpha\)</span>, long range interactions can
|
||||
modify the ‘effective dimension’ of thermodynamic systems <span
|
||||
class="citation"
|
||||
data-cites="angeliniRelationsShortrangeLongrange2014"> [<a
|
||||
href="#ref-angeliniRelationsShortrangeLongrange2014"
|
||||
role="doc-biblioref">40</a>]</span>.</p>
|
||||
<div id="fig:alpha_diagram" class="fignos">
|
||||
<p>The suppression of phase transitions is a common phenomena in one dimensional systems and the Ising model serves as a great illustration of this. In terms of classical spins <span class="math inline">\(S_i = \pm \frac{1}{2}\)</span> the standard Ising model reads</p>
|
||||
<p><span class="math display">\[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]</span></p>
|
||||
<p>Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">31</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">32</a>]</span> to be a consequence of the low energy penalty for domain walls in one dimensional systems.</p>
|
||||
<p>Following Peierls’ argument, consider the difference in free energy <span class="math inline">\(\Delta F = \Delta E - T\Delta S\)</span> between an ordered state and a state with single domain wall in a discrete order parameter. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size <span class="math inline">\(L\)</span> we see that short range interactions produce a constant energy penalty <span class="math inline">\(\Delta E\)</span> for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is <span class="math inline">\(\propto \ln L\)</span>. Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In two dimensions and above, the energy penalty of a domain wall scales like <span class="math inline">\(L^{d-1}\)</span> which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the one dimensional FK model.</p>
|
||||
<p>In contrast the long range Ising (LRI) model <span class="math inline">\(H_{\mathrm{LRI}}\)</span> can have an FTPT in one dimension.</p>
|
||||
<p><span class="math display">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]</span></p>
|
||||
<p>Renormalisation group analyses show that the LRI model has an ordered phase in 1D for <span class="math inline">\(1 < \alpha < 2\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">33</a>]</span>. Peierls’ argument can be extended <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>]</span> to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state <span class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span> and a domain wall state <span class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\)</span>. In the case of the LRI model, careful counting shows that this energy penalty is <span class="math display">\[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\]</span></p>
|
||||
<p>because each interaction between spins separated across the domain by a bond length <span class="math inline">\(n\)</span> can be drawn between <span class="math inline">\(n\)</span> equivalent pairs of sites. The behaviour then depends crucially on the sum scales with system size. Ruelle proved rigorously for a very general class of 1D systems, that if <span class="math inline">\(\Delta E\)</span> or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968"> [<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">34</a>]</span>. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on <span class="math inline">\(J(n)\)</span> <span class="citation" data-cites="dysonExistencePhasetransitionOnedimensional1969"> [<a href="#ref-dysonExistencePhasetransitionOnedimensional1969" role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>, there are a few cases to consider:</p>
|
||||
<p>For <span class="math inline">\(\alpha = 0\)</span> i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact <span class="citation" data-cites="lipkinValidityManybodyApproximation1965"> [<a href="#ref-lipkinValidityManybodyApproximation1965" role="doc-biblioref">35</a>]</span>. This limit is the same as the infinite dimensional limit.</p>
|
||||
<p>For <span class="math inline">\(\alpha \leq 1\)</span> we have very slowly decaying interactions. <span class="math inline">\(\Delta E\)</span> does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy <span class="citation" data-cites="grossNonextensiveHamiltonianSystems2002 lutskoQuestioningValidityNonextensive2011 wangCommentNonextensiveHamiltonian2003"> [<a href="#ref-grossNonextensiveHamiltonianSystems2002" role="doc-biblioref">36</a>–<a href="#ref-wangCommentNonextensiveHamiltonian2003" role="doc-biblioref">38</a>]</span> that we will not consider further here.</p>
|
||||
<p>For <span class="math inline">\(1 < \alpha < 2\)</span>, we get a phase transition to an ordered state at a finite temperature, this is what we want!</p>
|
||||
<p>For <span class="math inline">\(\alpha = 2\)</span>, the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">30</a>]</span>.</p>
|
||||
<p>Finally, for <span class="math inline">\(2 < \alpha\)</span> we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.</p>
|
||||
<p>One final complexity is that for <span class="math inline">\(\tfrac{3}{2} < \alpha < 2\)</span> renormalisation group methods show that the critical point has non-universal critical exponents that depend on <span class="math inline">\(\alpha\)</span> <span class="citation" data-cites="fisherCriticalExponentsLongRange1972"> [<a href="#ref-fisherCriticalExponentsLongRange1972" role="doc-biblioref">39</a>]</span>. To avoid this potential confounding factors we will park ourselves at <span class="math inline">\(\alpha = 1.25\)</span> when we apply these ideas to the FK model.</p>
|
||||
<p>Were we to extend this to arbitrary dimension <span class="math inline">\(d\)</span> we would find that thermodynamics properties generally both <span class="math inline">\(d\)</span> and <span class="math inline">\(\alpha\)</span>, long range interactions can modify the ‘effective dimension’ of thermodynamic systems <span class="citation" data-cites="angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">40</a>]</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/background_chapter/alpha_diagram.svg"
|
||||
data-short-caption="Long Range Ising Model Behaviour"
|
||||
style="width:100.0%"
|
||||
alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> The thermodynamic
|
||||
behaviour of the long range Ising model <span
|
||||
class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i -
|
||||
j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span
|
||||
class="math inline">\(\alpha\)</span> is varied.</figcaption>
|
||||
<img src="/assets/thesis/background_chapter/alpha_diagram.svg" id="fig:alpha_diagram" data-short-caption="Long Range Ising Model Behaviour" style="width:100.0%" alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
|
||||
<figcaption aria-hidden="true">Figure 3: The thermodynamic behaviour of the long range Ising model <span class="math inline">\(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\)</span> as the exponent of the interaction <span class="math inline">\(\alpha\)</span> is varied.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev
|
||||
Honeycomb Model</a></p>
|
||||
<p>Next Section: <a href="../2_Background/2.2_HKM_Model.html">The Kitaev Honeycomb Model</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
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|
||||
<div class="csl-left-margin">[40] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -27,19 +27,26 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#the-kitaev-honeycomb-model"
|
||||
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
|
||||
<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
|
||||
<ul>
|
||||
<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Model</a></li>
|
||||
<li><a href="#a-mapping-to-majorana-fermions"
|
||||
id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana
|
||||
Fermions</a></li>
|
||||
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a></li>
|
||||
<li><a href="#anyons-topology-and-the-chern-number"
|
||||
id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the
|
||||
Chern number</a></li>
|
||||
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</a></li>
|
||||
<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
|
||||
<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
|
||||
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
|
||||
<li><a href="#exact-solvability" id="toc-exact-solvability">Exact Solvability</a></li>
|
||||
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
|
||||
<li><a href="#mapping-to-a-majorana-hamiltonian" id="toc-mapping-to-a-majorana-hamiltonian">Mapping to a Majorana Hamiltonian</a>
|
||||
<ul>
|
||||
<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single spin</a></li>
|
||||
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors" id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning the Hilbert Space into Bond sectors</a></li>
|
||||
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace" id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping back from Bond Sectors to the Physical Subspace</a></li>
|
||||
<li><a href="#open-boundary-conditions" id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#emergent-gauge-fields" id="toc-emergent-gauge-fields">Emergent gauge fields</a>
|
||||
<ul>
|
||||
<li><a href="#ground-state-degeneracy" id="toc-ground-state-degeneracy">Ground State Degeneracy</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bg-the-ground-state" id="toc-bg-the-ground-state">The Ground State</a></li>
|
||||
<li><a href="#phases-of-the-kitaev-model" id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -54,19 +61,26 @@ Diagram</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#the-kitaev-honeycomb-model"
|
||||
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
|
||||
<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
|
||||
<ul>
|
||||
<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Model</a></li>
|
||||
<li><a href="#a-mapping-to-majorana-fermions"
|
||||
id="toc-a-mapping-to-majorana-fermions">A mapping to Majorana
|
||||
Fermions</a></li>
|
||||
<li><a href="#gauge-fields" id="toc-gauge-fields">Gauge Fields</a></li>
|
||||
<li><a href="#anyons-topology-and-the-chern-number"
|
||||
id="toc-anyons-topology-and-the-chern-number">Anyons, Topology and the
|
||||
Chern number</a></li>
|
||||
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</a></li>
|
||||
<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
|
||||
<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
|
||||
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
|
||||
<li><a href="#exact-solvability" id="toc-exact-solvability">Exact Solvability</a></li>
|
||||
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
|
||||
<li><a href="#mapping-to-a-majorana-hamiltonian" id="toc-mapping-to-a-majorana-hamiltonian">Mapping to a Majorana Hamiltonian</a>
|
||||
<ul>
|
||||
<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single spin</a></li>
|
||||
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors" id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning the Hilbert Space into Bond sectors</a></li>
|
||||
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace" id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping back from Bond Sectors to the Physical Subspace</a></li>
|
||||
<li><a href="#open-boundary-conditions" id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#emergent-gauge-fields" id="toc-emergent-gauge-fields">Emergent gauge fields</a>
|
||||
<ul>
|
||||
<li><a href="#ground-state-degeneracy" id="toc-ground-state-degeneracy">Ground State Degeneracy</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bg-the-ground-state" id="toc-bg-the-ground-state">The Ground State</a></li>
|
||||
<li><a href="#phases-of-the-kitaev-model" id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -78,82 +92,257 @@ Diagram</a></li>
|
||||
<p>2 Background</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="the-kitaev-honeycomb-model" class="level1">
|
||||
<section id="bg-hkm-model" class="level1">
|
||||
<h1>The Kitaev Honeycomb Model</h1>
|
||||
<p><strong>papers</strong> Jos on dynamics
|
||||
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p>
|
||||
<p><strong>intro</strong> - strong spin orbit coupling leads to
|
||||
anisotropic spin exchange (as opposed to isotropic exchange like the
|
||||
Heisenberg model) - geometrical frustration leads to QSL ground state
|
||||
with long range entanglement (not simple paramagnet)</p>
|
||||
<ul>
|
||||
<li>RuCl_3 is the classic QSL candidate material</li>
|
||||
<li>really follows the Kitaev-Heisenberg model</li>
|
||||
<li>experimental probes include inelastic neutron scattering, Raman
|
||||
scattering</li>
|
||||
</ul>
|
||||
<section id="bg-hkm-model" class="level2">
|
||||
<h2>The Model</h2>
|
||||
<div id="fig:intro_figure_by_hand" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg"
|
||||
data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%"
|
||||
alt="Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span>
|
||||
<strong>(a)</strong> The standard Kitaev model is defined on a honeycomb
|
||||
lattice. The special feature of the honeycomb lattice that makes the
|
||||
model solvable is that each vertex is joined by exactly three bonds,
|
||||
i.e. the lattice is trivalent. One of three labels is assigned to each
|
||||
<strong>(b)</strong>. We represent the antisymmetric gauge degree of
|
||||
freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> with arrows
|
||||
that point in the direction <span class="math inline">\(u_{jk} =
|
||||
+1\)</span> <strong>(c)</strong>. The Majorana transformation can be
|
||||
visualised as breaking each spin into four Majoranas which then pair
|
||||
along the bonds. The pairs of x,y and z Majoranas become part of the
|
||||
classical <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field
|
||||
<span class="math inline">\(u_{ij}\)</span>. This leavies a single
|
||||
Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig:intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom u_{jk} = \pm 1 that lives on each bond, the bond variables. These are antisymmetric, u_{jk} = -u_{kj}, so we represent them graphically with arrows on each bond that point in the direction that u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas b_i^x,\;b_i^y,\;b_i^z,\;c_i. The x, y and z Majoranas then pair along the bonds forming conserved \mathbb{Z}_2 bond operators u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle. The remaining c_i operators form an effective quadratic Hamiltonian H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j." />
|
||||
<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. <strong>(b)</strong>. After transforming to the Majorana representation we get an emergent gauge degree of freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> that lives on each bond, the bond variables. These are antisymmetric, <span class="math inline">\(u_{jk} = -u_{kj}\)</span>, so we represent them graphically with arrows on each bond that point in the direction that <span class="math inline">\(u_{jk} = +1\)</span> <strong>(c)</strong>. The Majorana transformation can be visualised as breaking each spin into four Majoranas <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span>. The x, y and z Majoranas then pair along the bonds forming conserved <span class="math inline">\(\mathbb{Z}_2\)</span> bond operators <span class="math inline">\(u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle\)</span>. The remaining <span class="math inline">\(c_i\)</span> operators form an effective quadratic Hamiltonian <span class="math inline">\(H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span>.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<section id="the-spin-hamiltonian" class="level2">
|
||||
<h2>The Spin Hamiltonian</h2>
|
||||
<p>This section introduces the seminal Kitaev honeycomb (KH) model. The KH model is an exactly solvable microscopic model of interacting spin<span class="math inline">\(-1/2\)</span> spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label <span class="math inline">\(\alpha \in \{ x, y, z\}\)</span> and that bond couple two spins along the <span class="math inline">\(\alpha\)</span> axis. See fig. <a href="#fig:intro_figure_by_hand">1</a> for a diagram.</p>
|
||||
<p>This gives us the Hamiltonian <span id="eq:bg-kh-model"><span class="math display">\[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + \Gamma \mathrm{three spin term}, \qquad{(1)}\]</span></span> where <span class="math inline">\(\sigma^\alpha_j\)</span> is a Pauli matrix acting on site <span class="math inline">\(j\)</span> and <span class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of nearest-neighbour indices connected by an <span class="math inline">\(\alpha\)</span>-bond with exchange coupling <span class="math inline">\(J^\alpha\)</span> <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The Kitaev Honeycomb model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span> can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-Jackeli2009" role="doc-biblioref">2</a>–<a href="#ref-Takagi2019" role="doc-biblioref">6</a>]</span>. The model is highly frustrated: energetically each spin would like to align along a different direction with each of its three neighbours. This cannot be achieved even classically. This frustration leads the the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">7</a>]</span>, the KH model was one of the first concrete models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. At finite temperature the model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">8</a>]</span>. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field and the remaining fermions are governed by a free fermion hamiltonian.</p>
|
||||
<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="../2_Background/2.2_HKM_Model.html#emergent-gauge-fields">emergent gauge fields</a> and the projector. We will then discuss the <a href="../2_Background/2.2_HKM_Model.html#bg-the-ground-state">ground state</a> found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.</p>
|
||||
<p>The <a href="../2_Background/2.3_Anyons.html#anyonic-statistics">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics.</p>
|
||||
</section>
|
||||
<section id="the-spin-model" class="level2">
|
||||
<h2>The Spin Model</h2>
|
||||
<p>Eq. <a href="#eq:bg-kh-model">1</a> shows</p>
|
||||
</section>
|
||||
<section id="the-majorana-model" class="level2">
|
||||
<h2>The Majorana Model</h2>
|
||||
<p>The Kitaev Honeycomb model <span class="math display">\[H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\]</span> is remarkable because it combines three key properties.</p>
|
||||
<p>First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as <span class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small corrections <span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">9</a>]</span>.</p>
|
||||
<p><strong>expand later: Why do we need spin orbit coupling and what will the corrections be?</strong></p>
|
||||
<p>Second, its ground state is the canonical example of the long sought after quantum spin liquid state. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations <span class="citation" data-cites="freedmanTopologicalQuantumComputation2003"> [<a href="#ref-freedmanTopologicalQuantumComputation2003" role="doc-biblioref">10</a>]</span>.</p>
|
||||
<p>Third, and perhaps most importantly, this model is a rare many body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many body ground states in terms of single particle states <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the Kitaev Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
|
||||
<p>“dynamical two spin correlation functions are identically zero beyond nearest neighbor separation in the Kitaev Model” <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">11</a>]</span></p>
|
||||
<ul>
|
||||
<li>strong spin orbit coupling yields spatial anisotropic spin exchange
|
||||
leading to compass models <span class="citation"
|
||||
data-cites="kugelJahnTellerEffectMagnetism1982"> [<a
|
||||
href="#ref-kugelJahnTellerEffectMagnetism1982"
|
||||
role="doc-biblioref">1</a>]</span></li>
|
||||
<li>strong spin orbit coupling yields spatial anisotropic spin exchange leading to compass models <span class="citation" data-cites="kugelJahnTellerEffectMagnetism1982"> [<a href="#ref-kugelJahnTellerEffectMagnetism1982" role="doc-biblioref">12</a>]</span></li>
|
||||
<li>spin model of the Kitaev model is one</li>
|
||||
<li>has extensively many conserved fluxes</li>
|
||||
<li></li>
|
||||
</ul>
|
||||
<p><strong>intro</strong> - strong spin orbit coupling leads to anisotropic spin exchange (as opposed to isotropic exchange like the Heisenberg model) - geometrical frustration leads to QSL ground state with long range entanglement (not simple paramagnet)</p>
|
||||
<ul>
|
||||
<li>RuCl_3 is the classic QSL candidate material</li>
|
||||
<li>really follows the Kitaev-Heisenberg model</li>
|
||||
<li>experimental probes include inelastic neutron scattering, Raman scattering</li>
|
||||
</ul>
|
||||
</section>
|
||||
<section id="a-mapping-to-majorana-fermions" class="level2">
|
||||
<h2>A mapping to Majorana Fermions</h2>
|
||||
<section id="exact-solvability" class="level2">
|
||||
<h2>Exact Solvability</h2>
|
||||
<p>For notational brevity, it is useful to introduce the spin bond operators <span class="math inline">\(K_{ij} = \sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span class="math inline">\(\alpha\)</span> is a function of <span class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
|
||||
<p>This Kitaev model has a set of conserved quantities that, in the spin language, take the form of Wilson loop operators <span class="math inline">\(W_p\)</span> winding around a closed path on the lattice. The direction does not matter, but we will keep to clockwise here. We will use the term plaquette and the symbol <span class="math inline">\(\phi\)</span> to refer to a Wilson loop operator that does not enclose any other sites, such as a single hexagon in a honeycomb lattice.</p>
|
||||
<p><span class="math display">\[W_p = \prod_{\mathrm{i,j}\; \in\; p} K_{ij}\]</span></p>
|
||||
<p>In closed loops, each site appears twice in the product with two of the three bond types. Applying <span class="math inline">\(\sigma^\alpha \sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq \beta\)</span> then gives us a product containing a single Pauli matrix associated with each site in the loop with the type of the <em>outward</em> pointing bond. Hence the <span class="math inline">\(W_p\)</span> associated with hexagons or shapes with an even number of sides all square to 1 and, hence, have eigenvalues <span class="math inline">\(\pm 1\)</span>. As the honeycomb lattice is bipartite, there are no closed loops that contain an odd number of edges. On other lattices <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">13</a>]</span>, plaquettes with an odd number of sides (odd plaquettes) have eigenvalues <span class="math inline">\(\pm i\)</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg" id="fig:regular_plaquettes" data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%" alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." />
|
||||
<figcaption aria-hidden="true">Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path.</figcaption>
|
||||
</figure>
|
||||
<p>Remarkably, all of the spin bond operators <span class="math inline">\(K_{ij}\)</span> commute with all the Wilson loop operators <span class="math inline">\(W_p\)</span>. <span class="math display">\[[W_p, K_{ij}] = 0\]</span> We can prove this by considering three cases: 1. neither <span class="math inline">\(i\)</span> nor <span class="math inline">\(j\)</span> is part of the loop 2. one of <span class="math inline">\(i\)</span> or <span class="math inline">\(j\)</span> are part of the loop 3. both are part of the loop</p>
|
||||
<p>The first case is trivial. The other two require some algebra, outlined in fig. <strong>¿fig:visual_kitaev_2?</strong>.</p>
|
||||
</section>
|
||||
<section id="gauge-fields" class="level2">
|
||||
<h2>Gauge Fields</h2>
|
||||
<section id="glossary" class="level2">
|
||||
<h2>Glossary</h2>
|
||||
<ul>
|
||||
<li><p>Lattice: The underlying graph on which the models are defined. Composed of sites (vertices), bonds (edges) and plaquettes (faces).</p></li>
|
||||
<li><p>The model : Used when I refer to properties of the the Kitaev model that do not depend on the particular lattice.</p></li>
|
||||
<li><p>The Honeycomb model : The Kitaev Model defined on the honeycomb lattice.</p></li>
|
||||
<li><p>The Amorphous model : The Kitaev Model defined on the amorphous lattices described here.</p></li>
|
||||
</ul>
|
||||
<p><strong>The Spin Hamiltonian</strong></p>
|
||||
<ul>
|
||||
<li>Spin Bond Operators: <span class="math inline">\(\hat{k}_{ij} = \sigma_i^\alpha \sigma_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} k_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
</ul>
|
||||
<p><strong>The Majorana Model</strong></p>
|
||||
<ul>
|
||||
<li>Majorana Operators on site <span class="math inline">\(i\)</span>: <span class="math inline">\(\hat{b}^x_i, \hat{b}^y_i, \hat{b}^z_i, \hat{c}_i\)</span></li>
|
||||
<li>Majorana Bond Operators: <span class="math inline">\(\hat{u}_{ij} = i b_i^\alpha b_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} u_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
<li>Gauge Operators: <span class="math inline">\(D_i = \hat{b}^x_i \hat{b}^y_i \hat{b}^z_i \hat{c}_i\)</span></li>
|
||||
<li>The Extended Hilbert space: The larger Hilbert space spanned by the Majorana operators.</li>
|
||||
<li>The physical subspace: The subspace of the extended Hilbert space that we identify with the Hilbert space of the original spin model.</li>
|
||||
<li>The Projector <span class="math inline">\(\hat{P}\)</span>: The projector onto the physical subspace.</li>
|
||||
</ul>
|
||||
<p><strong>Flux Sectors</strong></p>
|
||||
<ul>
|
||||
<li><p>Odd/Even Plaquettes: Plaquettes with an odd/even number of sides.</p></li>
|
||||
<li><p>Fluxes <span class="math inline">\(\phi_i\)</span>: The expectation values of the plaquette operators <span class="math inline">\(\pm 1\)</span> for even and <span class="math inline">\(\pm i\)</span> for odd plaquettes.</p></li>
|
||||
<li><p>Flux Sector: A subspace of Hilbert space in which the fluxes take particular values.</p></li>
|
||||
<li><p>Ground state flux sector: The Flux Sector containing the lowest energy many body state.</p></li>
|
||||
<li><p>Vortices: Flux excitations away from the ground state flux sector.</p></li>
|
||||
<li><p>Dual Loops: A set of <span class="math inline">\(u_{jk}\)</span> that correspond to loops on the dual lattice.</p></li>
|
||||
<li><p>non-contractible loops or dual loops: The two loops topologically distinct loops on the torus that cannot be smoothly deformed to a point.</p></li>
|
||||
<li><p>Topological Fluxes <span class="math inline">\(\Phi_{x}, \Phi_{y}\)</span>: The two fluxes associated with the two non-contractible loops.</p></li>
|
||||
<li><p>Topological Transport Operators: <span class="math inline">\(\mathcal{T}_{x}, \mathcal{T}_{y}\)</span>: The two vortex-pair operations associated with the non-contractible <em>dual</em> loops.</p></li>
|
||||
</ul>
|
||||
<p><strong>Phases</strong></p>
|
||||
<ul>
|
||||
<li>The A phase: The three anisotropic regions of the phase diagram <span class="math inline">\(A_x, A_y, A_z\)</span> where <span class="math inline">\(A_\alpha\)</span> means <span class="math inline">\(J_\alpha >> J_\beta, J_\gamma\)</span>.</li>
|
||||
<li>The B phase: The roughly isotropic region of the phase diagram.</li>
|
||||
</ul>
|
||||
<p><img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig:visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="A visual introduction to the Kitaev Model." /> <img src="/assets/thesis/amk_chapter/visual_kitaev_2.svg" id="fig:visual_kitaev_2" data-short-caption="Plaquette Operators are Conserved" style="width:100.0%" alt="Plaquette operators are conserved." /></p>
|
||||
<p>Since the Hamiltonian is a linear combination of bond operators, it commutes with the plaquette operators. This is helpful because it leads to a simultaneous eigenbasis for the Hamiltonian and the plaquette operators. We can, thus, work in <em>or “on”???</em> a basis in which the eigenvalues of the plaquette operators take on a definite value and, for all intents and purposes, act like classical degrees of freedom. These are the extensively many conserved quantities that make the model tractable.</p>
|
||||
<p>Plaquette operators measure flux. We will find that the ground state of the model corresponds to some particular choice of flux through each plaquette. We will refer to excitations which flip the expectation value of a plaquette operator away from the ground state as <strong>vortices</strong>.</p>
|
||||
<p>Thus, fixing a configuration of the vortices partitions the many-body Hilbert space into a set of ‘vortex sectors’ labelled by that particular flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm i\)</span>.</p>
|
||||
</section>
|
||||
<section id="anyons-topology-and-the-chern-number" class="level2">
|
||||
<h2>Anyons, Topology and the Chern number</h2>
|
||||
<section id="mapping-to-a-majorana-hamiltonian" class="level2">
|
||||
<h2>Mapping to a Majorana Hamiltonian</h2>
|
||||
<section id="for-a-single-spin" class="level3">
|
||||
<h3>For a single spin</h3>
|
||||
<p>Let us start by considering only one site and its <span class="math inline">\(\sigma^x, \sigma^y\)</span> and <span class="math inline">\(\sigma^z\)</span> operators which live in a two dimensional Hilbert space <span class="math inline">\(\mathcal{L}\)</span>.</p>
|
||||
<p>We will introduce two fermionic modes <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> that satisfy the canonical anticommutation relations along with their number operators <span class="math inline">\(n_f = f^\dagger f, n_g = g^\dagger g\)</span> and the total fermionic parity operator <span class="math inline">\(F_p = (2n_f - 1)(2n_g - 1)\)</span> which can be used to divide their Fock space up into even and odd parity subspaces. These subspaces are separated by the addition or removal of one fermion.</p>
|
||||
<p>From these two fermionic modes, we can build four Majorana operators: <span class="math display">\[\begin{aligned}
|
||||
b^x &= f + f^\dagger\\
|
||||
b^y &= -i(f - f^\dagger)\\
|
||||
b^z &= g + g^\dagger\\
|
||||
c &= -i(g - g^\dagger)
|
||||
\end{aligned}\]</span></p>
|
||||
<p>The Majoranas obey the usual commutation relations, squaring to one and anticommuting with each other. The fermions and Majorana live in a four dimensional Fock space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>. We can therefore identify the two dimensional space <span class="math inline">\(\mathcal{M}\)</span> with one of the parity subspaces of <span class="math inline">\(\mathcal{\tilde{L}}\)</span> which will be called the <em>physical subspace</em> <span class="math inline">\(\mathcal{\tilde{L}}_p\)</span>. Kitaev defines the operator <span class="math display">\[D = b^xb^yb^zc\]</span> which can be expanded to <span class="math display">\[D = -(2n_f - 1)(2n_g - 1) = -F_p\]</span> and labels the physical subspace as the space spanned by states for which <span class="math display">\[ D|\phi\rangle = |\phi\rangle\]</span></p>
|
||||
<p>We can also think of the physical subspace as whatever is left after applying the projector <span class="math display">\[P = \frac{1 - D}{2}\]</span> This formulation will be useful for taking states that span the extended space <span class="math inline">\(\mathcal{\tilde{M}}\)</span> and projecting them into the physical subspace.</p>
|
||||
<p>So now, with the caveat that we are working in the physical subspace, we can define new Pauli operators:</p>
|
||||
<p><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]</span></p>
|
||||
<p>These extended space Pauli operators satisfy all the usual commutation relations. The only difference is that if we evaluate <span class="math inline">\(\sigma^x \sigma^y \sigma^z = i\)</span>, we instead get <span class="math display">\[ \tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]</span></p>
|
||||
<p>This makes sense if we promise to confine ourselves to the physical subspace <span class="math inline">\(D = 1\)</span>.</p>
|
||||
<section id="for-multiple-spins" class="level4">
|
||||
<h4>For multiple spins</h4>
|
||||
<p>This construction easily generalises to the case of multiple spins. We get a set of 4 Majoranas <span class="math inline">\(b^x_j,\; b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j = b^x_jb^y_jb^z_jc_j\)</span> operator for every spin. For a state to be physical, we require that <span class="math inline">\(D_j |\psi\rangle = |\psi\rangle\)</span> for all <span class="math inline">\(j\)</span>.</p>
|
||||
<p>From these each Pauli operator can be constructed: <span class="math display">\[\tilde{\sigma}^\alpha_j = i b^\alpha_j c_j\]</span></p>
|
||||
<p>This is where the magic happens. We can promote the spin hamiltonian from <span class="math inline">\(\mathcal{L}\)</span> into the extended space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>, safe in the knowledge that nothing changes so long as we only actually work with physical states. The Hamiltonian <span class="math display">\[\begin{aligned}
|
||||
\tilde{H} &= - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\tilde{\sigma}_j^{\alpha}\tilde{\sigma}_k^{\alpha}\\
|
||||
&= \frac{i}{4} \sum_{\langle j,k\rangle_\alpha} 2J^{\alpha} (ib^\alpha_i b^\alpha_j) c_i c_j\\
|
||||
&= \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j
|
||||
\end{aligned}\]</span></p>
|
||||
<p>We can factor out the Majorana bond operators <span class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span>. Note that these bond operators are not equal to the spin bond operators <span class="math inline">\(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = - \hat{u}_{ij} c_i c_j\)</span>. In what follows, we will work much more frequently with the Majorana bond operators. Therefore, when we refer to bond operators without qualification, we are referring to the Majorana variety.</p>
|
||||
<p>Similarly to the argument with the spin bond operators <span class="math inline">\(K_{ij}\)</span>, we can quickly verify by considering three cases that the Majorana bond operators <span class="math inline">\(u_{ij}\)</span> all commute with one another. They square to one, so have eigenvalues <span class="math inline">\(\pm 1\)</span>. They also commute with the <span class="math inline">\(c_i\)</span> operators.</p>
|
||||
<p>Importantly, the operators <span class="math inline">\(D_i = b^x_i b^y_i b^z_i c_i\)</span> commute with <span class="math inline">\(K_{ij}\)</span> and, therefore, with <span class="math inline">\(\tilde{H}\)</span>. We will show later that the action of <span class="math inline">\(D_i\)</span> on a state is to flip the values of the three <span class="math inline">\(u_{ij}\)</span> bonds that connect to site <span class="math inline">\(i\)</span>. Physically, this indicates that <span class="math inline">\(u_{ij}\)</span> is a gauge field with a high degree of degeneracy.</p>
|
||||
<p>In summary, Majorana bond operators <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
|
||||
</section>
|
||||
<section id="phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Section: <a href="../2_Background/2.3_Disorder.html">Disorder
|
||||
and Localisation</a></p>
|
||||
</section>
|
||||
<section id="partitioning-the-hilbert-space-into-bond-sectors" class="level3">
|
||||
<h3>Partitioning the Hilbert Space into Bond sectors</h3>
|
||||
<p>Similarly to the story with the plaquette operators from the spin language, we can divide the Hilbert space <span class="math inline">\(\mathcal{L}\)</span> into sectors labelled by a set of choices <span class="math inline">\(\{\pm 1\}\)</span> for the value of each <span class="math inline">\(u_{ij}\)</span> operator which we denote by <span class="math inline">\(\mathcal{L}_u\)</span>. Since <span class="math inline">\(u_{ij} = -u_{ji}\)</span>, we can represent the <span class="math inline">\(u_{ij}\)</span> graphically with an arrow that points along each bond in the direction in which <span class="math inline">\(u_{ij} = 1\)</span>.</p>
|
||||
<p>Once confined to a particular <span class="math inline">\(\mathcal{L}_u\)</span>, we can ‘remove the hats’ from the <span class="math inline">\(\hat{u}_{ij}\)</span>. The hamiltonian becomes a quadratic, free fermion problem <span class="math display">\[\tilde{H_u} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> The ground state, <span class="math inline">\(|\psi_u\rangle\)</span> can be found easily as will be explained in the next part. At this point, we may wonder whether the <span class="math inline">\(\mathcal{L}_u\)</span> are confined entirely within the physical subspace <span class="math inline">\(\mathcal{L}_p\)</span> and, indeed, we will see that they are not. However, it will be helpful to first develop the theory of the Majorana Hamiltonian further.</p>
|
||||
<p><strong>The Majorana Hamiltonian</strong></p>
|
||||
<p>We now have a quadratic Hamiltonian <span class="math display">\[ \tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom have paired along bonds to become a classical gauge field <span class="math inline">\(u_{ij}\)</span>. What follows is relatively standard theory for quadratic Majorana Hamiltonians <span class="citation" data-cites="BlaizotRipka1986"> [<a href="#ref-BlaizotRipka1986" role="doc-biblioref">14</a>]</span>.</p>
|
||||
<p>Because of the antisymmetry of the matrix with entries <span class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in pairs <span class="math inline">\(\pm \epsilon_m\)</span>. This redundant information is a consequence of the doubling of the Hilbert space which occurred when we transformed to the Majorana representation.</p>
|
||||
<p>If we organise the eigenmodes of <span class="math inline">\(H\)</span> into pairs, such that <span class="math inline">\(b_m\)</span> and <span class="math inline">\(b_m'\)</span> have energies <span class="math inline">\(\epsilon_m\)</span> and <span class="math inline">\(-\epsilon_m\)</span>, we can construct the transformation <span class="math inline">\(Q\)</span> <span class="math display">\[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2, b_2' ... b_{N}, b_{N}')\]</span> and put the Hamiltonian into the form <span class="math display">\[\tilde{H}_u = \frac{i}{2} \sum_m \epsilon_m b_m b_m'\]</span></p>
|
||||
<p>The determinant of <span class="math inline">\(Q\)</span> will be useful later when we consider the projector from <span class="math inline">\(\mathcal{\tilde{L}}\)</span> to <span class="math inline">\(\mathcal{L}\)</span>. Otherwise, the <span class="math inline">\(b_m\)</span> are merely an intermediate step. From them, we form fermionic operators <span class="math display">\[ f_i = \tfrac{1}{2} (b_m + ib_m')\]</span> with their associated number operators <span class="math inline">\(n_i = f^\dagger_i f_i\)</span>. These let us write the Hamiltonian neatly as</p>
|
||||
<p><span class="math display">\[ \tilde{H}_u = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]</span></p>
|
||||
<p>The ground state <span class="math inline">\(|n_m = 0\rangle\)</span> of the many body system at fixed <span class="math inline">\(u\)</span> is then <span class="math display">\[E_{u,0} = -\frac{1}{2}\sum_m \epsilon_m \]</span> We can construct any state from a particular choice of <span class="math inline">\(n_m = 0,1\)</span>.</p>
|
||||
<p>If we only care about the value of <span class="math inline">\(E_{u,0}\)</span>, it is possible to skip forming the fermionic operators. The eigenvalues obtained directly from diagonalising <span class="math inline">\(J^{\alpha} u_{ij}\)</span> come in <span class="math inline">\(\pm \epsilon_m\)</span> pairs. We can take half the absolute value of the whole set to recover <span class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
|
||||
<p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond Sector.</p>
|
||||
</section>
|
||||
<section id="mapping-back-from-bond-sectors-to-the-physical-subspace" class="level3">
|
||||
<h3>Mapping back from Bond Sectors to the Physical Subspace</h3>
|
||||
<p>At this point, given a particular bond configuration <span class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> in the extended space and diagonalise it to find its ground state <span class="math inline">\(|\vec{u}, \vec{n} = 0\rangle\)</span>. This is not necessarily the ground state of the system as a whole, it is just the lowest energy state within the subspace <span class="math inline">\(\mathcal{L}_u\)</span></p>
|
||||
<p><strong>However, <span class="math inline">\(|u, n_m = 0\rangle\)</span> does not lie in the physical subspace</strong>. As an example, consider the lowest energy state associated with <span class="math inline">\(u_{ij} = +1\)</span>. This state satisfies <span class="math display">\[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle = |\vec{u}=1, \vec{n} = 0\rangle\]</span> for all bonds <span class="math inline">\(i,j\)</span>.</p>
|
||||
<p>If we act on it, this state with one of the gauge operators <span class="math inline">\(D_j = b_j^x b_j^y b_j^z c_j\)</span>, we see that <span class="math inline">\(D_j\)</span> flips the value of the three bonds <span class="math inline">\(u_{ij}\)</span> that surround site <span class="math inline">\(k\)</span>:</p>
|
||||
<p><span class="math display">\[ |u'\rangle = D_j |u=1, n_m = 0\rangle\]</span></p>
|
||||
<p><span class="math display">\[ \begin{aligned}
|
||||
\langle u'|u_{ij}|u'\rangle &= \langle u| b_j^x b_j^y b_j^z c_j \;ib^x_i b^x_j\; b_j^x b_j^y b_j^z c_j|u\rangle\\
|
||||
&= -1
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Since <span class="math inline">\(D_j\)</span> commutes with the Hamiltonian in the extended space <span class="math inline">\(\tilde{H}\)</span>, the fact that <span class="math inline">\(D_j\)</span> flips the value of bond operators indicates that there is a gauge degeneracy between the ground state of <span class="math inline">\(\tilde{H}_u\)</span> and the set of <span class="math inline">\(\tilde{H}_{u'}\)</span> related to it by gauge transformations <span class="math inline">\(D_j\)</span>. Thus, we can flip any three bonds around a vertex and the physics will stay the same.</p>
|
||||
<p>We can turn this into a symmetrisation procedure by taking a superposition of every possible gauge transformation. Every possible gauge transformation is just every possible subset of <span class="math inline">\({D_0, D_1 ... D_n}\)</span> which can be neatly expressed as <span class="math display">\[|\phi_w\rangle = \prod_i \left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\]</span> This is convenient because the quantity <span class="math inline">\(\frac{1 + D_i}{2}\)</span> is also the local projector onto the physical subspace. Here <span class="math inline">\(|\phi_w\rangle\)</span> is a gauge invariant state that lives in <span class="math inline">\(\mathcal{L}_p\)</span> which has been constructed from a set of states in different <span class="math inline">\(\mathcal{L}_u\)</span>.</p>
|
||||
<p>This gauge degeneracy leads us to the next topic of discussion, namely how to construct a set of gauge invariant quantities out of the <span class="math inline">\(u_{ij}\)</span>, these will turn out to just be the plaquette operators.</p>
|
||||
<p>Takeaway: The Bond Sectors overlap with the physical subspace but are not contained within it.</p>
|
||||
</section>
|
||||
<section id="open-boundary-conditions" class="level3">
|
||||
<h3>Open boundary conditions</h3>
|
||||
<p>Care must be taken when defining open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector.</p>
|
||||
<p>Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators could be performed.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="emergent-gauge-fields" class="level2">
|
||||
<h2>Emergent gauge fields</h2>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/torus.jpeg" id="fig:torus" data-short-caption="Loops on the Torus" style="width:86.0%" alt="Figure 3: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically, the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labelled x and y, which cannot be smoothly deformed to a point. These two non-contractible loops can be used to construct two special pairs of operators: The two topological fluxes \Phi_x and \Phi_y that are the expectation values of u_{jk} loops around each path. There are also two operators \hat{\mathcal{T}}_x and \hat{\mathcal{T}}_y that transform one half of a vortex pair around the loop before annihilating them together again, see later." />
|
||||
<figcaption aria-hidden="true">Figure 3: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically, the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labelled <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>, which cannot be smoothly deformed to a point. These two non-contractible loops can be used to construct two special pairs of operators: The two topological fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> that are the expectation values of <span class="math inline">\(u_{jk}\)</span> loops around each path. There are also two operators <span class="math inline">\(\hat{\mathcal{T}}_x\)</span> and <span class="math inline">\(\hat{\mathcal{T}}_y\)</span> that transform one half of a vortex pair around the loop before annihilating them together again, see later.</figcaption>
|
||||
</figure>
|
||||
<section id="ground-state-degeneracy" class="level3">
|
||||
<h3>Ground State Degeneracy</h3>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg" id="fig:loops_and_dual_loops" data-short-caption="Topological Loops and Dual Loops" style="width:100.0%" alt="Figure 4: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again." />
|
||||
<figcaption aria-hidden="true">Figure 4: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again.</figcaption>
|
||||
</figure>
|
||||
<p>More general arguments <span class="citation" data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a href="#ref-chungExplicitMonodromyMoore2007" role="doc-biblioref">15</a>,<a href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" role="doc-biblioref">16</a>]</span> imply that <span class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an interesting relationship to the topological fluxes. In the non-Abelian phase, we expect that it will change sign in exactly one of the four topological sectors.</p>
|
||||
<p>This means that the lowest state in three of the topological sectors contain no fermions, while in one of them there must be one fermion to preserve product of fermion vortex parity. So overall the non-Abelian model has a three-fold degenerate ground state rather than the fourfold of the Abelian case (and of my intuition!). In the Abelian phase, this does not happen and we get a fourfold degenerate ground state. <strong>Whether this analysis generalises to the amorphous case is unclear.</strong></p>
|
||||
<p>An alternative way to view this is to imagine we start in one state of the ground state manifold. We then attempt to construct other ground states by creating vortex pairs, transporting one vortex around one or both non-contractible loops and then annihilating them. This works for either of the two non-contractible loops but when we try to do it for <em>both</em> something strange happens. When we transport a vortex around <strong>both</strong> the major and minor axes of the torus this changes its fusion channel. Normally two vortices fuse to the vacuum but after this operation they fuse into a fermion excitation. And hence our attempt to construct that last ground state doesn’t yield a ground state at all, leaving us with just three.</p>
|
||||
<p><strong>NOTE to self: This argument seems to involve adiabatic insertion of the fluxes <span class="math inline">\(\Phi_{x,y}\)</span> as the operations that undo vortex transport around the lattice. I don’t understand why that part is necessary</strong></p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/threefold_degeneracy.png" id="fig:threefold_degeneracy" data-short-caption="Ground State Degeneracy in the Abelian and Non-Abelian Phases" style="width:86.0%" alt="Figure 5: In the non-Abelian phase one of the lowest energy state in one of the topological sectors contains a fermion and hence is slightly higher in energy than the other three. This manifests as a fourfold ground state degeneracy in the Abelian phase and a threefold degeneracy in the non-Abelian phase." />
|
||||
<figcaption aria-hidden="true">Figure 5: In the non-Abelian phase one of the lowest energy state in one of the topological sectors contains a fermion and hence is slightly higher in energy than the other three. This manifests as a fourfold ground state degeneracy in the Abelian phase and a threefold degeneracy in the non-Abelian phase.</figcaption>
|
||||
</figure>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bg-the-ground-state" class="level2">
|
||||
<h2>The Ground State</h2>
|
||||
<p>Discuss Lieb’s theorem and generalisations for other lattices</p>
|
||||
</section>
|
||||
<section id="phases-of-the-kitaev-model" class="level2">
|
||||
<h2>Phases of the Kitaev Model</h2>
|
||||
<p>discuss the Abelian A phase / toric code phase / anisotropic phase</p>
|
||||
<p>the isotropic gapless phase of the standard model</p>
|
||||
<p>The isotropic gapped phase with the addition of a magnetic field </i,j></i,j></p>
|
||||
<p>Next Section: <a href="../2_Background/2.3_Anyons.html">Anyonic Statistics</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
|
||||
I. Kugel’ and D. I. Khomskiĭ, <em><a
|
||||
href="https://doi.org/10.1070/PU1982v025n04ABEH004537">The Jahn-Teller
|
||||
Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys.
|
||||
Usp. <strong>25</strong>, 231 (1982).</div>
|
||||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A. Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons in an Exactly Solved Model and Beyond</a></em>, Annals of Physics <strong>321</strong>, 2 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">G. Jackeli and G. Khaliullin, <em><a href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models</a></em>, Physical Review Letters <strong>102</strong>, 017205 (2009).</div>
|
||||
</div>
|
||||
<div id="ref-HerrmannsAnRev2018" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M. Hermanns, I. Kimchi, and J. Knolle, <em><a href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections</a></em>, Annual Review of Condensed Matter Physics <strong>9</strong>, 17 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-Winter2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, <em>Models and Materials for Generalized Kitaev Magnetism</em>, Journal of Physics: Condensed Matter <strong>29</strong>, 493002 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-TrebstPhysRep2022" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">S. Trebst and C. Hickey, <em><a href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev Materials</a></em>, Physics Reports <strong>950</strong>, 1 (2022).</div>
|
||||
</div>
|
||||
<div id="ref-Takagi2019" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, <em>Concept and Realization of Kitaev Quantum Spin Liquids</em>, Nature Reviews Physics <strong>1</strong>, 264 (2019).</div>
|
||||
</div>
|
||||
<div id="ref-andersonResonatingValenceBonds1973" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">P. W. Anderson, <em><a href="https://doi.org/10.1016/0025-5408(73)90167-0">Resonating Valence Bonds: A New Kind of Insulator?</a></em>, Materials Research Bulletin <strong>8</strong>, 153 (1973).</div>
|
||||
</div>
|
||||
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">C. N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B <strong>99</strong>, 045142 (2019).</div>
|
||||
</div>
|
||||
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">A. Banerjee et al., <em><a href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater <strong>15</strong>, 733 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-freedmanTopologicalQuantumComputation2003" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31 (2003).</div>
|
||||
</div>
|
||||
<div id="ref-baskaranExactResultsSpin2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">G. Baskaran, S. Mandal, and R. Shankar, <em><a href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys. Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">K. I. Kugel’ and D. I. Khomskii, <em><a href="https://doi.org/10.1070/PU1982v025n04ABEH004537">The Jahn-Teller Effect and Magnetism: Transition Metal Compounds</a></em>, Sov. Phys. Usp. <strong>25</strong>, 231 (1982).</div>
|
||||
</div>
|
||||
<div id="ref-yaoExactChiralSpin2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">H. Yao and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett. <strong>99</strong>, 247203 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory of Finite Systems</em> (The MIT Press, 1986).</div>
|
||||
</div>
|
||||
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">S. B. Chung and M. Stone, <em><a href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor. <strong>40</strong>, 4923 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">M. Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy of Non-Abelian States for Dummies</a></em>, Annals of Physics <strong>322</strong>, 1477 (2007).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
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<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic Statistics</a></li>
|
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<li><a href="#what-is-so-great-about-two-dimensions" id="toc-what-is-so-great-about-two-dimensions">What is so great about two dimensions?</a></li>
|
||||
<li><a href="#topology-chirality-and-edge-modes" id="toc-topology-chirality-and-edge-modes">Topology chirality and edge modes</a></li>
|
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<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic Statistics</a></li>
|
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<li><a href="#what-is-so-great-about-two-dimensions" id="toc-what-is-so-great-about-two-dimensions">What is so great about two dimensions?</a></li>
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<li><a href="#topology-chirality-and-edge-modes" id="toc-topology-chirality-and-edge-modes">Topology chirality and edge modes</a></li>
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|
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<p>2 Background</p>
|
||||
<hr />
|
||||
</div>
|
||||
<p>Anyons are exotic two dimensional particles that are intermediate between bosons and fermions. Abelian anyons pick up an arbitrary phase <span class="math inline">\(e^{i\phi}\)</span> up interchange. The Kitaev model is also topologically non-trivial, supporting a degenerate ground state manifold of varying size. The interchange of non-Abelian anyons corresponds to arbitrary rotations within the ground state manifold, operations which may not commute and thus form a non-Abelian group.</p>
|
||||
<section id="anyonic-statistics" class="level3">
|
||||
<h3>Anyonic Statistics</h3>
|
||||
<p><strong>NB: I’m thinking about moving this section to the overall intro, but it’s nice to be able to refer to specifics of the Kitaev model also so I’m not sure. It currently repeats a discussion of the ground state degeneracy from the projector section.</strong></p>
|
||||
<p>In dimensions greater than two, the quantum state of a system must pick up a factor of <span class="math inline">\(-1\)</span> or <span class="math inline">\(+1\)</span> if two identical particles are swapped. We call these Fermions and Bosons.</p>
|
||||
<p>This argument is predicated on the idea that performing two swaps is equivalent to doing nothing. Doing nothing should not change the quantum state at all. Therefore, doing one swap can at most multiply it by <span class="math inline">\(\pm 1\)</span>.</p>
|
||||
<p>However, there are many hidden parts to this argument. First, this argument does not present the whole story. For instance, if you want to know why Fermions have half integer spin, you have to go to field theory.</p>
|
||||
<p>Second, why does this argument only work in dimensions greater than two? When we say that two swaps do nothing, we in fact say that the world lines of two particles that have been swapped twice can be untangled without crossing. Why can’t they cross? Because if they cross, the particles can interact and the quantum state could change in an arbitrary way. We are implicitly using the locality of physics to argue that, if the worldlines stay well separated, the overall quantum state cannot change.</p>
|
||||
<p>In two dimensions, we cannot untangle the worldlines of two particles that have swapped places. They are braided together (see fig. <a href="#fig:braiding">1</a>).</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig:braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 1: Worldlines of particles in two dimensions can become tangled or braided with one another." />
|
||||
<figcaption aria-hidden="true">Figure 1: Worldlines of particles in two dimensions can become tangled or <em>braided</em> with one another.</figcaption>
|
||||
</figure>
|
||||
<p>From this fact flows a whole of behaviours. The quantum state can acquire a phase factor <span class="math inline">\(e^{i\phi}\)</span> upon exchange of two identical particles, which we now call Anyons.</p>
|
||||
<p>The Kitaev Model is a good demonstration of the connection between Anyons and topological degeneracy. In the Kitaev model, we can create a pair of vortices, move one around a non-contractable loop <span class="math inline">\(\mathcal{T}_{x/y}\)</span> and then annihilate them together. Without topology, this should leave the quantum state unchanged. Instead, we move towards another ground state in a topologically degenerate ground state subspace. Practically speaking, it flips a dual line of bonds <span class="math inline">\(u_{jk}\)</span> going around the loop which we cannot undo with any gauge transformation made from <span class="math inline">\(D_j\)</span> operators.</p>
|
||||
<p>If the ground state subspace is multidimensional, quasiparticle exchange can move us around in the space with an action corresponding to a matrix. In general, these matrices do not commute so these are known as non-Abelian anyons.</p>
|
||||
<p>From here, the situation becomes even more complex. The Kitaev model has a non-Abelian phase when exposed to a magnetic field. The amorphous Kitaev Model has a non-Abelian phase because of its broken chiral symmetry.</p>
|
||||
<p>By subdividing the Kitaev model into vortex sectors, we neatly separate between vortices and fermionic excitations. However, if we looked at the full many body picture, we would see that a vortex carries with it a cloud of bound Majorana states.</p>
|
||||
<p>Consider two processes</p>
|
||||
<ol type="1">
|
||||
<li><p>We transport one half of a vortex pair around either the x or y loops of the torus before annihilating back to the ground state vortex sector <span class="math inline">\(\mathcal{T}_{x,y}\)</span>.</p></li>
|
||||
<li><p>We flip a line of bond operators corresponding to measuring the flux through either the major or minor axes of the torus <span class="math inline">\(\mathcal{\Phi}_{x,y}\)</span></p></li>
|
||||
</ol>
|
||||
<p>The plaquette operators <span class="math inline">\(\phi_i\)</span> are associated with fluxes. Wilson loops that wind the torus are associated with the fluxes through its two diameters <span class="math inline">\(\mathcal{\Phi}_{x,y}\)</span>.</p>
|
||||
<p>In the Abelian phase, we can move a vortex along any path at will before bringing them back together. They will annihilate back to the vacuum, where we understand ‘the vacuum’ to refer to one of the ground states. However, this will not necessarily be the same ground state we started in. We can use this to get from the <span class="math inline">\((\Phi_x, \Phi_y) = (+1, +1)\)</span> ground state and construct the set <span class="math inline">\((+1, +1), (+1, -1), (-1, +1), (-1, -1)\)</span>.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig:topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Figure 2: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make [1]." />
|
||||
<figcaption aria-hidden="true">Figure 2: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make <span class="citation" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis" role="doc-biblioref">1</a>]</span>.</figcaption>
|
||||
</figure>
|
||||
<p>However, in the non-Abelian phase we have to wrangle with monodromy <span class="citation" data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a href="#ref-chungExplicitMonodromyMoore2007" role="doc-biblioref">2</a>,<a href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" role="doc-biblioref">3</a>]</span>. Monodromy is the behaviour of objects as they move around a singularity. This manifests here in that the identity of a vortex and cloud of Majoranas can change as we wind them around the torus in such a way that, rather than annihilating to the vacuum, we annihilate them to create an excited state instead of a ground state. This means that we end up with only three degenerate ground states in the non-Abelian phase <span class="math inline">\((+1, +1), (+1, -1), (-1, +1)\)</span> <span class="citation" data-cites="Chung_Topological_quantum2010 yaoAlgebraicSpinLiquid2009"> [<a href="#ref-Chung_Topological_quantum2010" role="doc-biblioref">4</a>,<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">5</a>]</span>. Concretely, this is because the projector enforces both flux and fermion parity. When we wind a vortex around both non-contractible loops of the torus, it flips the flux parity. Therefore, we have to introduce a fermionic excitation to make the state physical. Hence, the process does not give a fourth ground state.</p>
|
||||
<p>Recently, the topology has notably gained interest because of proposals to use this ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations <span class="citation" data-cites="kitaev_fault-tolerant_2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">6</a>–<a href="#ref-hastingsDynamicallyGeneratedLogical2021" role="doc-biblioref">8</a>]</span>.</p>
|
||||
</section>
|
||||
<section id="what-is-so-great-about-two-dimensions" class="level2">
|
||||
<h2>What is so great about two dimensions?</h2>
|
||||
</section>
|
||||
<section id="topology-chirality-and-edge-modes" class="level2">
|
||||
<h2>Topology chirality and edge modes</h2>
|
||||
<p>Most thermodynamic and quantum phases studied can be characterised by a local order parameter. That is, a function or operator that only requires knowledge about some fixed sized patch of the system that does not scale with system size.</p>
|
||||
<p>However, there are quantum phases that cannot be characterised by such a local order parameter. These phases are instead said to possess ‘topological order’.</p>
|
||||
<p>One easily observable property of topological order is that the ground state degeneracy depends on the topology of the manifold that we put the system on to. This is referred to as topological degeneracy to distinguish it from standard symmetry breaking.</p>
|
||||
<p>The Kitaev model is a good example. We have already looked at it defined on a graph that is embedded either into the plane or onto the torus. The extension to surfaces like the torus but with more than one handle is relatively easy.</p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Section: <a href="../2_Background/2.4_Disorder.html">Disorder and Localisation</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-parkerWhyDoesThis" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline"><em><a href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon Have -1 Holes?</a></em> (n.d.).</div>
|
||||
</div>
|
||||
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">S. B. Chung and M. Stone, <em><a href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor. <strong>40</strong>, 4923 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M. Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy of Non-Abelian States for Dummies</a></em>, Annals of Physics <strong>322</strong>, 1477 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-Chung_Topological_quantum2010" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S. B. Chung, H. Yao, T. L. Hughes, and E.-A. Kim, <em><a href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at Finite Temperature</a></em>, Phys. Rev. B <strong>81</strong>, 060403 (2010).</div>
|
||||
</div>
|
||||
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">H. Yao, S.-C. Zhang, and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. <strong>102</strong>, 217202 (2009).</div>
|
||||
</div>
|
||||
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A. Yu. Kitaev, <em><a href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant Quantum Computation by Anyons</a></em>, Annals of Physics <strong>303</strong>, 2 (2003).</div>
|
||||
</div>
|
||||
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">D. Poulin, <em><a href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev. Lett. <strong>95</strong>, 230504 (2005).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">M. B. Hastings and J. Haah, <em><a href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
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<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
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<ul>
|
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<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
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<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
|
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<!-- Table of Contents -->
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<li><a href="#bg-disorder-and-localisation" id="toc-bg-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#localisation-anderson-many-body-and-disorder-free" id="toc-localisation-anderson-many-body-and-disorder-free">Localisation: Anderson, Many Body and Disorder-Free</a></li>
|
||||
<li><a href="#disorder-and-spin-liquids" id="toc-disorder-and-spin-liquids">Disorder and Spin liquids</a></li>
|
||||
<li><a href="#amorphous-magnetism" id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<li><a href="#localisation" id="toc-localisation">Localisation</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
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</nav>
|
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|
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|
||||
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|
||||
<div id="page-header">
|
||||
<p>2 Background</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="bg-disorder-and-localisation" class="level1">
|
||||
<h1>Disorder and Localisation</h1>
|
||||
<section id="localisation-anderson-many-body-and-disorder-free" class="level2">
|
||||
<h2>Localisation: Anderson, Many Body and Disorder-Free</h2>
|
||||
</section>
|
||||
<section id="disorder-and-spin-liquids" class="level2">
|
||||
<h2>Disorder and Spin liquids</h2>
|
||||
</section>
|
||||
<section id="amorphous-magnetism" class="level2">
|
||||
<h2>Amorphous Magnetism</h2>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</section>
|
||||
<section id="localisation" class="level2">
|
||||
<h2>Localisation</h2>
|
||||
<p>The discovery of localisation in quantum systems surprising at the time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesis <span class="citation" data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994"> [<a href="#ref-abaninRecentProgressManybody2017" role="doc-biblioref">1</a>,<a href="#ref-srednickiChaosQuantumThermalization1994" role="doc-biblioref">2</a>]</span>, allowing quantum systems to avoid to retain memory of their initial conditions in the face of thermal noise.</p>
|
||||
<p>The simplest and first discovered kind is Anderson localisation, first studied in 1958 <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">3</a>]</span> in the context of non-interacting fermions subject to a static or quenched disorder potential <span class="math inline">\(V_j\)</span> drawn uniformly from the interval <span class="math inline">\([-W,W]\)</span></p>
|
||||
<p><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
|
||||
\]</span></p>
|
||||
<p>this model exhibits exponentially localised eigenfunctions <span class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\)</span> which cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can exist <span class="citation" data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012"> [<a href="#ref-izrailevLocalizationMobilityEdge1999" role="doc-biblioref">4</a>–<a href="#ref-izrailevAnomalousLocalizationLowDimensional2012" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>Later localisation was found in interacting many-body systems with quenched disorder:</p>
|
||||
<p><span class="math display">\[
|
||||
H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k
|
||||
\]</span></p>
|
||||
<p>where the number operators <span class="math inline">\(n_j = c^\dagger_j c_j\)</span>. Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL) <span class="citation" data-cites="imbrieManyBodyLocalizationQuantum2016"> [<a href="#ref-imbrieManyBodyLocalizationQuantum2016" role="doc-biblioref">7</a>]</span>.</p>
|
||||
<p>Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder necessary to generate localisation is generated entirely from the dynamics of the model. This contracts with typical models of disordered systems in which disorder is explicitly introduced into the Hamilton or the initial state.</p>
|
||||
<p>The concept of disorder-free localisation was first proposed in the context of Helium mixtures <span class="citation" data-cites="kagan1984localization"> [<a href="#ref-kagan1984localization" role="doc-biblioref">8</a>]</span> and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>,<a href="#ref-schiulazDynamicsManybodyLocalized2015" role="doc-biblioref">10</a>]</span> instead find that the models thermalise exponentially slowly in system size, which Ref. <span class="citation" data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016"> [<a href="#ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" role="doc-biblioref">9</a>]</span> dubs Quasi-MBL.</p>
|
||||
<p>True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities <span class="citation" data-cites="smithDisorderFreeLocalization2017"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">11</a>]</span>. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit.</p>
|
||||
<p>-link to the FK model</p>
|
||||
<p>-link to the Kitaev Model</p>
|
||||
<p>-link to the physics of amorphous systems</p>
|
||||
<p>Next Chapter: <a href="../3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-abaninRecentProgressManybody2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">D. A. Abanin and Z. Papić, <em><a href="https://doi.org/10.1002/andp.201700169">Recent Progress in Many-Body Localization</a></em>, ANNALEN DER PHYSIK <strong>529</strong>, 1700169 (2017).</div>
|
||||
</div>
|
||||
<div id="ref-srednickiChaosQuantumThermalization1994" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">M. Srednicki, <em><a href="https://doi.org/10.1103/PhysRevE.50.888">Chaos and Quantum Thermalization</a></em>, Phys. Rev. E <strong>50</strong>, 888 (1994).</div>
|
||||
</div>
|
||||
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">P. W. Anderson, <em><a href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492 (1958).</div>
|
||||
</div>
|
||||
<div id="ref-izrailevLocalizationMobilityEdge1999" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">F. M. Izrailev and A. A. Krokhin, <em><a href="https://doi.org/10.1103/PhysRevLett.82.4062">Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder</a></em>, Phys. Rev. Lett. <strong>82</strong>, 4062 (1999).</div>
|
||||
</div>
|
||||
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-izrailevAnomalousLocalizationLowDimensional2012" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">F. M. Izrailev, A. A. Krokhin, and N. M. Makarov, <em><a href="https://doi.org/10.1016/j.physrep.2011.11.002">Anomalous Localization in Low-Dimensional Systems with Correlated Disorder</a></em>, Physics Reports <strong>512</strong>, 125 (2012).</div>
|
||||
</div>
|
||||
<div id="ref-imbrieManyBodyLocalizationQuantum2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J. Z. Imbrie, <em><a href="https://doi.org/10.1007/s10955-016-1508-x">On Many-Body Localization for Quantum Spin Chains</a></em>, J Stat Phys <strong>163</strong>, 998 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-kagan1984localization" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">Y. Kagan and L. Maksimov, <em>Localization in a System of Interacting Particles Diffusing in a Regular Crystal</em>, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki <strong>87</strong>, 348 (1984).</div>
|
||||
</div>
|
||||
<div id="ref-yaoQuasiManyBodyLocalizationTranslationInvariant2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore, <em><a href="https://doi.org/10.1103/PhysRevLett.117.240601">Quasi-Many-Body Localization in Translation-Invariant Systems</a></em>, Phys. Rev. Lett. <strong>117</strong>, 240601 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-schiulazDynamicsManybodyLocalized2015" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">M. Schiulaz, A. Silva, and M. Müller, <em><a href="https://doi.org/10.1103/PhysRevB.91.184202">Dynamics in Many-Body Localized Quantum Systems Without Disorder</a></em>, Phys. Rev. B <strong>91</strong>, 184202 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-smithDisorderFreeLocalization2017" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, <em><a href="https://doi.org/10.1103/PhysRevLett.118.266601">Disorder-Free Localization</a></em>, Phys. Rev. Lett. <strong>118</strong>, 266601 (2017).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
</main>
|
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|
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|
||||
@ -27,6 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#chap:3-the-long-range-falikov-kimball-model" id="toc-chap:3-the-long-range-falikov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -41,6 +42,7 @@ image:
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#chap:3-the-long-range-falikov-kimball-model" id="toc-chap:3-the-long-range-falikov-kimball-model">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<li><a href="#fk-model" id="toc-fk-model">The Model</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -52,469 +54,131 @@ image:
|
||||
<p>3 The Long Range Falikov-Kimball Model</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="chap:3-the-long-range-falikov-kimball-model" class="level1">
|
||||
<h1>3 The Long Range Falikov-Kimball Model</h1>
|
||||
<p>This chapter expands on work presented in</p>
|
||||
<p> <span class="citation"
|
||||
data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a
|
||||
href="#ref-hodsonOnedimensionalLongrangeFalikovKimball2021"
|
||||
role="doc-biblioref">1</a>]</span> <a
|
||||
href="https://link.aps.org/doi/10.1103/PhysRevB.104.045116">One-dimensional
|
||||
long-range Falikov-Kimball model: Thermal phase transition and
|
||||
disorder-free localization</a>, Hodson, T. and Willsher, J. and Knolle,
|
||||
J., Phys. Rev. B, <strong>104</strong>, 4, 2021,</p>
|
||||
<p>the code for which is available is available at <span
|
||||
class="citation" data-cites="hodsonMCMCFKModel2021"> [<a
|
||||
href="#ref-hodsonMCMCFKModel2021"
|
||||
role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p> <span class="citation" data-cites="hodsonOnedimensionalLongrangeFalikovKimball2021"> [<a href="#ref-hodsonOnedimensionalLongrangeFalikovKimball2021" role="doc-biblioref">1</a>]</span> <a href="https://link.aps.org/doi/10.1103/PhysRevB.104.045116">One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization</a>, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, <strong>104</strong>, 4, 2021,</p>
|
||||
<p>the code for which is available is available at <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p><strong>Contributions</strong></p>
|
||||
<p>Johannes had the initial idea to use a long range Ising term to
|
||||
stablise order in a one dimension Falikov-Kimball model. Josef developed
|
||||
a proof of concept during a summer project at Imperial along with
|
||||
Alexander Belcik. I wrote the simulation code and performed all the
|
||||
analysis presented here.</p>
|
||||
<p>Johannes had the initial idea to use a long range Ising term to stablise order in a one dimension Falikov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.</p>
|
||||
<p><strong>Chapter Summary</strong></p>
|
||||
<p>The paper is organised as follows. First, we introduce the model and
|
||||
present its phase diagram. Second, we present the methods used to solve
|
||||
it numerically. Last, we investigate the model’s localisation properties
|
||||
and conclude.</p>
|
||||
<p>The paper is organised as follows. First, we introduce the model and present its phase diagram. Second, we present the methods used to solve it numerically. Last, we investigate the model’s localisation properties and conclude.</p>
|
||||
</section>
|
||||
<section id="fk-model" class="level1">
|
||||
<h1>The Model</h1>
|
||||
<p>Dimensionality is crucial for the physics of both localisation and
|
||||
FTPTs. In 1D, disorder generally dominates, even the weakest disorder
|
||||
exponentially localises <em>all</em> single particle eigenstates. Only
|
||||
longer-range correlations of the disorder potential can potentially
|
||||
induce delocalisation <span class="citation"
|
||||
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
|
||||
href="#ref-aubryAnalyticityBreakingAnderson1980"
|
||||
role="doc-biblioref">3</a>–<a
|
||||
href="#ref-dunlapAbsenceLocalizationRandomdimer1990"
|
||||
role="doc-biblioref">5</a>]</span>. Thermodynamically, short-range
|
||||
interactions cannot overcome thermal defects in 1D which prevents
|
||||
ordered phases at nonzero temperature <span class="citation"
|
||||
data-cites="andersonAbsenceDiffusionCertain1958 goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a
|
||||
href="#ref-andersonAbsenceDiffusionCertain1958"
|
||||
role="doc-biblioref">6</a>–<a
|
||||
href="#ref-kramerLocalizationTheoryExperiment1993"
|
||||
role="doc-biblioref">9</a>]</span>. However, the absence of an FTPT in
|
||||
the short ranged FK chain is far from obvious because the
|
||||
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the
|
||||
fermions <span class="citation"
|
||||
data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a
|
||||
href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">10</a>–<a
|
||||
href="#ref-yosidaMagneticPropertiesCuMn1957"
|
||||
role="doc-biblioref">13</a>]</span> decays as <span
|
||||
class="math inline">\(r^{-1}\)</span> in 1D <span class="citation"
|
||||
data-cites="rusinCalculationRKKYRange2017"> [<a
|
||||
href="#ref-rusinCalculationRKKYRange2017"
|
||||
role="doc-biblioref">14</a>]</span>. This could in principle induce the
|
||||
necessary long-range interactions for the classical Ising
|
||||
background <span class="citation"
|
||||
data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">15</a>,<a
|
||||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||||
role="doc-biblioref">16</a>]</span>. However, Kennedy and Lieb
|
||||
established rigorously that at half-filling a CDW phase only exists at
|
||||
<span class="math inline">\(T = 0\)</span> for the 1D FK model <span
|
||||
class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>Here, we construct a generalised one-dimensional FK model with
|
||||
long-range interactions which induces the otherwise forbidden CDW phase
|
||||
at non-zero temperature. We find a rich phase diagram with a CDW FTPT
|
||||
and interaction-tuned Anderson versus Mott localized phases similar to
|
||||
the 2D FK model <span class="citation"
|
||||
data-cites="antipovInteractionTunedAndersonMott2016"> [<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">18</a>]</span>. We explore the localization
|
||||
properties of the fermionic sector and find that the localisation
|
||||
lengths vary dramatically across the phases and for different energies.
|
||||
Although moderate system sizes indicate the coexistence of localized and
|
||||
delocalized states within the CDW phase, we find quantitatively similar
|
||||
behaviour in a model of uncorrelated binary disorder on a CDW
|
||||
background. For large system sizes, i.e. for our 1D disorder model we
|
||||
can treat linear sizes of several thousand sites, we find that all
|
||||
states are eventually localized with a localization length which
|
||||
diverges towards zero temperature.</p>
|
||||
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits
|
||||
a finite temperature phase transition to an ordered charge density wave
|
||||
(CDW) phase <span class="citation"
|
||||
data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||||
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||||
role="doc-biblioref">19</a>]</span>. In this phase, the ions are
|
||||
confined to one of the two sublattices, breaking the <span
|
||||
class="math inline">\(\mathbb{Z}_2\)</span> symmetry.</p>
|
||||
<p>In 1D, however, Periel’s argument <span class="citation"
|
||||
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-peierlsIsingModelFerromagnetism1936"
|
||||
role="doc-biblioref">16</a>,<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">17</a>]</span> states that domain walls only
|
||||
introduce a constant energy penalty into the free energy while bringing
|
||||
a entropic contribution logarithmic in system size. Hence the 1D model
|
||||
does not have a finite temperature phase transition. However 1D systems
|
||||
are much easier to study numerically and admit simpler realisations
|
||||
experimentally. We therefore introduce a long range coupling between the
|
||||
ions in order to stabilise a CDW phase in 1D. This leads to a disordered
|
||||
system that is gaped by the CDW background but with correlated
|
||||
fluctuations leading to a disorder-free correlation induced mobility
|
||||
edge in one dimension.</p>
|
||||
<p>The presence of the classical field makes the model amenable to an
|
||||
exact numerical treatment at finite temperature via a sign problem free
|
||||
MCMC algorithm <span class="citation"
|
||||
data-cites="devriesGapsDensitiesStates1993 devriesSimplifiedHubbardModel1993 antipovInteractionTunedAndersonMott2016 debskiPossibilityDetectionFinite2016 herrmannSpreadingCorrelationsFalicovKimball2018 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">18</a>–<a
|
||||
href="#ref-herrmannSpreadingCorrelationsFalicovKimball2018"
|
||||
role="doc-biblioref">22</a>,<a
|
||||
href="#ref-debskiPossibilityDetectionFinite2016"
|
||||
role="doc-biblioref"><strong>debskiPossibilityDetectionFinite2016?</strong></a>]</span>.
|
||||
The MCMC treatment motivates a view of the classical background field as
|
||||
a disorder potential, which suggests an intimate link to localisation
|
||||
physics. Indeed, thermal fluctuations of the classical sector act as
|
||||
disorder potentials drawn from a thermal distribution and the emergence
|
||||
of disorder in a translationally invariant Hamiltonian links the FK
|
||||
model to recent interest in disorder-free localisation <span
|
||||
class="citation"
|
||||
data-cites="smithDisorderFreeLocalization2017 smithDynamicalLocalizationMathbbZ2018 brenesManyBodyLocalizationDynamics2018"> [<a
|
||||
href="#ref-smithDisorderFreeLocalization2017"
|
||||
role="doc-biblioref">23</a>–<a
|
||||
href="#ref-brenesManyBodyLocalizationDynamics2018"
|
||||
role="doc-biblioref">25</a>]</span>.</p>
|
||||
<p>To evaluate thermodynamic averages we perform a classical Markov
|
||||
Chain Monte Carlo random walk over the space of ionic configurations, at
|
||||
each step diagonalising the effective electronic Hamiltonian <span
|
||||
class="citation"
|
||||
data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||||
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||||
role="doc-biblioref">19</a>]</span>. Using a binder-cumulant
|
||||
method <span class="citation"
|
||||
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
|
||||
href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">26</a>,<a
|
||||
href="#ref-musialMonteCarloSimulations2002"
|
||||
role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>,
|
||||
we demonstrate the model has a finite temperature phase transition when
|
||||
the interaction is sufficiently long ranged. We then estimate the
|
||||
density of states and the inverse participation ratio as a function of
|
||||
energy to diagnose localisation properties.</p>
|
||||
<p>We interpret the FK model as a model of spinless fermions, <span
|
||||
class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice
|
||||
against a classical Ising spin background, <span
|
||||
class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions
|
||||
couple to the spins via an onsite interaction with strength <span
|
||||
class="math inline">\(U\)</span> which we supplement by a long-range
|
||||
interaction, <span class="math inline">\(J_{ij} = 4\kappa J (-1)^{|i-j|}
|
||||
|i-j|^{-\alpha}\)</span>, between the spins. The normalisation, <span
|
||||
class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>,
|
||||
renders the 0th order mean field critical temperature independent of
|
||||
system size. The hopping strength of the electrons, <span
|
||||
class="math inline">\(t = 1\)</span>, sets the overall energy scale and
|
||||
we concentrate throughout on the particle-hole symmetric point at zero
|
||||
chemical potential and half filling <span class="citation"
|
||||
data-cites="gruberFalicovKimballModelReview1996"> [<a
|
||||
href="#ref-gruberFalicovKimballModelReview1996"
|
||||
role="doc-biblioref">27</a>]</span>. <span
|
||||
class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
|
||||
\tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
|
||||
<p>Dimensionality is crucial for the physics of both localisation and FTPTs. In 1D, disorder generally dominates, even the weakest disorder exponentially localises <em>all</em> single particle eigenstates. Only longer-range correlations of the disorder potential can potentially induce delocalisation <span class="citation" data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a href="#ref-aubryAnalyticityBreakingAnderson1980" role="doc-biblioref">3</a>–<a href="#ref-dunlapAbsenceLocalizationRandomdimer1990" role="doc-biblioref">5</a>]</span>. Thermodynamically, short-range interactions cannot overcome thermal defects in 1D which prevents ordered phases at nonzero temperature <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958 goldshteinPurePointSpectrum1977 abrahamsScalingTheoryLocalization1979 kramerLocalizationTheoryExperiment1993"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">6</a>–<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">9</a>]</span>. However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions <span class="citation" data-cites="kasuyaTheoryMetallicFerro1956 rudermanIndirectExchangeCoupling1954 vanvleckNoteInteractionsSpins1962 yosidaMagneticPropertiesCuMn1957"> [<a href="#ref-kasuyaTheoryMetallicFerro1956" role="doc-biblioref">10</a>–<a href="#ref-yosidaMagneticPropertiesCuMn1957" role="doc-biblioref">13</a>]</span> decays as <span class="math inline">\(r^{-1}\)</span> in 1D <span class="citation" data-cites="rusinCalculationRKKYRange2017"> [<a href="#ref-rusinCalculationRKKYRange2017" role="doc-biblioref">14</a>]</span>. This could in principle induce the necessary long-range interactions for the classical Ising background <span class="citation" data-cites="thoulessLongRangeOrderOneDimensional1969 peierlsIsingModelFerromagnetism1936"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">15</a>,<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">16</a>]</span>. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at <span class="math inline">\(T = 0\)</span> for the 1D FK model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>Here, we construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. We find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>]</span>. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.</p>
|
||||
<p>On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. In this phase, the ions are confined to one of the two sublattices, breaking the <span class="math inline">\(\mathbb{Z}_2\)</span> symmetry.</p>
|
||||
<p>In 1D, however, Periel’s argument <span class="citation" data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a href="#ref-peierlsIsingModelFerromagnetism1936" role="doc-biblioref">16</a>,<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span> states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long range coupling between the ions in order to stabilise a CDW phase in 1D. This leads to a disordered system that is gaped by the CDW background but with correlated fluctuations leading to a disorder-free correlation induced mobility edge in one dimension.</p>
|
||||
<p>The presence of the classical field makes the model amenable to an exact numerical treatment at finite temperature via a sign problem free MCMC algorithm <span class="citation" data-cites="devriesGapsDensitiesStates1993 devriesSimplifiedHubbardModel1993 antipovInteractionTunedAndersonMott2016 debskiPossibilityDetectionFinite2016 herrmannSpreadingCorrelationsFalicovKimball2018 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>–<a href="#ref-herrmannSpreadingCorrelationsFalicovKimball2018" role="doc-biblioref">23</a>]</span>. The MCMC treatment motivates a view of the classical background field as a disorder potential, which suggests an intimate link to localisation physics. Indeed, thermal fluctuations of the classical sector act as disorder potentials drawn from a thermal distribution and the emergence of disorder in a translationally invariant Hamiltonian links the FK model to recent interest in disorder-free localisation <span class="citation" data-cites="smithDisorderFreeLocalization2017 smithDynamicalLocalizationMathbbZ2018 brenesManyBodyLocalizationDynamics2018"> [<a href="#ref-smithDisorderFreeLocalization2017" role="doc-biblioref">24</a>–<a href="#ref-brenesManyBodyLocalizationDynamics2018" role="doc-biblioref">26</a>]</span>.</p>
|
||||
<p>To evaluate thermodynamic averages we perform a classical Markov Chain Monte Carlo random walk over the space of ionic configurations, at each step diagonalising the effective electronic Hamiltonian <span class="citation" data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. Using a binder-cumulant method <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">27</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">28</a>]</span>, we demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged. We then estimate the density of states and the inverse participation ratio as a function of energy to diagnose localisation properties.</p>
|
||||
<p>We interpret the FK model as a model of spinless fermions, <span class="math inline">\(c^\dagger_{i}\)</span>, hopping on a 1D lattice against a classical Ising spin background, <span class="math inline">\(S_i \in {\pm \frac{1}{2}}\)</span>. The fermions couple to the spins via an onsite interaction with strength <span class="math inline">\(U\)</span> which we supplement by a long-range interaction, <span class="math inline">\(J_{ij} = 4\kappa J (-1)^{|i-j|} |i-j|^{-\alpha}\)</span>, between the spins. The normalisation, <span class="math inline">\(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\)</span>, renders the 0th order mean field critical temperature independent of system size. The hopping strength of the electrons, <span class="math inline">\(t = 1\)</span>, sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling <span class="citation" data-cites="gruberFalicovKimballModelReview1996"> [<a href="#ref-gruberFalicovKimballModelReview1996" role="doc-biblioref">29</a>]</span>. <span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\
|
||||
& + \sum_{i, j}^{N} J_{ij} S_i S_j \nonumber
|
||||
\label{eq:HFK}\end{aligned}\]</span></p>
|
||||
<p>In two or more dimensions, the <span
|
||||
class="math inline">\(J\!=0\!\)</span> FK model has a FTPT to the CDW
|
||||
phase with non-zero staggered magnetisation <span
|
||||
class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> and
|
||||
fermionic order parameter <span class="math inline">\(f = 2
|
||||
N^{-1}|\sum_i (-1)^i \; \expval{c^\dagger_{i}c_{i}}|\)</span> <span
|
||||
class="citation"
|
||||
data-cites="antipovInteractionTunedAndersonMott2016 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">18</a>,<a
|
||||
href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006"
|
||||
role="doc-biblioref">19</a>]</span>. This only exists at zero
|
||||
temperature in the short ranged 1D model <span class="citation"
|
||||
data-cites="kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">17</a>]</span>. To study the CDW phase at finite
|
||||
temperature in 1D, we add an additional coupling that is both
|
||||
long-ranged and staggered by a factor <span
|
||||
class="math inline">\((-1)^{|i-j|}\)</span>. The additional coupling
|
||||
stabilises the Antiferromagnetic (AFM) order of the Ising spins which
|
||||
promotes the finite temperature CDW phase of the fermionic sector.</p>
|
||||
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples
|
||||
the spins from the fermions, which gives a spin sector governed by a
|
||||
classical LRI model. Note, the transformation of the spins <span
|
||||
class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model
|
||||
to the FM one. We recall that Peierls’ classic argument can be extended
|
||||
to show that, for the 1D LRI model, a power law decay of <span
|
||||
class="math inline">\(\alpha < 2\)</span> is required for a FTPT as
|
||||
the energy of defect domain then scales with the system size and can
|
||||
overcome the entropic contribution. A renormalisation group analysis
|
||||
supports this finding and shows that the critical exponents are only
|
||||
universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span
|
||||
class="citation"
|
||||
data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">15</a>,<a
|
||||
href="#ref-ruelleStatisticalMechanicsOnedimensional1968"
|
||||
role="doc-biblioref">28</a>,<a
|
||||
href="#ref-angeliniRelationsShortrangeLongrange2014"
|
||||
role="doc-biblioref">29</a>]</span>. In the following, we choose <span
|
||||
class="math inline">\(\alpha = 5/4\)</span> to avoid this additional
|
||||
complexity.</p>
|
||||
<p>To improve the scaling of finite size effects, we make the
|
||||
replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow
|
||||
|f(i - j)|^{-\alpha}\)</span>, in both <span
|
||||
class="math inline">\(J_{ij}\)</span> and <span
|
||||
class="math inline">\(\kappa\)</span>, where <span
|
||||
class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>,
|
||||
which is smooth across the circular boundary <span class="citation"
|
||||
data-cites="fukuiOrderNClusterMonte2009"> [<a
|
||||
href="#ref-fukuiOrderNClusterMonte2009"
|
||||
role="doc-biblioref">30</a>]</span>. We only consider even system sizes
|
||||
given that odd system sizes are not commensurate with a CDW state.</p>
|
||||
<p>Next Section: <a
|
||||
href="../3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
|
||||
<p>In two or more dimensions, the <span class="math inline">\(J\!=0\!\)</span> FK model has a FTPT to the CDW phase with non-zero staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \expval{c^\dagger_{i}c_{i}}|\)</span> <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016 maskaThermodynamicsTwodimensionalFalicovKimball2006"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">18</a>,<a href="#ref-maskaThermodynamicsTwodimensionalFalicovKimball2006" role="doc-biblioref">19</a>]</span>. This only exists at zero temperature in the short ranged 1D model <span class="citation" data-cites="kennedyItinerantElectronModel1986"> [<a href="#ref-kennedyItinerantElectronModel1986" role="doc-biblioref">17</a>]</span>. To study the CDW phase at finite temperature in 1D, we add an additional coupling that is both long-ranged and staggered by a factor <span class="math inline">\((-1)^{|i-j|}\)</span>. The additional coupling stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.</p>
|
||||
<p>Taking the limit <span class="math inline">\(U = 0\)</span> decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins <span class="math inline">\(S_i \to (-1)^{i} S_i\)</span> maps the AFM model to the FM one. We recall that Peierls’ classic argument can be extended to show that, for the 1D LRI model, a power law decay of <span class="math inline">\(\alpha < 2\)</span> is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for <span class="math inline">\(\alpha \leq 3/2\)</span> <span class="citation" data-cites="ruelleStatisticalMechanicsOnedimensional1968 thoulessLongRangeOrderOneDimensional1969 angeliniRelationsShortrangeLongrange2014"> [<a href="#ref-thoulessLongRangeOrderOneDimensional1969" role="doc-biblioref">15</a>,<a href="#ref-ruelleStatisticalMechanicsOnedimensional1968" role="doc-biblioref">30</a>,<a href="#ref-angeliniRelationsShortrangeLongrange2014" role="doc-biblioref">31</a>]</span>. In the following, we choose <span class="math inline">\(\alpha = 5/4\)</span> to avoid this additional complexity.</p>
|
||||
<p>To improve the scaling of finite size effects, we make the replacement <span class="math inline">\(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\)</span>, in both <span class="math inline">\(J_{ij}\)</span> and <span class="math inline">\(\kappa\)</span>, where <span class="math inline">\(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\)</span>, which is smooth across the circular boundary <span class="citation" data-cites="fukuiOrderNClusterMonte2009"> [<a href="#ref-fukuiOrderNClusterMonte2009" role="doc-biblioref">32</a>]</span>. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html">Methods</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
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||||
States of Two Hubbard-Like Models</a></em>, Phys. Rev. Lett.
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||||
<strong>70</strong>, 2463 (1993).</div>
|
||||
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|
||||
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|
||||
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||||
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||||
de Vries, K. Michielsen, and H. De Raedt, <em><a
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||||
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||||
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J. Herrmann, A. E. Antipov, and P. Werner, <em><a
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||||
Correlations in the Falicov-Kimball Model</a></em>, Phys. Rev. B
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||||
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||||
</div>
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||||
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Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin, <em><a
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href="https://doi.org/10.1103/PhysRevB.97.245137">Dynamical Localization
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in $\Mathbb{Z}_2$ Lattice Gauge Theories</a></em>, Phys. Rev. B
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<strong>97</strong>, 245137 (2018).</div>
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||||
</div>
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||||
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Brenes, M. Dalmonte, M. Heyl, and A. Scardicchio, <em><a
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href="https://doi.org/10.1103/PhysRevLett.120.030601">Many-Body
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||||
Localization Dynamics from Gauge Invariance</a></em>, Phys. Rev. Lett.
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<strong>120</strong>, 030601 (2018).</div>
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<div id="ref-smithDynamicalLocalizationMathbbZ2018" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[25] </div><div class="csl-right-inline">A. Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin, <em><a href="https://doi.org/10.1103/PhysRevB.97.245137">Dynamical Localization in $\Mathbb{Z}_2$ Lattice Gauge Theories</a></em>, Phys. Rev. B <strong>97</strong>, 245137 (2018).</div>
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||||
</div>
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<div class="csl-left-margin">[26] </div><div class="csl-right-inline">K.
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Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
|
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Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
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Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
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<div class="csl-left-margin">[26] </div><div class="csl-right-inline">M. Brenes, M. Dalmonte, M. Heyl, and A. Scardicchio, <em><a href="https://doi.org/10.1103/PhysRevLett.120.030601">Many-Body Localization Dynamics from Gauge Invariance</a></em>, Phys. Rev. Lett. <strong>120</strong>, 030601 (2018).</div>
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||||
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<div class="csl-left-margin">[27] </div><div class="csl-right-inline">C.
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Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact
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||||
Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>,
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(1996).</div>
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<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[27] </div><div class="csl-right-inline">K. Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size Scaling Analysis of Ising Model Block Distribution Functions</a></em>, Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
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class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[28] </div><div class="csl-right-inline">D.
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Ruelle, <em><a
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href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical
|
||||
Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys.
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<strong>9</strong>, 267 (1968).</div>
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<div class="csl-left-margin">[28] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
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</div>
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<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[29] </div><div class="csl-right-inline">M.
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C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a
|
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href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between
|
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Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E
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<strong>89</strong>, 062120 (2014).</div>
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<div id="ref-gruberFalicovKimballModelReview1996" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[29] </div><div class="csl-right-inline">C. Gruber and N. Macris, <em>The Falicov-Kimball Model: A Review of Exact Results and Extensions</em>, Helvetica Physica Acta <strong>69</strong>, (1996).</div>
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</div>
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<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[30] </div><div class="csl-right-inline">K.
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Fukui and S. Todo, <em><a
|
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href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte
|
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Carlo Method for Spin Systems with Long-Range Interactions</a></em>,
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Journal of Computational Physics <strong>228</strong>, 2629
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(2009).</div>
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<div id="ref-ruelleStatisticalMechanicsOnedimensional1968" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[30] </div><div class="csl-right-inline">D. Ruelle, <em><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0234697">Statistical Mechanics of a One-Dimensional Lattice Gas</a></em>, Comm. Math. Phys. <strong>9</strong>, 267 (1968).</div>
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</div>
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<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[31] </div><div class="csl-right-inline">M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, <em><a href="https://doi.org/10.1103/PhysRevE.89.062120">Relations Between Short-Range and Long-Range Ising Models</a></em>, Phys. Rev. E <strong>89</strong>, 062120 (2014).</div>
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||||
</div>
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<div id="ref-fukuiOrderNClusterMonte2009" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[32] </div><div class="csl-right-inline">K. Fukui and S. Todo, <em><a href="https://doi.org/10.1016/j.jcp.2008.12.022">Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions</a></em>, Journal of Computational Physics <strong>228</strong>, 2629 (2009).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -29,31 +29,18 @@ image:
|
||||
<ul>
|
||||
<li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model"
|
||||
id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK
|
||||
Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder"
|
||||
id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain
|
||||
Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model"
|
||||
id="toc-application-to-the-fk-model">Application to the FK
|
||||
Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step
|
||||
Trick</a></li>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model" id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder" id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model" id="toc-application-to-the-fk-model">Application to the FK Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder
|
||||
Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation"
|
||||
id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation" id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#inverse-participation-ratio"
|
||||
id="toc-inverse-participation-ratio">Inverse Participation
|
||||
Ratio</a></li>
|
||||
<li><a href="#inverse-participation-ratio" id="toc-inverse-participation-ratio">Inverse Participation Ratio</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence
|
||||
Time</a></li>
|
||||
<li><a href="#auto-correlation-time"
|
||||
id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -70,31 +57,18 @@ id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
<ul>
|
||||
<li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model"
|
||||
id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK
|
||||
Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder"
|
||||
id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain
|
||||
Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model"
|
||||
id="toc-application-to-the-fk-model">Application to the FK
|
||||
Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step
|
||||
Trick</a></li>
|
||||
<li><a href="#thermodynamics-of-the-lrfk-model" id="toc-thermodynamics-of-the-lrfk-model">Thermodynamics of the LRFK Model</a></li>
|
||||
<li><a href="#markov-chain-monte-carlo-and-emergent-disorder" id="toc-markov-chain-monte-carlo-and-emergent-disorder">Markov Chain Monte Carlo and Emergent Disorder</a></li>
|
||||
<li><a href="#application-to-the-fk-model" id="toc-application-to-the-fk-model">Application to the FK Model</a></li>
|
||||
<li><a href="#two-step-trick" id="toc-two-step-trick">Two Step Trick</a></li>
|
||||
<li><a href="#scaling" id="toc-scaling">Scaling</a></li>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder
|
||||
Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation"
|
||||
id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<li><a href="#binder-cumulants" id="toc-binder-cumulants">Binder Cumulants</a></li>
|
||||
<li><a href="#diagnostics-of-localisation" id="toc-diagnostics-of-localisation">Diagnostics of Localisation</a>
|
||||
<ul>
|
||||
<li><a href="#inverse-participation-ratio"
|
||||
id="toc-inverse-participation-ratio">Inverse Participation
|
||||
Ratio</a></li>
|
||||
<li><a href="#inverse-participation-ratio" id="toc-inverse-participation-ratio">Inverse Participation Ratio</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence
|
||||
Time</a></li>
|
||||
<li><a href="#auto-correlation-time"
|
||||
id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
<li><a href="#convergence-time" id="toc-convergence-time">Convergence Time</a></li>
|
||||
<li><a href="#auto-correlation-time" id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -110,597 +84,152 @@ id="toc-auto-correlation-time">Auto-correlation Time</a></li>
|
||||
<h1>Methods</h1>
|
||||
<section id="thermodynamics-of-the-lrfk-model" class="level2">
|
||||
<h2>Thermodynamics of the LRFK Model</h2>
|
||||
<p>The results for the phase diagram were obtained with a classical
|
||||
Markov Chain Monte Carlo (MCMC) method which we discuss in the
|
||||
following. It allows us to solve our long-range FK model efficiently,
|
||||
yielding unbiased estimates of thermal expectation values and linking it
|
||||
to disorder physics in a translationally invariant setting.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be
|
||||
split into a classical spin part <span
|
||||
class="math inline">\(H_s\)</span> and an operator valued part <span
|
||||
class="math inline">\(H_c\)</span>.</p>
|
||||
<p>The results for the phase diagram were obtained with a classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following. It allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values and linking it to disorder physics in a translationally invariant setting.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} +
|
||||
c^\dagger_{i+1}c_{i}) \end{aligned}\]</span></p>
|
||||
<p>The partition function can then be written as a sum over spin
|
||||
configurations, <span class="math inline">\(\vec{S} = (S_0,
|
||||
S_1...S_{N-1})\)</span>:</p>
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}) \end{aligned}\]</span></p>
|
||||
<p>The partition function can then be written as a sum over spin configurations, <span class="math inline">\(\vec{S} = (S_0, S_1...S_{N-1})\)</span>:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s}
|
||||
\mathrm{Tr}_c e^{-\beta H_c} .\end{aligned}\]</span></p>
|
||||
<p>The contribution of <span class="math inline">\(H_c\)</span> to the
|
||||
grand canonical partition function can be obtained by performing the sum
|
||||
over eigenstate occupation numbers giving <span
|
||||
class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta
|
||||
\epsilon_k})}\)</span> where <span
|
||||
class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues
|
||||
of the matrix representation of <span class="math inline">\(H_c\)</span>
|
||||
determined through exact diagonalisation. This gives a partition
|
||||
function containing a classical energy which corresponds to the
|
||||
long-range interaction of the spins, and a free energy which corresponds
|
||||
to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta
|
||||
F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta
|
||||
E[\vec{S}]}\end{aligned}\]</span></p>
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .\end{aligned}\]</span></p>
|
||||
<p>The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]</span></p>
|
||||
</section>
|
||||
<section id="markov-chain-monte-carlo-and-emergent-disorder"
|
||||
class="level2">
|
||||
<section id="markov-chain-monte-carlo-and-emergent-disorder" class="level2">
|
||||
<h2>Markov Chain Monte Carlo and Emergent Disorder</h2>
|
||||
<p>Classical MCMC defines a weighted random walk over the spin states
|
||||
<span class="math inline">\((\vec{S}_0, \vec{S}_1, \vec{S}_2,
|
||||
...)\)</span>, such that the likelihood of visiting a particular state
|
||||
converges to its Boltzmann probability <span
|
||||
class="math inline">\(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta
|
||||
E}\)</span> <span class="citation"
|
||||
data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a
|
||||
href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">1</a>–<a
|
||||
href="#ref-wolffMonteCarloErrors2004"
|
||||
role="doc-biblioref">3</a>]</span>. Hence, any observable can be
|
||||
estimated as a mean over the states visited by the walk, <span
|
||||
class="math display">\[\begin{aligned}
|
||||
<p>Classical MCMC defines a weighted random walk over the spin states <span class="math inline">\((\vec{S}_0, \vec{S}_1, \vec{S}_2, ...)\)</span>, such that the likelihood of visiting a particular state converges to its Boltzmann probability <span class="math inline">\(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta E}\)</span>. Hence, any observable can be estimated as a mean over the states visited by the walk <span class="citation" data-cites="binderGuidePracticalWork1988 kerteszAdvancesComputerSimulation1998 wolffMonteCarloErrors2004"> [<a href="#ref-binderGuidePracticalWork1988" role="doc-biblioref">1</a>–<a href="#ref-wolffMonteCarloErrors2004" role="doc-biblioref">3</a>]</span>, <span class="math display">\[\begin{aligned}
|
||||
\label{eq:thermal_expectation}
|
||||
\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O
|
||||
\rangle_{\vec{S}}\\
|
||||
& = \sum_{i = 0}^{M} \langle O\rangle_{\vec{S}_i}
|
||||
\pm \mathcal{O}(\tfrac{1}{\sqrt{M}})
|
||||
\end{aligned}\]</span> where the former sum runs over the entire state
|
||||
space while the later runs over all the state visited by a particular
|
||||
MCMC run.</p>
|
||||
\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O \rangle_{\vec{S}}\\
|
||||
& = \sum_{i = 0}^{M} \langle O\rangle_{\vec{S}_i} \pm \mathcal{O}(\tfrac{1}{\sqrt{M}})
|
||||
\end{aligned}\]</span> where the former sum runs over the entire state space while the later runs over all the state visited by a particular MCMC run.</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\langle O \rangle_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu})
|
||||
\langle O \rangle{\nu}
|
||||
\langle O \rangle_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) \langle O \rangle{\nu}
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Where <span class="math inline">\(\nu\)</span> runs over the
|
||||
eigenstates of <span class="math inline">\(H_c\)</span> for a particular
|
||||
spin configuration and <span class="math inline">\(n_F(\epsilon) =
|
||||
\left(e^{-\beta\epsilon} + 1\right)^{-1}\)</span> is the Fermi
|
||||
function.</p>
|
||||
<p>The choice of the transition function for MCMC is under-determined as
|
||||
one only needs to satisfy a set of balance conditions for which there
|
||||
are many solutions <span class="citation"
|
||||
data-cites="kellyReversibilityStochasticNetworks1981"> [<a
|
||||
href="#ref-kellyReversibilityStochasticNetworks1981"
|
||||
role="doc-biblioref">4</a>]</span>. Here, we incorporate a modification
|
||||
to the standard Metropolis-Hastings algorithm <span class="citation"
|
||||
data-cites="hastingsMonteCarloSampling1970"> [<a
|
||||
href="#ref-hastingsMonteCarloSampling1970"
|
||||
role="doc-biblioref">5</a>]</span> gleaned from Krauth <span
|
||||
class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a
|
||||
href="#ref-krauthIntroductionMonteCarlo1998"
|
||||
role="doc-biblioref">6</a>]</span>. Let us first recall the standard
|
||||
algorithm which decomposes the transition probability into <span
|
||||
class="math inline">\(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to
|
||||
b)\)</span>. Here, <span class="math inline">\(p\)</span> is the
|
||||
proposal distribution that we can directly sample from while <span
|
||||
class="math inline">\(\mathcal{A}\)</span> is the acceptance
|
||||
probability. The standard Metropolis-Hastings choice is <span
|
||||
class="math display">\[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to
|
||||
a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</span> with <span
|
||||
class="math inline">\(\Delta E = E_b - E_a\)</span>. The walk then
|
||||
proceeds by sampling a state <span class="math inline">\(b\)</span> from
|
||||
<span class="math inline">\(p\)</span> and moving to <span
|
||||
class="math inline">\(b\)</span> with probability <span
|
||||
class="math inline">\(\mathcal{A}(a \to b)\)</span>. The latter
|
||||
operation is typically implemented by performing a transition if a
|
||||
uniform random sample from the unit interval is less than <span
|
||||
class="math inline">\(\mathcal{A}(a \to b)\)</span> and otherwise
|
||||
repeating the current state as the next step in the random walk. The
|
||||
proposal distribution is often symmetric so does not appear in <span
|
||||
class="math inline">\(\mathcal{A}\)</span>. Here, we flip a small number
|
||||
of sites in <span class="math inline">\(b\)</span> at random to generate
|
||||
proposals, which is indeed symmetric.</p>
|
||||
<p>In our computations <span class="citation"
|
||||
data-cites="hodsonMCMCFKModel2021"> [<a
|
||||
href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we
|
||||
employ a modification of the algorithm which is based on the observation
|
||||
that the free energy of the <span data-acronym-label="FK"
|
||||
data-acronym-form="singular+short">FK</span> system is composed of a
|
||||
classical part which is much quicker to compute than the quantum part.
|
||||
Hence, we can obtain a computational speedup by first considering the
|
||||
value of the classical energy difference <span
|
||||
class="math inline">\(\Delta H_s\)</span> and rejecting the transition
|
||||
if the former is too high. We only compute the quantum energy difference
|
||||
<span class="math inline">\(\Delta F_c\)</span> if the transition is
|
||||
accepted. We then perform a second rejection sampling step based upon
|
||||
it. This corresponds to two nested comparisons with the majority of the
|
||||
work only occurring if the first test passes and has the acceptance
|
||||
function <span class="math display">\[\mathcal{A}(a \to b) =
|
||||
\min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta
|
||||
F_c}\right)\;.\]</span></p>
|
||||
<p>See Appendix <a href="#app:balance" data-reference-type="ref"
|
||||
data-reference="app:balance">[app:balance]</a> for a proof that this
|
||||
satisfies the detailed balance condition.</p>
|
||||
<p>For the model parameters used in Fig. <a href="#fig:indiv_IPR"
|
||||
data-reference-type="ref" data-reference="fig:indiv_IPR">1</a>, we find
|
||||
that with our new scheme the matrix diagonalisation is skipped around
|
||||
30% of the time at <span class="math inline">\(T = 2.5\)</span> and up
|
||||
to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that
|
||||
for <span class="math inline">\(N = 50\)</span>, the matrix
|
||||
diagonalisation, if it occurs, occupies around 60% of the total
|
||||
computation time for a single step. This rises to 90% at N = 300 and
|
||||
further increases for larger N. We therefore get the greatest speedup
|
||||
for large system sizes at low temperature where many prospective
|
||||
transitions are rejected at the classical stage and the matrix
|
||||
computation takes up the greatest fraction of the total computation
|
||||
time. The upshot is that we find a speedup of up to a factor of 10 at
|
||||
the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common
|
||||
method for speeding up <span data-acronym-label="MCMC"
|
||||
data-acronym-form="singular+short">MCMC</span> which is to add asymmetry
|
||||
to the proposal distribution to make it as similar as possible to <span
|
||||
class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>.
|
||||
This reduces the number of rejected states, which brings the algorithm
|
||||
closer in efficiency to a direct sampling method. However it comes at
|
||||
the expense of requiring a way to directly sample from this complex
|
||||
distribution, a problem which <span data-acronym-label="MCMC"
|
||||
data-acronym-form="singular+short">MCMC</span> was employed to solve in
|
||||
the first place. For example, recent work trains restricted Boltzmann
|
||||
machines (RBMs) to generate samples for the proposal distribution of the
|
||||
<span data-acronym-label="FK"
|
||||
data-acronym-form="singular+short">FK</span> model <span
|
||||
class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a
|
||||
href="#ref-huangAcceleratedMonteCarlo2017"
|
||||
role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a
|
||||
parametrisation of the proposal distribution that can be efficiently
|
||||
sampled from while offering sufficient flexibility that they can be
|
||||
adjusted to match the target distribution. Our proposed method is
|
||||
considerably simpler and does not require training while still reaping
|
||||
some of the benefits of reduced computation.</p>
|
||||
<p>Where <span class="math inline">\(\nu\)</span> runs over the eigenstates of <span class="math inline">\(H_c\)</span> for a particular spin configuration and <span class="math inline">\(n_F(\epsilon) = \left(e^{-\beta\epsilon} + 1\right)^{-1}\)</span> is the Fermi function.</p>
|
||||
<p>The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions <span class="citation" data-cites="kellyReversibilityStochasticNetworks1981"> [<a href="#ref-kellyReversibilityStochasticNetworks1981" role="doc-biblioref">4</a>]</span>. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">5</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">6</a>]</span>. Let us first recall the standard algorithm which decomposes the transition probability into <span class="math inline">\(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to b)\)</span>. Here, <span class="math inline">\(p\)</span> is the proposal distribution that we can directly sample from while <span class="math inline">\(\mathcal{A}\)</span> is the acceptance probability. The standard Metropolis-Hastings choice is <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]</span> with <span class="math inline">\(\Delta E = E_b - E_a\)</span>. The walk then proceeds by sampling a state <span class="math inline">\(b\)</span> from <span class="math inline">\(p\)</span> and moving to <span class="math inline">\(b\)</span> with probability <span class="math inline">\(\mathcal{A}(a \to b)\)</span>. The latter operation is typically implemented by performing a transition if a uniform random sample from the unit interval is less than <span class="math inline">\(\mathcal{A}(a \to b)\)</span> and otherwise repeating the current state as the next step in the random walk. The proposal distribution is often symmetric so does not appear in <span class="math inline">\(\mathcal{A}\)</span>. Here, we flip a small number of sites in <span class="math inline">\(b\)</span> at random to generate proposals, which is indeed symmetric.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we employ a modification of the algorithm which is based on the observation that the free energy of the <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speedup by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes and has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters used in Fig. [1], we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
</section>
|
||||
<section id="application-to-the-fk-model" class="level2">
|
||||
<h2>Application to the FK Model</h2>
|
||||
<p>We will work in the grand canonical ensemble of fixed temperature,
|
||||
chemical potential and volume.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be
|
||||
split into a classical spin part <span
|
||||
class="math inline">\(H_s\)</span> and an operator valued part <span
|
||||
class="math inline">\(H_c\)</span>. <span
|
||||
class="math display">\[\begin{aligned}
|
||||
<p>We will work in the grand canonical ensemble of fixed temperature, chemical potential and volume.</p>
|
||||
<p>Since the spin configurations are classical, the Hamiltonian can be split into a classical spin part <span class="math inline">\(H_s\)</span> and an operator valued part <span class="math inline">\(H_c\)</span>. <span class="math display">\[\begin{aligned}
|
||||
H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} +
|
||||
c^\dagger_{i+1}c_{i})
|
||||
\end{aligned}\]</span> The partition function can then be written as a
|
||||
sum over spin configurations, <span class="math inline">\(\vec{S} =
|
||||
(S_0, S_1...S_{N-1})\)</span>: <span
|
||||
class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s}
|
||||
\mathrm{Tr}_c e^{-\beta H_c} .
|
||||
H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i})
|
||||
\end{aligned}\]</span> The partition function can then be written as a sum over spin configurations, <span class="math inline">\(\vec{S} = (S_0, S_1...S_{N-1})\)</span>: <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .
|
||||
\end{aligned}
|
||||
\]</span> The contribution of <span class="math inline">\(H_c\)</span>
|
||||
to the grand canonical partition function can be obtained by performing
|
||||
the sum over eigenstate occupation numbers giving <span
|
||||
class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta
|
||||
\epsilon_k})}\)</span> where <span
|
||||
class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues
|
||||
of the matrix representation of <span class="math inline">\(H_c\)</span>
|
||||
determined through exact diagonalisation. This gives a partition
|
||||
function containing a classical energy which corresponds to the
|
||||
long-range interaction of the spins, and a free energy which corresponds
|
||||
to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta
|
||||
F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
|
||||
\]</span> The contribution of <span class="math inline">\(H_c\)</span> to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving <span class="math inline">\(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\)</span> where <span class="math inline">\({\epsilon_k[\vec{S}]}\)</span> are the eigenvalues of the matrix representation of <span class="math inline">\(H_c\)</span> determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem. <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}
|
||||
\end{aligned}\]</span></p>
|
||||
</section>
|
||||
<section id="two-step-trick" class="level2">
|
||||
<h2>Two Step Trick</h2>
|
||||
<p>Here, we incorporate a modification to the standard
|
||||
Metropolis-Hastings algorithm <span class="citation"
|
||||
data-cites="hastingsMonteCarloSampling1970"> [<a
|
||||
href="#ref-hastingsMonteCarloSampling1970"
|
||||
role="doc-biblioref">5</a>]</span> gleaned from Krauth <span
|
||||
class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a
|
||||
href="#ref-krauthIntroductionMonteCarlo1998"
|
||||
role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>In our computations <span class="citation"
|
||||
data-cites="hodsonMCMCFKModel2021"> [<a
|
||||
href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we
|
||||
employ a modification of the algorithm which is based on the observation
|
||||
that the free energy of the FK system is composed of a classical part
|
||||
which is much quicker to compute than the quantum part. Hence, we can
|
||||
obtain a computational speedup by first considering the value of the
|
||||
classical energy difference <span class="math inline">\(\Delta
|
||||
H_s\)</span> and rejecting the transition if the former is too high. We
|
||||
only compute the quantum energy difference <span
|
||||
class="math inline">\(\Delta F_c\)</span> if the transition is accepted.
|
||||
We then perform a second rejection sampling step based upon it. This
|
||||
corresponds to two nested comparisons with the majority of the work only
|
||||
occurring if the first test passes and has the acceptance function <span
|
||||
class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta
|
||||
\Delta H_s}\right)\min\left(1, e^{-\beta \Delta
|
||||
F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters <span class="math inline">\(U=2/5, T = 1.5 /
|
||||
2.5, J = 5,\;\alpha = 1.25\)</span>, we find that with our new scheme
|
||||
the matrix diagonalisation is skipped around 30% of the time at <span
|
||||
class="math inline">\(T = 2.5\)</span> and up to 80% at <span
|
||||
class="math inline">\(T = 1.5\)</span>. We observe that for <span
|
||||
class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it
|
||||
occurs, occupies around 60% of the total computation time for a single
|
||||
step. This rises to 90% at N = 300 and further increases for larger N.
|
||||
We therefore get the greatest speedup for large system sizes at low
|
||||
temperature where many prospective transitions are rejected at the
|
||||
classical stage and the matrix computation takes up the greatest
|
||||
fraction of the total computation time. The upshot is that we find a
|
||||
speedup of up to a factor of 10 at the cost of very little extra
|
||||
algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common
|
||||
method for speeding up MCMC which is to add asymmetry to the proposal
|
||||
distribution to make it as similar as possible to <span
|
||||
class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>.
|
||||
This reduces the number of rejected states, which brings the algorithm
|
||||
closer in efficiency to a direct sampling method. However it comes at
|
||||
the expense of requiring a way to directly sample from this complex
|
||||
distribution, a problem which MCMC was employed to solve in the first
|
||||
place. For example, recent work trains restricted Boltzmann machines
|
||||
(RBMs) to generate samples for the proposal distribution of the FK
|
||||
model <span class="citation"
|
||||
data-cites="huangAcceleratedMonteCarlo2017"> [<a
|
||||
href="#ref-huangAcceleratedMonteCarlo2017"
|
||||
role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a
|
||||
parametrisation of the proposal distribution that can be efficiently
|
||||
sampled from while offering sufficient flexibility that they can be
|
||||
adjusted to match the target distribution. Our proposed method is
|
||||
considerably simpler and does not require training while still reaping
|
||||
some of the benefits of reduced computation.</p>
|
||||
<p>Here, we incorporate a modification to the standard Metropolis-Hastings algorithm <span class="citation" data-cites="hastingsMonteCarloSampling1970"> [<a href="#ref-hastingsMonteCarloSampling1970" role="doc-biblioref">5</a>]</span> gleaned from Krauth <span class="citation" data-cites="krauthIntroductionMonteCarlo1998"> [<a href="#ref-krauthIntroductionMonteCarlo1998" role="doc-biblioref">6</a>]</span>.</p>
|
||||
<p>In our computations <span class="citation" data-cites="hodsonMCMCFKModel2021"> [<a href="#ref-hodsonMCMCFKModel2021" role="doc-biblioref">7</a>]</span> we employ a modification of the algorithm which is based on the observation that the free energy of the FK system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speedup by first considering the value of the classical energy difference <span class="math inline">\(\Delta H_s\)</span> and rejecting the transition if the former is too high. We only compute the quantum energy difference <span class="math inline">\(\Delta F_c\)</span> if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes and has the acceptance function <span class="math display">\[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]</span></p>
|
||||
<p>For the model parameters <span class="math inline">\(U=2/5, T = 1.5 / 2.5, J = 5,\;\alpha = 1.25\)</span>, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at <span class="math inline">\(T = 2.5\)</span> and up to 80% at <span class="math inline">\(T = 1.5\)</span>. We observe that for <span class="math inline">\(N = 50\)</span>, the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.</p>
|
||||
<p>Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to <span class="math inline">\(\min\left(1, e^{-\beta \Delta E}\right)\)</span>. This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model <span class="citation" data-cites="huangAcceleratedMonteCarlo2017"> [<a href="#ref-huangAcceleratedMonteCarlo2017" role="doc-biblioref">8</a>]</span>. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.</p>
|
||||
</section>
|
||||
<section id="scaling" class="level2">
|
||||
<h2>Scaling</h2>
|
||||
<p>In order to reduce the effects of the boundary conditions and the
|
||||
finite size of the system we redefine and normalise the coupling matrix
|
||||
to have 0 derivative at its furthest extent rather than cutting off
|
||||
abruptly.</p>
|
||||
<p>In order to reduce the effects of the boundary conditions and the finite size of the system we redefine and normalise the coupling matrix to have 0 derivative at its furthest extent rather than cutting off abruptly.</p>
|
||||
<p><span class="math display">\[
|
||||
\begin{aligned}
|
||||
J'(x) &= \abs{\frac{L}{\pi}\sin \frac{\pi x}{L}}^{-\alpha} \\
|
||||
J(x) &= \frac{J_0 J'(x)}{\sum_y J'(y)}
|
||||
\end{aligned}\]</span> % The scaling ensures that, in the ordered phase,
|
||||
the overall potential felt by each site due to the rest of the system is
|
||||
independent of system size.</p>
|
||||
\end{aligned}\]</span> % The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.</p>
|
||||
</section>
|
||||
<section id="binder-cumulants" class="level2">
|
||||
<h2>Binder Cumulants</h2>
|
||||
<p>The Binder cumulant is defined as: <span class="math display">\[U_B =
|
||||
1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\]</span> % where <span
|
||||
class="math display">\[\mu_n = \tex{(m - \tex{m})^n}\]</span> % are the
|
||||
central moments of the order parameter m: <span class="math display">\[m
|
||||
= \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated
|
||||
against temperature can be used as a diagnostic for the existence of a
|
||||
phase transition. If multiple such curves are plotted for different
|
||||
system sizes, a crossing indicates the location of a critical
|
||||
point <span class="citation"
|
||||
data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a
|
||||
href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">9</a>,<a
|
||||
href="#ref-musialMonteCarloSimulations2002"
|
||||
role="doc-biblioref"><strong>musialMonteCarloSimulations2002?</strong></a>]</span>.</p>
|
||||
<p>The Binder cumulant is defined as: <span class="math display">\[U_B = 1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\]</span> % where <span class="math display">\[\mu_n = \tex{(m - \tex{m})^n}\]</span> % are the central moments of the order parameter m: <span class="math display">\[m = \sum_i (-1)^i (2n_i - 1) / N\]</span> % The Binder cumulant evaluated against temperature can be used as a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point <span class="citation" data-cites="binderFiniteSizeScaling1981 musialMonteCarloSimulations2002"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">9</a>,<a href="#ref-musialMonteCarloSimulations2002" role="doc-biblioref">10</a>]</span>.</p>
|
||||
</section>
|
||||
<section id="diagnostics-of-localisation" class="level2">
|
||||
<h2>Diagnostics of Localisation</h2>
|
||||
<section id="inverse-participation-ratio" class="level3">
|
||||
<h3>Inverse Participation Ratio</h3>
|
||||
<p>The inverse participation ratio is defined for a normalised wave
|
||||
function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i
|
||||
\abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation"
|
||||
data-cites="kramerLocalizationTheoryExperiment1993"> [<a
|
||||
href="#ref-kramerLocalizationTheoryExperiment1993"
|
||||
role="doc-biblioref">10</a>]</span>: <span class="math display">\[
|
||||
<p>The inverse participation ratio is defined for a normalised wave function <span class="math inline">\(\psi_i = \psi(x_i), \sum_i \abs{\psi_i}^2 = 1\)</span> as its fourth moment <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">11</a>]</span>: <span class="math display">\[
|
||||
P^{-1} = \sum_i \abs{\psi_i}^4
|
||||
\]</span> % It acts as a measure of the portion of space occupied by the
|
||||
wave function. For localised states it will be independent of system
|
||||
size while for plane wave states in d dimensions $P = L^d $. States may
|
||||
also be intermediate between localised and extended, described by their
|
||||
fractal dimensionality <span class="math inline">\(d > d* >
|
||||
0\)</span>: <span class="math display">\[
|
||||
\]</span> % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system size while for plane wave states in d dimensions $P = L^d $. States may also be intermediate between localised and extended, described by their fractal dimensionality <span class="math inline">\(d > d* > 0\)</span>: <span class="math display">\[
|
||||
P(L) \goeslike L^{d*}
|
||||
\]</span> % For extended states <span class="math inline">\(d* =
|
||||
0\)</span> while for localised ones <span class="math inline">\(d* =
|
||||
0\)</span>. In this work we take use an energy resolved IPR <span
|
||||
class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
|
||||
href="#ref-andersonAbsenceDiffusionCertain1958"
|
||||
role="doc-biblioref">11</a>]</span>: <span class="math display">\[
|
||||
\]</span> % For extended states <span class="math inline">\(d* = 0\)</span> while for localised ones <span class="math inline">\(d* = 0\)</span>. In this work we take use an energy resolved IPR <span class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a href="#ref-andersonAbsenceDiffusionCertain1958" role="doc-biblioref">12</a>]</span>: <span class="math display">\[
|
||||
DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n)
|
||||
\abs{\psi_{n,i}}^4
|
||||
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the
|
||||
wavefunction corresponding to the energy <span
|
||||
class="math inline">\(\epsilon_n\)</span> at the ith site. In practice
|
||||
we bin the energies and IPRs into a fine energy grid and use Lorentzian
|
||||
smoothing if necessary.</p>
|
||||
<div id="fig:raw" class="fignos">
|
||||
IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4
|
||||
\]</span> Where <span class="math inline">\(\psi_{n,i}\)</span> is the wavefunction corresponding to the energy <span class="math inline">\(\epsilon_n\)</span> at the ith site. In practice we bin the energies and IPRs into a fine energy grid and use Lorentzian smoothing if necessary.</p>
|
||||
<figure>
|
||||
<embed src="figs/lsr/raw_steps_single_flip.pdf" />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span> An MCMC walk
|
||||
starting from the staggered charge density wave ground state for a
|
||||
system with <span class="math inline">\(N = 100\)</span> sites and
|
||||
10,000 MCMC steps. In this simulation only a single spin can be flipped
|
||||
per step according to the Metropolis-Hastings Algorithm. The staggered
|
||||
magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \;
|
||||
S_i\)</span> order parameter is plotted below. At this temperature the
|
||||
thermal average of m is zero, while the initial state has m = 1. We see
|
||||
that it takes about 1000 steps for the system to converge, after which
|
||||
it moves about the m = 0 average with a finite auto-correlation time.
|
||||
<span class="math inline">\(t = 1, \alpha = 1.25, T = 3, J = U =
|
||||
5\)</span> <span id="fig:raw"
|
||||
label="fig:raw">[fig:raw]</span></figcaption>
|
||||
<embed src="figs/lsr/raw_steps_single_flip.pdf" id="fig:raw" />
|
||||
<figcaption aria-hidden="true">Figure 1: An MCMC walk starting from the staggered charge density wave ground state for a system with <span class="math inline">\(N = 100\)</span> sites and 10,000 MCMC steps. In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\)</span> order parameter is plotted below. At this temperature the thermal average of m is zero, while the initial state has m = 1. We see that it takes about 1000 steps for the system to converge, after which it moves about the m = 0 average with a finite auto-correlation time. <span class="math inline">\(t = 1, \alpha = 1.25, T = 3, J = U = 5\)</span> <span id="fig:raw" label="fig:raw">[fig:raw]</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p><span data-acronym-label="MCMC"
|
||||
data-acronym-form="singular+short">MCMC</span> sidesteps these issues by
|
||||
defining a random walk that focuses on the states with the greatest
|
||||
Boltzmann weight. At low temperatures this means we need only visit a
|
||||
few low energy states to make good estimates while at high temperatures
|
||||
the weights become uniform so a small number of samples distributed
|
||||
across the state space suffice. However we will see that the method is
|
||||
not without difficulties of its own.</p>
|
||||
<div id="fig:single" class="fignos">
|
||||
<p><span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> sidesteps these issues by defining a random walk that focuses on the states with the greatest Boltzmann weight. At low temperatures this means we need only visit a few low energy states to make good estimates while at high temperatures the weights become uniform so a small number of samples distributed across the state space suffice. However we will see that the method is not without difficulties of its own.</p>
|
||||
<figure>
|
||||
<embed src="figs/lsr/single.pdf" />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> Two MCMC chains
|
||||
starting from the same initial state for a system with <span
|
||||
class="math inline">\(N = 90\)</span> sites and 1000 MCMC steps. In this
|
||||
simulation the MCMC step is defined differently: an attempt is made to
|
||||
flip n spins, where n is drawn from Uniform(1,N). This is repeated <span
|
||||
class="math inline">\(N^2/100\)</span> times for each step. This trades
|
||||
off computation time for storage space, as it makes the samples less
|
||||
correlated, giving smaller statistical error for a given number of
|
||||
stored samples. These simulations therefore have the potential to
|
||||
necessitate <span class="math inline">\(N^2/100\)</span> matrix
|
||||
diagonalisations for every MCMC sample, though this can be cut down with
|
||||
caching and other tricks. <span class="math inline">\(t = 1, \alpha =
|
||||
1.25, T = 2.2, J = U = 5\)</span> <span id="fig:single"
|
||||
label="fig:single">[fig:single]</span></figcaption>
|
||||
<embed src="figs/lsr/single.pdf" id="fig:single" />
|
||||
<figcaption aria-hidden="true">Figure 2: Two MCMC chains starting from the same initial state for a system with <span class="math inline">\(N = 90\)</span> sites and 1000 MCMC steps. In this simulation the MCMC step is defined differently: an attempt is made to flip n spins, where n is drawn from Uniform(1,N). This is repeated <span class="math inline">\(N^2/100\)</span> times for each step. This trades off computation time for storage space, as it makes the samples less correlated, giving smaller statistical error for a given number of stored samples. These simulations therefore have the potential to necessitate <span class="math inline">\(N^2/100\)</span> matrix diagonalisations for every MCMC sample, though this can be cut down with caching and other tricks. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.2, J = U = 5\)</span> <span id="fig:single" label="fig:single">[fig:single]</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>In implementation <span data-acronym-label="MCMC"
|
||||
data-acronym-form="singular+short">MCMC</span> can be boiled down to
|
||||
choosing a transition function <span
|
||||
class="math inline">\(\mathcal{T}(\s_{t} \rightarrow \s_t+1)\)</span>
|
||||
where <span class="math inline">\(\s\)</span> are vectors representing
|
||||
classical spin configurations. We start in some initial state <span
|
||||
class="math inline">\(\s_0\)</span> and then repeatedly jump to new
|
||||
states according to the probabilities given by <span
|
||||
class="math inline">\(\mathcal{T}\)</span>. This defines a set of random
|
||||
walks <span class="math inline">\(\{\s_0\ldots \s_i\ldots
|
||||
\s_N\}\)</span>. Fig. <a href="#fig:single" data-reference-type="ref"
|
||||
data-reference="fig:single">2</a> shows this in practice: we have a
|
||||
(rather small) ensemble of <span class="math inline">\(M = 2\)</span>
|
||||
walkers starting at the same point in state space and then spreading
|
||||
outwards by flipping spins along the way.</p>
|
||||
<p>In pseudo-code one could write the MCMC simulation for a single
|
||||
walker as:</p>
|
||||
<p>In implementation <span data-acronym-label="MCMC" data-acronym-form="singular+short">MCMC</span> can be boiled down to choosing a transition function <span class="math inline">\(\mathcal{T}(\s_{t} \rightarrow \s_t+1)\)</span> where <span class="math inline">\(\s\)</span> are vectors representing classical spin configurations. We start in some initial state <span class="math inline">\(\s_0\)</span> and then repeatedly jump to new states according to the probabilities given by <span class="math inline">\(\mathcal{T}\)</span>. This defines a set of random walks <span class="math inline">\(\{\s_0\ldots \s_i\ldots \s_N\}\)</span>. Fig. <a href="#fig:single" data-reference-type="ref" data-reference="fig:single">2</a> shows this in practice: we have a (rather small) ensemble of <span class="math inline">\(M = 2\)</span> walkers starting at the same point in state space and then spreading outwards by flipping spins along the way.</p>
|
||||
<p>In pseudo-code one could write the MCMC simulation for a single walker as:</p>
|
||||
<div class="markdown">
|
||||
<p>“’python current_state = initial_state</p>
|
||||
<p>for i in range(N_steps): new_state = sample_T(current_state)
|
||||
states[i] = current_state “’</p>
|
||||
<p>for i in range(N_steps): new_state = sample_T(current_state) states[i] = current_state “’</p>
|
||||
</div>
|
||||
<p>Where the <code>sample_T</code> function here produces a state with
|
||||
probability determined by the <code>current_state</code> and the
|
||||
transition function <span
|
||||
class="math inline">\(\mathcal{T}\)</span>.</p>
|
||||
<p>If we ran many such walkers in parallel we could then approximate the
|
||||
distribution <span class="math inline">\(p_t(\s; \s_0)\)</span> which
|
||||
tells us where the walkers are likely to be after they’ve evolved for
|
||||
<span class="math inline">\(t\)</span> steps from an initial state <span
|
||||
class="math inline">\(\s_0\)</span>. We need to carefully choose <span
|
||||
class="math inline">\(\mathcal{T}\)</span> such that after a large
|
||||
number of steps <span class="math inline">\(k\)</span> (the convergence
|
||||
time) the probability <span class="math inline">\(p_t(\s;\s_0)\)</span>
|
||||
approaches the thermal distribution <span class="math inline">\(P(\s;
|
||||
\beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to
|
||||
be quite easy to achieve using the Metropolis-Hasting algorithm.</p>
|
||||
<p>Where the <code>sample_T</code> function here produces a state with probability determined by the <code>current_state</code> and the transition function <span class="math inline">\(\mathcal{T}\)</span>.</p>
|
||||
<p>If we ran many such walkers in parallel we could then approximate the distribution <span class="math inline">\(p_t(\s; \s_0)\)</span> which tells us where the walkers are likely to be after they’ve evolved for <span class="math inline">\(t\)</span> steps from an initial state <span class="math inline">\(\s_0\)</span>. We need to carefully choose <span class="math inline">\(\mathcal{T}\)</span> such that after a large number of steps <span class="math inline">\(k\)</span> (the convergence time) the probability <span class="math inline">\(p_t(\s;\s_0)\)</span> approaches the thermal distribution <span class="math inline">\(P(\s; \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\)</span>. This turns out to be quite easy to achieve using the Metropolis-Hasting algorithm.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="convergence-time" class="level2">
|
||||
<h2>Convergence Time</h2>
|
||||
<p>Considering <span class="math inline">\(p(\s)\)</span> as a vector
|
||||
<span class="math inline">\(\vec{p}\)</span> whose jth entry is the
|
||||
probability of the jth state <span class="math inline">\(p_j =
|
||||
p(\s_j)\)</span>, and writing <span
|
||||
class="math inline">\(\mathcal{T}\)</span> as the matrix with entries
|
||||
<span class="math inline">\(T_{ij} = \mathcal{T}(\s_j \rightarrow
|
||||
\s_i)\)</span> we can write the update rule for the ensemble probability
|
||||
as: <span class="math display">\[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t
|
||||
\implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\]</span> where <span
|
||||
class="math inline">\(\vec{p}_0\)</span> is vector which is one on the
|
||||
starting state and zero everywhere else. Since all states must
|
||||
transition to somewhere with probability one: <span
|
||||
class="math inline">\(\sum_i T_{ij} = 1\)</span>.</p>
|
||||
<p>Matrices that satisfy this are called stochastic matrices exactly
|
||||
because they model these kinds of Markov processes. It can be shown that
|
||||
they have real eigenvalues, and ordering them by magnitude, that <span
|
||||
class="math inline">\(\lambda_0 = 1\)</span> and <span
|
||||
class="math inline">\(0 < \lambda_{i\neq0} < 1\)</span>. Assuming
|
||||
<span class="math inline">\(\mathcal{T}\)</span> has been chosen
|
||||
correctly, its single eigenvector with eigenvalue 1 will be the thermal
|
||||
distribution [^3] so repeated application of the transition function
|
||||
eventually leads there, while memory of the initial conditions decays
|
||||
exponentially with a convergence time <span
|
||||
class="math inline">\(k\)</span> determined by <span
|
||||
class="math inline">\(\lambda_1\)</span>. In practice this means that
|
||||
one throws away the data from the beginning of the random walk in order
|
||||
reduce the dependence on the initial conditions and be close enough to
|
||||
the target distribution.</p>
|
||||
<p>Considering <span class="math inline">\(p(\s)\)</span> as a vector <span class="math inline">\(\vec{p}\)</span> whose jth entry is the probability of the jth state <span class="math inline">\(p_j = p(\s_j)\)</span>, and writing <span class="math inline">\(\mathcal{T}\)</span> as the matrix with entries <span class="math inline">\(T_{ij} = \mathcal{T}(\s_j \rightarrow \s_i)\)</span> we can write the update rule for the ensemble probability as: <span class="math display">\[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t \implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\]</span> where <span class="math inline">\(\vec{p}_0\)</span> is vector which is one on the starting state and zero everywhere else. Since all states must transition to somewhere with probability one: <span class="math inline">\(\sum_i T_{ij} = 1\)</span>.</p>
|
||||
<p>Matrices that satisfy this are called stochastic matrices exactly because they model these kinds of Markov processes. It can be shown that they have real eigenvalues, and ordering them by magnitude, that <span class="math inline">\(\lambda_0 = 1\)</span> and <span class="math inline">\(0 < \lambda_{i\neq0} < 1\)</span>. Assuming <span class="math inline">\(\mathcal{T}\)</span> has been chosen correctly, its single eigenvector with eigenvalue 1 will be the thermal distribution [^3] so repeated application of the transition function eventually leads there, while memory of the initial conditions decays exponentially with a convergence time <span class="math inline">\(k\)</span> determined by <span class="math inline">\(\lambda_1\)</span>. In practice this means that one throws away the data from the beginning of the random walk in order reduce the dependence on the initial conditions and be close enough to the target distribution.</p>
|
||||
</section>
|
||||
<section id="auto-correlation-time" class="level2">
|
||||
<h2>Auto-correlation Time</h2>
|
||||
<div id="fig:m_autocorr" class="fignos">
|
||||
<figure>
|
||||
<img src="figs/lsr/m_autocorr.png"
|
||||
alt="Figure 3: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5 [fig:m_autocorr]" />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> (Upper) 10 MCMC
|
||||
chains starting from the same initial state for a system with <span
|
||||
class="math inline">\(N = 150\)</span> sites and 3000 MCMC steps. At
|
||||
each MCMC step, n spins are flipped where n is drawn from Uniform(1,N)
|
||||
and this is repeated <span class="math inline">\(N^2/100\)</span> times.
|
||||
The simulations therefore have the potential to necessitate <span
|
||||
class="math inline">\(10*N^2\)</span> matrix diagonalisations for each
|
||||
100 MCMC steps. (Lower) The normalised auto-correlation <span
|
||||
class="math inline">\((\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i})
|
||||
/ Var(m_i))\)</span> averaged over <span
|
||||
class="math inline">\(i\)</span>. It can be seen that even with each
|
||||
MCMC step already being composed of many individual flip attempts, the
|
||||
auto-correlation is still non negligible and must be taken into account
|
||||
in the statistics. <span class="math inline">\(t = 1, \alpha = 1.25, T =
|
||||
2.2, J = U = 5\)</span> <span id="fig:m_autocorr"
|
||||
label="fig:m_autocorr">[fig:m_autocorr]</span></figcaption>
|
||||
<img src="figs/lsr/m_autocorr.png" id="fig:m_autocorr" alt="Figure 3: (Upper) 10 MCMC chains starting from the same initial state for a system with N = 150 sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated N^2/100 times. The simulations therefore have the potential to necessitate 10*N^2 matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation (\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / Var(m_i)) averaged over i. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. t = 1, \alpha = 1.25, T = 2.2, J = U = 5 [fig:m_autocorr]" />
|
||||
<figcaption aria-hidden="true">Figure 3: (Upper) 10 MCMC chains starting from the same initial state for a system with <span class="math inline">\(N = 150\)</span> sites and 3000 MCMC steps. At each MCMC step, n spins are flipped where n is drawn from Uniform(1,N) and this is repeated <span class="math inline">\(N^2/100\)</span> times. The simulations therefore have the potential to necessitate <span class="math inline">\(10*N^2\)</span> matrix diagonalisations for each 100 MCMC steps. (Lower) The normalised auto-correlation <span class="math inline">\((\expval{m_i m_{i-j}} - \expval{m_i}\expval{m_i}) / Var(m_i))\)</span> averaged over <span class="math inline">\(i\)</span>. It can be seen that even with each MCMC step already being composed of many individual flip attempts, the auto-correlation is still non negligible and must be taken into account in the statistics. <span class="math inline">\(t = 1, \alpha = 1.25, T = 2.2, J = U = 5\)</span> <span id="fig:m_autocorr" label="fig:m_autocorr">[fig:m_autocorr]</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>At this stage one might think we’re done. We can indeed draw
|
||||
independent samples from <span class="math inline">\(P(\s;
|
||||
\beta)\)</span> by starting from some arbitrary initial state and doing
|
||||
<span class="math inline">\(k\)</span> steps to arrive at a sample.
|
||||
However a key insight is that after the convergence time, every state
|
||||
generated is a sample from <span class="math inline">\(P(\s;
|
||||
\beta)\)</span>! They are not, however, independent samples. In Fig. <a
|
||||
href="#fig:raw" data-reference-type="ref" data-reference="fig:raw">1</a>
|
||||
it is already clear that the samples of the order parameter m have some
|
||||
auto-correlation because only a few spins are flipped each step but even
|
||||
when the number of spins flipped per step is increased, Fig. <a
|
||||
href="#fig:m_autocorr" data-reference-type="ref"
|
||||
data-reference="fig:m_autocorr">3</a> shows that it can be an important
|
||||
effect near the phase transition. Let’s define the auto-correlation time
|
||||
<span class="math inline">\(\tau(O)\)</span> informally as the number of
|
||||
MCMC samples of some observable O that are statistically equal to one
|
||||
independent sample. [^4] The auto-correlation time is generally shorter
|
||||
than the convergence time so it therefore makes sense from an efficiency
|
||||
standpoint to run a single walker for many MCMC steps rather than to run
|
||||
a huge ensemble for <span class="math inline">\(k\)</span> steps
|
||||
each.</p>
|
||||
<p>Once the random walk has been carried out for many steps, the
|
||||
expectation values of <span class="math inline">\(O\)</span> can be
|
||||
estimated from the MCMC samples <span
|
||||
class="math inline">\(\s_i\)</span>: <span
|
||||
class="math display">\[\tex{O} = \sum_{i = 0}^{N} O(\s_i) +
|
||||
\mathcal{O}(\frac{1}{\sqrt{N}})\]</span> The the samples are correlated
|
||||
so the N of them effectively contains less information than <span
|
||||
class="math inline">\(N\)</span> independent samples would, in fact
|
||||
roughly <span class="math inline">\(N/\tau\)</span> effective samples.
|
||||
As a consequence the variance is larger than the <span
|
||||
class="math inline">\(\qex{O^2} - \qex{O}^2\)</span> form it would have
|
||||
if the estimates were uncorrelated. There are many methods in the
|
||||
literature for estimating the true variance of <span
|
||||
class="math inline">\(\qex{O}\)</span> and deciding how many steps are
|
||||
needed but my approach has been to run a small number of parallel
|
||||
chains, which are independent, in order to estimate the statistical
|
||||
error produced. This is a slightly less computationally efficient
|
||||
because it requires throwing away those <span
|
||||
class="math inline">\(k\)</span> steps generated before convergence
|
||||
multiple times but it is a conceptually simple workaround.</p>
|
||||
<p>In summary, to do efficient simulations we want to reduce both the
|
||||
convergence time and the auto-correlation time as much as possible. In
|
||||
order to explain how, we need to introduce the Metropolis-Hasting (MH)
|
||||
algorithm and how it gives an explicit form for the transition
|
||||
function.</p>
|
||||
<p>Next Section: <a
|
||||
href="../3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html">Results</a></p>
|
||||
<p>At this stage one might think we’re done. We can indeed draw independent samples from <span class="math inline">\(P(\s; \beta)\)</span> by starting from some arbitrary initial state and doing <span class="math inline">\(k\)</span> steps to arrive at a sample. However a key insight is that after the convergence time, every state generated is a sample from <span class="math inline">\(P(\s; \beta)\)</span>! They are not, however, independent samples. In Fig. <a href="#fig:raw" data-reference-type="ref" data-reference="fig:raw">1</a> it is already clear that the samples of the order parameter m have some auto-correlation because only a few spins are flipped each step but even when the number of spins flipped per step is increased, Fig. <a href="#fig:m_autocorr" data-reference-type="ref" data-reference="fig:m_autocorr">3</a> shows that it can be an important effect near the phase transition. Let’s define the auto-correlation time <span class="math inline">\(\tau(O)\)</span> informally as the number of MCMC samples of some observable O that are statistically equal to one independent sample. [^4] The auto-correlation time is generally shorter than the convergence time so it therefore makes sense from an efficiency standpoint to run a single walker for many MCMC steps rather than to run a huge ensemble for <span class="math inline">\(k\)</span> steps each.</p>
|
||||
<p>Once the random walk has been carried out for many steps, the expectation values of <span class="math inline">\(O\)</span> can be estimated from the MCMC samples <span class="math inline">\(\s_i\)</span>: <span class="math display">\[\tex{O} = \sum_{i = 0}^{N} O(\s_i) + \mathcal{O}(\frac{1}{\sqrt{N}})\]</span> The the samples are correlated so the N of them effectively contains less information than <span class="math inline">\(N\)</span> independent samples would, in fact roughly <span class="math inline">\(N/\tau\)</span> effective samples. As a consequence the variance is larger than the <span class="math inline">\(\qex{O^2} - \qex{O}^2\)</span> form it would have if the estimates were uncorrelated. There are many methods in the literature for estimating the true variance of <span class="math inline">\(\qex{O}\)</span> and deciding how many steps are needed but my approach has been to run a small number of parallel chains, which are independent, in order to estimate the statistical error produced. This is a slightly less computationally efficient because it requires throwing away those <span class="math inline">\(k\)</span> steps generated before convergence multiple times but it is a conceptually simple workaround.</p>
|
||||
<p>In summary, to do efficient simulations we want to reduce both the convergence time and the auto-correlation time as much as possible. In order to explain how, we need to introduce the Metropolis-Hasting (MH) algorithm and how it gives an explicit form for the transition function.</p>
|
||||
<p>Next Section: <a href="../3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html">Results</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
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Carlo Sampling Methods Using Markov Chains and Their
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Applications</a></em>, Biometrika <strong>57</strong>, 97 (1970).</div>
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Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To
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Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation:
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Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20
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July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
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<div class="csl-left-margin">[6] </div><div class="csl-right-inline">W. Krauth, <em><a href="https://doi.org/10.1007/BFb0105456">Introduction To Monte Carlo Algorithms</a></em>, in <em>Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996</em> (Springer-Verlag, Berlin Heidelberg, 1998).</div>
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||||
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Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov
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Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
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<div id="ref-hodsonMCMCFKModel2021" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[7] </div><div class="csl-right-inline">T. Hodson, <em><a href="https://doi.org/10.5281/zenodo.4593904">Markov Chain Monte Carlo for the Kitaev Model</a></em>, (2021).</div>
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">L.
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||||
Huang and L. Wang, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte
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||||
Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys.
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||||
Rev. B <strong>95</strong>, 035105 (2017).</div>
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||||
<div id="ref-huangAcceleratedMonteCarlo2017" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[8] </div><div class="csl-right-inline">L. Huang and L. Wang, <em><a href="https://doi.org/10.1103/PhysRevB.95.035105">Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines</a></em>, Phys. Rev. B <strong>95</strong>, 035105 (2017).</div>
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||||
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||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
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||||
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<div class="csl-left-margin">[9] </div><div class="csl-right-inline">K.
|
||||
Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
|
||||
Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
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||||
Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
|
||||
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<div class="csl-left-margin">[9] </div><div class="csl-right-inline">K. Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size Scaling Analysis of Ising Model Block Distribution Functions</a></em>, Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
|
||||
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|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">B.
|
||||
Kramer and A. MacKinnon, <em><a
|
||||
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
|
||||
and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
|
||||
(1993).</div>
|
||||
<div id="ref-musialMonteCarloSimulations2002" class="csl-entry" role="doc-biblioentry">
|
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<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G. Musiał, L. Dȩbski, and G. Kamieniarz, <em><a href="https://doi.org/10.1103/PhysRevB.66.012407">Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model</a></em>, Phys. Rev. B <strong>66</strong>, 012407 (2002).</div>
|
||||
</div>
|
||||
<div id="ref-andersonAbsenceDiffusionCertain1958" class="csl-entry"
|
||||
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|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">P.
|
||||
W. Anderson, <em><a
|
||||
href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in
|
||||
Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492
|
||||
(1958).</div>
|
||||
<div id="ref-kramerLocalizationTheoryExperiment1993" class="csl-entry" role="doc-biblioentry">
|
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<div class="csl-left-margin">[11] </div><div class="csl-right-inline">B. Kramer and A. MacKinnon, <em><a href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469 (1993).</div>
|
||||
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|
||||
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|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">P. W. Anderson, <em><a href="https://doi.org/10.1103/PhysRev.109.1492">Absence of Diffusion in Certain Random Lattices</a></em>, Phys. Rev. <strong>109</strong>, 1492 (1958).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -29,13 +29,10 @@ image:
|
||||
<ul>
|
||||
<li><a href="#fk-results" id="toc-fk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#lrfk-results-phase-diagram"
|
||||
id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
|
||||
<li><a href="#localisation-properties"
|
||||
id="toc-localisation-properties">Localisation Properties</a></li>
|
||||
<li><a href="#lrfk-results-phase-diagram" id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
|
||||
<li><a href="#localisation-properties" id="toc-localisation-properties">Localisation Properties</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and
|
||||
Conclusion</a></li>
|
||||
<li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and Conclusion</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -51,13 +48,10 @@ Conclusion</a></li>
|
||||
<ul>
|
||||
<li><a href="#fk-results" id="toc-fk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#lrfk-results-phase-diagram"
|
||||
id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
|
||||
<li><a href="#localisation-properties"
|
||||
id="toc-localisation-properties">Localisation Properties</a></li>
|
||||
<li><a href="#lrfk-results-phase-diagram" id="toc-lrfk-results-phase-diagram">Phase Diagram</a></li>
|
||||
<li><a href="#localisation-properties" id="toc-localisation-properties">Localisation Properties</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and
|
||||
Conclusion</a></li>
|
||||
<li><a href="#fk-conclusion" id="toc-fk-conclusion">Discussion and Conclusion</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -70,500 +64,98 @@ Conclusion</a></li>
|
||||
</div>
|
||||
<section id="fk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<div id="fig:phase_diagram" class="fignos">
|
||||
<p>Phase diagrams of the long-range 1D FK model. (a) The TJ plane at <span class="math inline">\(U = 5\)</span>: the CDW ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature <span class="math inline">\(T_c\)</span>, linear in J. (b) The TU plane at <span class="math inline">\(J = 5\)</span>: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at <span class="math inline">\(E=0\)</span> in the single particle energy spectrum. <span class="math inline">\(U_c\)</span> is independent of temperature. At <span class="math inline">\(U = 0\)</span> the fermions are decoupled from the spins forming a simple Fermi gas. (c) The order parameters, <span class="math inline">\(\langle m^2 \rangle\)</span>(solid) and <span class="math inline">\(1 - f\)</span> (dashed) describing the onset of the <span data-acronym-label="CDW" data-acronym-form="singular+short">CDW</span> phase of the long-range 1D <span data-acronym-label="FK" data-acronym-form="singular+short">FK</span> model at low temperature with staggered magnetisation <span class="math inline">\(m = N^{-1} \sum_i (-1)^i S_i\)</span> and fermionic order parameter <span class="math inline">\(f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle\)</span> . (d) The crossing of the Binder cumulant, <span class="math inline">\(B = \langle m^4 \rangle / \langle m^2 \rangle^2\)</span>, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in (a) and (b). All plots use system sizes <span class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and lines are coloured from <span class="math inline">\(N = 10\)</span> in dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span> except where explicitly varied.</p>
|
||||
<figure>
|
||||
<img src="pdf_figs/phase_diagram.svg"
|
||||
alt="Figure 1: Phase diagrams of the long-range 1D FK model. (a) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature T_c, linear in J. (b) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas. (c) The order parameters, \tex{m^2}(solid) and 1 - f (dashed) describing the onset of the CDW phase of the long-range 1D FK model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}} . (d) The crossing of the Binder cumulant, B = \tex{m^4} / \tex{m^2}^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in (a) and (b). All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly varied." />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span> Phase diagrams of
|
||||
the long-range 1D FK model. (a) The TJ plane at <span
|
||||
class="math inline">\(U = 5\)</span>: the <span data-acronym-label="CDW"
|
||||
data-acronym-form="singular+short">CDW</span> ordered phase is separated
|
||||
from a disordered Mott insulating (MI) phase by a critical temperature
|
||||
<span class="math inline">\(T_c\)</span>, linear in J. (b) The TU plane
|
||||
at <span class="math inline">\(J = 5\)</span>: the disordered phase is
|
||||
split into two: at large/small U there’s a MI/Anderson phase
|
||||
characterised by the presence/absence of a gap at <span
|
||||
class="math inline">\(E=0\)</span> in the single particle energy
|
||||
spectrum. <span class="math inline">\(U_c\)</span> is independent of
|
||||
temperature. At <span class="math inline">\(U = 0\)</span> the fermions
|
||||
are decoupled from the spins forming a simple Fermi gas. (c) The order
|
||||
parameters, <span class="math inline">\(\tex{m^2}\)</span>(solid) and
|
||||
<span class="math inline">\(1 - f\)</span> (dashed) describing the onset
|
||||
of the <span data-acronym-label="CDW"
|
||||
data-acronym-form="singular+short">CDW</span> phase of the long-range 1D
|
||||
<span data-acronym-label="FK"
|
||||
data-acronym-form="singular+short">FK</span> model at low temperature
|
||||
with staggered magnetisation <span class="math inline">\(m = N^{-1}
|
||||
\sum_i (-1)^i S_i\)</span> and fermionic order parameter <span
|
||||
class="math inline">\(f = 2 N^{-1}\abs{\sum_i (-1)^i \;
|
||||
\expval{c^\dag_{i}c_{i}}}\)</span> . (d) The crossing of the Binder
|
||||
cumulant, <span class="math inline">\(B = \tex{m^4} /
|
||||
\tex{m^2}^2\)</span>, with system size provides a diagnostic that the
|
||||
phase transition is not a finite size effect, it’s used to estimate the
|
||||
critical lines shown in (a) and (b). All plots use system sizes <span
|
||||
class="math inline">\(N = [10,20,30,50,70,110,160,250]\)</span> and
|
||||
lines are coloured from <span class="math inline">\(N = 10\)</span> in
|
||||
dark blue to <span class="math inline">\(N = 250\)</span> in yellow. The
|
||||
parameter values <span class="math inline">\(U = 5,\;J = 5,\;\alpha =
|
||||
1.25\)</span> except where explicitly varied.</figcaption>
|
||||
<img src="/assets/thesis/fk_chapter/binder.png" id="fig:binder" data-short-caption="no title" style="width:100.0%" alt="Figure 1: Hello I am the figure caption!" />
|
||||
<figcaption aria-hidden="true">Figure 1: Hello I am the figure caption!</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<div id="fig:binder" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/fk_chapter/binder.png"
|
||||
data-short-caption="no title" style="width:100.0%"
|
||||
alt="Figure 2: Hello I am the figure caption!" />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> Hello I am the
|
||||
figure caption!</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<section id="lrfk-results-phase-diagram" class="level2">
|
||||
<h2>Phase Diagram</h2>
|
||||
<p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref"
|
||||
data-reference="fig:phase_diagram">1</a>a] and [<a
|
||||
href="#fig:phase_diagram" data-reference-type="ref"
|
||||
data-reference="fig:phase_diagram">1</a>b] show the phase diagram for
|
||||
constant <span class="math inline">\(U=5\)</span> and constant <span
|
||||
class="math inline">\(J=5\)</span>, respectively. We determined the
|
||||
transition temperatures from the crossings of the Binder cumulants <span
|
||||
class="math inline">\(B_4 = \tex{m^4}/\tex{m^2}^2\)</span> <span
|
||||
class="citation" data-cites="binderFiniteSizeScaling1981"> [<a
|
||||
href="#ref-binderFiniteSizeScaling1981"
|
||||
role="doc-biblioref">1</a>]</span>. For a representative set of
|
||||
parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref"
|
||||
data-reference="fig:phase_diagram">1</a>c] shows the order parameter
|
||||
<span class="math inline">\(\tex{m}^2\)</span>. Fig. [<a
|
||||
href="#fig:phase_diagram" data-reference-type="ref"
|
||||
data-reference="fig:phase_diagram">1</a>d] shows the Binder cumulants,
|
||||
both as functions of system size and temperature. The crossings confirm
|
||||
that the system has a FTPT and that the ordered phase is not a finite
|
||||
size effect.</p>
|
||||
<p>The CDW transition temperature is largely independent from the
|
||||
strength of the interaction <span class="math inline">\(U\)</span>. This
|
||||
demonstrates that the phase transition is driven by the long-range term
|
||||
<span class="math inline">\(J\)</span> with little effect from the
|
||||
coupling to the fermions <span class="math inline">\(U\)</span>. The
|
||||
physics of the spin sector in our long-range FK model mimics that of the
|
||||
LRI model and is not significantly altered by the presence of the
|
||||
fermions, which shows that the long range tail expected from a basic
|
||||
fermion mediated RKKY interaction between the Ising spins is absent.</p>
|
||||
<p>Our main interest concerns the additional structure of the fermionic
|
||||
sector in the high temperature phase. Following Ref. <span
|
||||
class="citation"
|
||||
data-cites="antipovInteractionTunedAndersonMott2016"> [<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">2</a>]</span>, we can distinguish between the Mott
|
||||
and Anderson insulating phases. The former is characterised by a gapped
|
||||
DOS in the absence of a CDW. Thus, the opening of a gap for large <span
|
||||
class="math inline">\(U\)</span> is distinct from the gap-opening
|
||||
induced by the translational symmetry breaking in the CDW state below
|
||||
<span class="math inline">\(T_c\)</span>, see also Fig. [<a
|
||||
href="#fig:band_opening" data-reference-type="ref"
|
||||
data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless
|
||||
but, as we explain below, shows localised fermionic eigenstates.</p>
|
||||
<p>Figs. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>a] and [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>b] show the phase diagram for constant <span class="math inline">\(U=5\)</span> and constant <span class="math inline">\(J=5\)</span>, respectively. We determined the transition temperatures from the crossings of the Binder cumulants <span class="math inline">\(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)</span> <span class="citation" data-cites="binderFiniteSizeScaling1981"> [<a href="#ref-binderFiniteSizeScaling1981" role="doc-biblioref">1</a>]</span>. For a representative set of parameters, Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>c] shows the order parameter <span class="math inline">\(\rangle m \langle^2\)</span>. Fig. [<a href="#fig:phase_diagram" data-reference-type="ref" data-reference="fig:phase_diagram">1</a>d] shows the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.</p>
|
||||
<p>The CDW transition temperature is largely independent from the strength of the interaction <span class="math inline">\(U\)</span>. This demonstrates that the phase transition is driven by the long-range term <span class="math inline">\(J\)</span> with little effect from the coupling to the fermions <span class="math inline">\(U\)</span>. The physics of the spin sector in our long-range FK model mimics that of the LRI model and is not significantly altered by the presence of the fermions, which shows that the long range tail expected from a basic fermion mediated RKKY interaction between the Ising spins is absent.</p>
|
||||
<p>Our main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref. <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">2</a>]</span>, we can distinguish between the Mott and Anderson insulating phases. The former is characterised by a gapped DOS in the absence of a CDW. Thus, the opening of a gap for large <span class="math inline">\(U\)</span> is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below <span class="math inline">\(T_c\)</span>, see also Fig. [<a href="#fig:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</a>a]. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates.</p>
|
||||
</section>
|
||||
<section id="localisation-properties" class="level2">
|
||||
<h2>Localisation Properties</h2>
|
||||
<p>The MCMC formulation suggests viewing the spin configurations as a
|
||||
form of annealed binary disorder whose probability distribution is given
|
||||
by the Boltzmann weight <span class="math inline">\(e^{-\beta
|
||||
H_S[\vec{S}] - \beta F_c[\vec{S}]}\)</span>. This makes apparent the
|
||||
link to the study of disordered systems and Anderson localisation. While
|
||||
these systems are typically studied by defining the probability
|
||||
distribution for the quenched disorder potential externally, here we
|
||||
have a translation invariant system with disorder as a natural
|
||||
consequence of the Ising background field conserved under the
|
||||
dynamics.</p>
|
||||
<p>In the limits of zero and infinite temperature, our model becomes a
|
||||
simple tight-binding model for the fermions. At zero temperature, the
|
||||
spin background is in one of the two translation invariant AFM ground
|
||||
states with two gapped fermionic CDW bands at energies <span
|
||||
class="math display">\[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos
|
||||
ka)^2}\;.\]</span></p>
|
||||
<p>At infinite temperature, all the spin configurations become equally
|
||||
likely and the fermionic model reduces to one of binary uncorrelated
|
||||
disorder in which all eigenstates are Anderson localised <span
|
||||
class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a
|
||||
href="#ref-abrahamsScalingTheoryLocalization1979"
|
||||
role="doc-biblioref">3</a>]</span>. An Anderson localised state centered
|
||||
around <span class="math inline">\(r_0\)</span> has magnitude that drops
|
||||
exponentially over some localisation length <span
|
||||
class="math inline">\(\xi\)</span> i.e <span
|
||||
class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r -
|
||||
r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span>
|
||||
directly is numerically demanding. Therefore, we determine if a given
|
||||
state is localised via the energy-resolved IPR and the DOS defined as
|
||||
<span class="math display">\[\begin{aligned}
|
||||
\mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i -
|
||||
\omega)\\
|
||||
\mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S},
|
||||
\omega)^{-1} \sum_{i,j} \delta(\epsilon_i -
|
||||
\omega)\;\psi^{4}_{i,j}\end{aligned}\]</span> where <span
|
||||
class="math inline">\(\epsilon_i\)</span> and <span
|
||||
class="math inline">\(\psi_{i,j}\)</span> are the <span
|
||||
class="math inline">\(i\)</span>th energy level and <span
|
||||
class="math inline">\(j\)</span>th element of the corresponding
|
||||
eigenfunction, both dependent on the background spin configuration <span
|
||||
class="math inline">\(\vec{S}\)</span>.</p>
|
||||
<p>The scaling of the IPR with system size <span
|
||||
class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span> depends
|
||||
on the localisation properties of states at that energy. For delocalised
|
||||
states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is
|
||||
the physical dimension. For fully localised states <span
|
||||
class="math inline">\(\tau\)</span> goes to zero in the thermodynamic
|
||||
limit. However, for special types of disorder such as binary disorder,
|
||||
the localisation lengths can be large comparable to the system size at
|
||||
hand, which can make it difficult to extract the correct scaling. An
|
||||
additional complication arises from the fact that the scaling exponent
|
||||
may display intermediate behaviours for correlated disorder and in the
|
||||
vicinity of a localisation-delocalisation transition <span
|
||||
class="citation"
|
||||
data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a
|
||||
href="#ref-kramerLocalizationTheoryExperiment1993"
|
||||
role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008"
|
||||
role="doc-biblioref">5</a>]</span>. The thermal defects of the CDW phase
|
||||
lead to a binary disorder potential with a finite correlation length,
|
||||
which in principle could result in delocalized eigenstates.</p>
|
||||
<p>The key question for our system is then: How is the <span
|
||||
class="math inline">\(T=0\)</span> CDW phase with fully delocalized
|
||||
fermionic states connected to the fully localized phase at high
|
||||
temperatures?</p>
|
||||
<div id="fig:indiv_IPR" class="fignos">
|
||||
<p>The MCMC formulation suggests viewing the spin configurations as a form of annealed binary disorder whose probability distribution is given by the Boltzmann weight <span class="math inline">\(e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]}\)</span>. This makes apparent the link to the study of disordered systems and Anderson localisation. While these systems are typically studied by defining the probability distribution for the quenched disorder potential externally, here we have a translation invariant system with disorder as a natural consequence of the Ising background field conserved under the dynamics.</p>
|
||||
<p>In the limits of zero and infinite temperature, our model becomes a simple tight-binding model for the fermions. At zero temperature, the spin background is in one of the two translation invariant AFM ground states with two gapped fermionic CDW bands at energies <span class="math display">\[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos ka)^2}\;.\]</span></p>
|
||||
<p>At infinite temperature, all the spin configurations become equally likely and the fermionic model reduces to one of binary uncorrelated disorder in which all eigenstates are Anderson localised <span class="citation" data-cites="abrahamsScalingTheoryLocalization1979"> [<a href="#ref-abrahamsScalingTheoryLocalization1979" role="doc-biblioref">3</a>]</span>. An Anderson localised state centered around <span class="math inline">\(r_0\)</span> has magnitude that drops exponentially over some localisation length <span class="math inline">\(\xi\)</span> i.e <span class="math inline">\(|\psi(r)|^2 \sim \exp{-\abs{r - r_0}/\xi}\)</span>. Calculating <span class="math inline">\(\xi\)</span> directly is numerically demanding. Therefore, we determine if a given state is localised via the energy-resolved IPR and the DOS defined as <span class="math display">\[\begin{aligned}
|
||||
\mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\
|
||||
\mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\]</span> where <span class="math inline">\(\epsilon_i\)</span> and <span class="math inline">\(\psi_{i,j}\)</span> are the <span class="math inline">\(i\)</span>th energy level and <span class="math inline">\(j\)</span>th element of the corresponding eigenfunction, both dependent on the background spin configuration <span class="math inline">\(\vec{S}\)</span>.</p>
|
||||
<p>The scaling of the IPR with system size <span class="math display">\[\mathrm{IPR} \propto N^{-\tau}\]</span> depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, <span class="math inline">\(\tau\)</span> is the physical dimension. For fully localised states <span class="math inline">\(\tau\)</span> goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition <span class="citation" data-cites="kramerLocalizationTheoryExperiment1993 eversAndersonTransitions2008"> [<a href="#ref-kramerLocalizationTheoryExperiment1993" role="doc-biblioref">4</a>,<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">5</a>]</span>. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.</p>
|
||||
<p>The key question for our system is then: How is the <span class="math inline">\(T=0\)</span> CDW phase with fully delocalized fermionic states connected to the fully localized phase at high temperatures?</p>
|
||||
<figure>
|
||||
<img src="pdf_figs/indiv_IPR.svg"
|
||||
alt="Figure 3: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> Energy resolved
|
||||
DOS(<span class="math inline">\(\omega\)</span>) and <span
|
||||
class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span
|
||||
class="math inline">\(\omega\)</span>) against system size <span
|
||||
class="math inline">\(N\)</span>). The left column shows the Anderson
|
||||
phase <span class="math inline">\(U = 2\)</span> at high <span
|
||||
class="math inline">\(T = 2.5\)</span> and the CDW phase at low <span
|
||||
class="math inline">\(T = 1.5\)</span> temperature. IPRs are evaluated
|
||||
for one of the in-gap states <span class="math inline">\(\omega_0/U =
|
||||
0.057\)</span> and the center of the band <span
|
||||
class="math inline">\(\omega_1\)</span> <span class="math inline">\(U =
|
||||
0.81\)</span>. The right column shows instead the Mott and CDW phases at
|
||||
<span class="math inline">\(U = 5\)</span> with <span
|
||||
class="math inline">\(\omega_0/U = 0.24\)</span> and <span
|
||||
class="math inline">\(\omega_1/U = 0.571\)</span>. For all the plots
|
||||
<span class="math inline">\(J = 5,\;\alpha = 1.25\)</span> and the fits
|
||||
for <span class="math inline">\(\tau\)</span> use system sizes greater
|
||||
than 60. The measured <span class="math inline">\(\tau_0,\tau_1\)</span>
|
||||
for each figure are: (a) <span class="math inline">\((0.06\pm0.01,
|
||||
0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02,
|
||||
0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03,
|
||||
0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04,
|
||||
0.15\pm0.05\)</span> We show later that the apparent scaling of the IPR
|
||||
with system size can be explained by the changing defect density with
|
||||
system size rather than due to delocalisation of the
|
||||
states.</figcaption>
|
||||
<img src="pdf_figs/indiv_IPR.svg" id="fig:indiv_IPR" alt="Figure 2: Energy resolved DOS(\omega) and \tau (the scaling exponent of IPR(\omega) against system size N). The left column shows the Anderson phase U = 2 at high T = 2.5 and the CDW phase at low T = 1.5 temperature. IPRs are evaluated for one of the in-gap states \omega_0/U = 0.057 and the center of the band \omega_1 U = 0.81. The right column shows instead the Mott and CDW phases at U = 5 with \omega_0/U = 0.24 and \omega_1/U = 0.571. For all the plots J = 5,\;\alpha = 1.25 and the fits for \tau use system sizes greater than 60. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states." />
|
||||
<figcaption aria-hidden="true">Figure 2: Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> (the scaling exponent of IPR(<span class="math inline">\(\omega\)</span>) against system size <span class="math inline">\(N\)</span>). The left column shows the Anderson phase <span class="math inline">\(U = 2\)</span> at high <span class="math inline">\(T = 2.5\)</span> and the CDW phase at low <span class="math inline">\(T = 1.5\)</span> temperature. IPRs are evaluated for one of the in-gap states <span class="math inline">\(\omega_0/U = 0.057\)</span> and the center of the band <span class="math inline">\(\omega_1\)</span> <span class="math inline">\(U = 0.81\)</span>. The right column shows instead the Mott and CDW phases at <span class="math inline">\(U = 5\)</span> with <span class="math inline">\(\omega_0/U = 0.24\)</span> and <span class="math inline">\(\omega_1/U = 0.571\)</span>. For all the plots <span class="math inline">\(J = 5,\;\alpha = 1.25\)</span> and the fits for <span class="math inline">\(\tau\)</span> use system sizes greater than 60. The measured <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\((0.06\pm0.01, 0.02\pm0.01\)</span> (b) <span class="math inline">\(0.04\pm0.02, 0.00\pm0.01\)</span> (c) <span class="math inline">\(0.05\pm0.03, 0.30\pm0.03\)</span> (d) <span class="math inline">\(0.06\pm0.04, 0.15\pm0.05\)</span> We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalisation of the states.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<div id="fig:band_opening" class="fignos">
|
||||
<figure>
|
||||
<img src="pdf_figs/gap_openingboth.svg"
|
||||
alt="Figure 4: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true"><span>Figure 4:</span> The DOS (a and c)
|
||||
and scaling exponent <span class="math inline">\(\tau\)</span> (b and d)
|
||||
as a function of energy and temperature. (a) and (b) show the system
|
||||
transitioning from the CDW phase to the gapless Anderson insulating one
|
||||
at <span class="math inline">\(U=2\)</span> while (c) and (d) show the
|
||||
CDW to gapped Mott phase transition at <span
|
||||
class="math inline">\(U=5\)</span>. Regions where the DOS is close to
|
||||
zero are shown a white. The scaling exponent <span
|
||||
class="math inline">\(\tau\)</span> is obtained from fits to <span
|
||||
class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of
|
||||
system sizes. <span class="math inline">\(U = 5,\;J = 5,\;\alpha =
|
||||
1.25\)</span></figcaption>
|
||||
<img src="pdf_figs/gap_openingboth.svg" id="fig:band_opening" alt="Figure 3: The DOS (a and c) and scaling exponent \tau (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at U=2 while (c) and (d) show the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown a white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. U = 5,\;J = 5,\;\alpha = 1.25" />
|
||||
<figcaption aria-hidden="true">Figure 3: The DOS (a and c) and scaling exponent <span class="math inline">\(\tau\)</span> (b and d) as a function of energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at <span class="math inline">\(U=2\)</span> while (c) and (d) show the CDW to gapped Mott phase transition at <span class="math inline">\(U=5\)</span>. Regions where the DOS is close to zero are shown a white. The scaling exponent <span class="math inline">\(\tau\)</span> is obtained from fits to <span class="math inline">\(IPR(N) = A N^{-\lambda}\)</span> for a range of system sizes. <span class="math inline">\(U = 5,\;J = 5,\;\alpha = 1.25\)</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p><img src="../figure_code/fk_chapter/gap_opening_high_U.svg"
|
||||
id="fig:gap_opening_high_U" data-short-caption="no title"
|
||||
style="width:100.0%"
|
||||
alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" />
|
||||
<img src="../figure_code/fk_chapter/gap_opening_low_U.svg"
|
||||
id="fig:gap_opening_low_U" data-short-caption="no title"
|
||||
style="width:100.0%"
|
||||
alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" /></p>
|
||||
<div id="fig:indiv_IPR_disorder" class="fignos">
|
||||
<p><img src="../figure_code/fk_chapter/gap_opening_high_U.svg" id="fig:gap_opening_high_U" data-short-caption="no title" style="width:100.0%" alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" /> <img src="../figure_code/fk_chapter/gap_opening_low_U.svg" id="fig:gap_opening_low_U" data-short-caption="no title" style="width:100.0%" alt="The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25" /></p>
|
||||
<figure>
|
||||
<img src="pdf_figs/indiv_IPR_disorder.svg"
|
||||
alt="Figure 5: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true"><span>Figure 5:</span> A comparison of
|
||||
the full FK model to a simple binary disorder model (DM) with a CDW wave
|
||||
background perturbed by uncorrelated defects at density <span
|
||||
class="math inline">\(0 < \rho < 1\)</span> matched to the largest
|
||||
corresponding FK model. As in Fig <a href="#fig:indiv_IPR"
|
||||
data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, the
|
||||
Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and
|
||||
<span class="math inline">\(\tau\)</span> are shown. The DOSs match well
|
||||
and this data makes clear that the apparent scaling of IPR with system
|
||||
size is a finite size effect due to weak localisation <span
|
||||
class="citation"
|
||||
data-cites="antipovInteractionTunedAndersonMott2016"> [<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">2</a>]</span>, hence all the states are indeed
|
||||
localised as one would expect in 1D. The disorder model <span
|
||||
class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a)
|
||||
<span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span
|
||||
class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span
|
||||
class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span
|
||||
class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are
|
||||
fit on system sizes <span class="math inline">\(N >
|
||||
400\)</span></figcaption>
|
||||
<img src="pdf_figs/indiv_IPR_disorder.svg" id="fig:indiv_IPR_disorder" alt="Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the largest corresponding FK model. As in Fig 2, the Energy resolved DOS(\omega) and \tau are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation [2], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400" />
|
||||
<figcaption aria-hidden="true">Figure 4: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density <span class="math inline">\(0 < \rho < 1\)</span> matched to the largest corresponding FK model. As in Fig <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a>, the Energy resolved DOS(<span class="math inline">\(\omega\)</span>) and <span class="math inline">\(\tau\)</span> are shown. The DOSs match well and this data makes clear that the apparent scaling of IPR with system size is a finite size effect due to weak localisation <span class="citation" data-cites="antipovInteractionTunedAndersonMott2016"> [<a href="#ref-antipovInteractionTunedAndersonMott2016" role="doc-biblioref">2</a>]</span>, hence all the states are indeed localised as one would expect in 1D. The disorder model <span class="math inline">\(\tau_0,\tau_1\)</span> for each figure are: (a) <span class="math inline">\(0.01\pm0.05, -0.02\pm0.06\)</span> (b) <span class="math inline">\(0.01\pm0.04, -0.01\pm0.04\)</span> (c) <span class="math inline">\(0.05\pm0.06, 0.04\pm0.06\)</span> (d) <span class="math inline">\(-0.03\pm0.06, 0.01\pm0.06\)</span>. The lines are fit on system sizes <span class="math inline">\(N > 400\)</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>Fig. <a href="#fig:indiv_IPR" data-reference-type="ref"
|
||||
data-reference="fig:indiv_IPR">2</a> shows the DOS and <span
|
||||
class="math inline">\(\tau\)</span>, the scaling exponent of the IPR
|
||||
with system size, for a representative set of parameters covering all
|
||||
three phases. The DOS is symmetric about <span
|
||||
class="math inline">\(0\)</span> because of the particle hole symmetry
|
||||
of the model. At high temperatures, all of the eigenstates are localised
|
||||
in both the Mott and Anderson phases (with <span
|
||||
class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We
|
||||
also checked that the states are localised by direct inspection. Note
|
||||
that there are in-gap states for instance at <span
|
||||
class="math inline">\(\omega_0\)</span>, below the upper band which are
|
||||
localized and smoothly connected across the phase transition.</p>
|
||||
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and
|
||||
<span class="math inline">\(U=5\)</span>, we find for the states within
|
||||
the gapped CDW bands, e.g. at <span
|
||||
class="math inline">\(\omega_1\)</span>, scaling exponents <span
|
||||
class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span
|
||||
class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This
|
||||
surprising finding suggests that the CDW bands are partially delocalised
|
||||
with multi-fractal behaviour of the wavefunctions <span class="citation"
|
||||
data-cites="eversAndersonTransitions2008"> [<a
|
||||
href="#ref-eversAndersonTransitions2008"
|
||||
role="doc-biblioref">5</a>]</span>. This phenomenon would be unexpected
|
||||
in a 1D model as they generally do not support delocalisation in the
|
||||
presence of disorder except as the result of correlations in the
|
||||
emergent disorder potential <span class="citation"
|
||||
data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a
|
||||
href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a
|
||||
href="#ref-goldshteinPurePointSpectrum1977"
|
||||
role="doc-biblioref">7</a>]</span>. However, we later show by comparison
|
||||
to an uncorrelated Anderson model that these nonzero exponents are a
|
||||
finite size effect and the states are localised with a finite <span
|
||||
class="math inline">\(\xi\)</span> similar to the system size. As a
|
||||
result, the IPR does not scale correctly until the system size has grown
|
||||
much larger than <span class="math inline">\(\xi\)</span>. Fig. [<a
|
||||
href="#fig:indiv_IPR_disorder" data-reference-type="ref"
|
||||
data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of
|
||||
the IPR in the CDW phase does flatten out eventually.</p>
|
||||
<p>Next, we use the DOS and the scaling exponent <span
|
||||
class="math inline">\(\tau\)</span> to explore the localisation
|
||||
properties over the energy-temperature plane in Fig. <a
|
||||
href="#fig:band_opening" data-reference-type="ref"
|
||||
data-reference="fig:band_opening">3</a>. Gapped areas are shown in
|
||||
white, which highlights the distinction between the gapped Mott phase
|
||||
and the ungapped Anderson phase. In-gap states appear just below the
|
||||
critical point, smoothly filling the bandgap in the Anderson phase and
|
||||
forming islands in the Mott phase. As in the finite <span
|
||||
class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
|
||||
href="#ref-zondaGaplessRegimeCharge2019"
|
||||
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>
|
||||
and infinite dimensional <span class="citation"
|
||||
data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a
|
||||
href="#ref-hassanSpectralPropertiesChargedensitywave2007"
|
||||
role="doc-biblioref">8</a>]</span> cases, the in-gap states merge and
|
||||
are pushed to lower energy for decreasing U as the <span
|
||||
class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the
|
||||
presence of in-gap states can be understood as a result of domain wall
|
||||
fluctuations away from the AFM ordered background. These domain walls
|
||||
act as local potentials for impurity-like bound states <span
|
||||
class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a
|
||||
href="#ref-zondaGaplessRegimeCharge2019"
|
||||
role="doc-biblioref"><strong>zondaGaplessRegimeCharge2019?</strong></a>]</span>.</p>
|
||||
<p>In order to understand the localization properties we can compare the
|
||||
behaviour of our model with that of a simpler Anderson disorder model
|
||||
(DM) in which the spins are replaced by a CDW background with
|
||||
uncorrelated binary defect potentials, see Appendix <a
|
||||
href="#app:disorder_model" data-reference-type="ref"
|
||||
data-reference="app:disorder_model">[app:disorder_model]</a>. Fig. [<a
|
||||
href="#fig:indiv_IPR_disorder" data-reference-type="ref"
|
||||
data-reference="fig:indiv_IPR_disorder">4</a>] compares the FK model to
|
||||
the disorder model at different system sizes, matching the defect
|
||||
densities of the disorder model to the FK model at <span
|
||||
class="math inline">\(N = 270\)</span> above and below the CDW
|
||||
transition. We find very good, even quantitative, agreement between the
|
||||
FK and disorder models, which suggests that correlations in the spin
|
||||
sector do not play a significant role. As we can sample directly from
|
||||
the disorder model, rather than through MCMC, the samples are
|
||||
uncorrelated. Hence we can evaluate much larger system sizes with the
|
||||
disorder model which enables us to pin down the correct localisation
|
||||
effects. In particular, what appear to be delocalized states for small
|
||||
system sizes eventually turn out to be states with large localization
|
||||
length. The localization length diverges towards the ordered zero
|
||||
temperature CDW state. Overall, we see that the interplay of
|
||||
interactions, here manifest as a peculiar binary potential, and
|
||||
localization can be very intricate and the added advantage of our 1D
|
||||
model is that we can explore very large system sizes for a complete
|
||||
understanding.</p>
|
||||
<p>Fig. <a href="#fig:indiv_IPR" data-reference-type="ref" data-reference="fig:indiv_IPR">2</a> shows the DOS and <span class="math inline">\(\tau\)</span>, the scaling exponent of the IPR with system size, for a representative set of parameters covering all three phases. The DOS is symmetric about <span class="math inline">\(0\)</span> because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with <span class="math inline">\(\tau \leq 0.07\)</span> for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at <span class="math inline">\(\omega_0\)</span>, below the upper band which are localized and smoothly connected across the phase transition.</p>
|
||||
<p>In the CDW phases at <span class="math inline">\(U=2\)</span> and <span class="math inline">\(U=5\)</span>, we find for the states within the gapped CDW bands, e.g. at <span class="math inline">\(\omega_1\)</span>, scaling exponents <span class="math inline">\(\tau = 0.30\pm0.03\)</span> and <span class="math inline">\(\tau = 0.15\pm0.05\)</span>, respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions <span class="citation" data-cites="eversAndersonTransitions2008"> [<a href="#ref-eversAndersonTransitions2008" role="doc-biblioref">5</a>]</span>. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential <span class="citation" data-cites="croyAndersonLocalization1D2011 goldshteinPurePointSpectrum1977"> [<a href="#ref-croyAndersonLocalization1D2011" role="doc-biblioref">6</a>,<a href="#ref-goldshteinPurePointSpectrum1977" role="doc-biblioref">7</a>]</span>. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite <span class="math inline">\(\xi\)</span> similar to the system size. As a result, the IPR does not scale correctly until the system size has grown much larger than <span class="math inline">\(\xi\)</span>. Fig. [<a href="#fig:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</a>] shows that the scaling of the IPR in the CDW phase does flatten out eventually.</p>
|
||||
<p>Next, we use the DOS and the scaling exponent <span class="math inline">\(\tau\)</span> to explore the localisation properties over the energy-temperature plane in Fig. <a href="#fig:band_opening" data-reference-type="ref" data-reference="fig:band_opening">3</a>. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">8</a>]</span> and infinite dimensional <span class="citation" data-cites="hassanSpectralPropertiesChargedensitywave2007"> [<a href="#ref-hassanSpectralPropertiesChargedensitywave2007" role="doc-biblioref">9</a>]</span> cases, the in-gap states merge and are pushed to lower energy for decreasing U as the <span class="math inline">\(T=0\)</span> CDW gap closes. Intuitively, the presence of in-gap states can be understood as a result of domain wall fluctuations away from the AFM ordered background. These domain walls act as local potentials for impurity-like bound states <span class="citation" data-cites="zondaGaplessRegimeCharge2019"> [<a href="#ref-zondaGaplessRegimeCharge2019" role="doc-biblioref">8</a>]</span>.</p>
|
||||
<p>In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials, see Appendix ??. Fig. [<a href="#fig:indiv_IPR_disorder" data-reference-type="ref" data-reference="fig:indiv_IPR_disorder">4</a>] compares the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at <span class="math inline">\(N = 270\)</span> above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role. As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. Overall, we see that the interplay of interactions, here manifest as a peculiar binary potential, and localization can be very intricate and the added advantage of our 1D model is that we can explore very large system sizes for a complete understanding.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="fk-conclusion" class="level1">
|
||||
<h1>Discussion and Conclusion</h1>
|
||||
<p>The FK model is one of the simplest non-trivial models of interacting
|
||||
fermions. We studied its thermodynamic and localisation properties
|
||||
brought down in dimensionality to 1D by adding a novel long-ranged
|
||||
coupling designed to stabilise the CDW phase present in dimension two
|
||||
and above. Our hybrid MCMC approach elucidates a disorder-free
|
||||
localization mechanism within our translationally invariant system.
|
||||
Further, we demonstrate a significant speedup over the naive method. We
|
||||
show that our long-range FK in 1D retains much of the rich phase diagram
|
||||
of its higher dimensional cousins. Careful scaling analysis indicates
|
||||
that all the single particle eigenstates eventually localise at nonzero
|
||||
temperature albeit only for very large system sizes of several
|
||||
thousand.</p>
|
||||
<p>Our work raises a number of interesting questions for future
|
||||
research. A straightforward but numerically challenging problem is to
|
||||
pin down the model’s behaviour closer to the critical point where
|
||||
correlations in the spin sector would become significant. Would this
|
||||
modify the localisation behaviour? Similar to other soluble models of
|
||||
disorder-free localisation, we expect intriguing out-of equilibrium
|
||||
physics, for example slow entanglement dynamics akin to more generic
|
||||
interacting systems <span class="citation"
|
||||
data-cites="hartLogarithmicEntanglementGrowth2020"> [<a
|
||||
href="#ref-hartLogarithmicEntanglementGrowth2020"
|
||||
role="doc-biblioref">9</a>]</span>. One could also investigate whether
|
||||
the rich ground state phenomenology of the FK model as a function of
|
||||
filling <span class="citation"
|
||||
data-cites="gruberGroundStatesSpinless1990"> [<a
|
||||
href="#ref-gruberGroundStatesSpinless1990"
|
||||
role="doc-biblioref">10</a>]</span> such as the devil’s staircase <span
|
||||
class="citation"
|
||||
data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a
|
||||
href="#ref-michelettiCompleteDevilTextquotesingles1997"
|
||||
role="doc-biblioref">11</a>]</span> could be stabilised at finite
|
||||
temperature. In a broader context, we envisage that long-range
|
||||
interactions can also be used to gain a deeper understanding of the
|
||||
temperature evolution of topological phases. One example would be a
|
||||
long-ranged FK version of the celebrated Su-Schrieffer-Heeger model
|
||||
where one could explore the interplay of topological bound states and
|
||||
thermal domain wall defects. Finally, the rich physics of our model
|
||||
should be realizable in systems with long-range interactions, such as
|
||||
trapped ion quantum simulators, where one can also explore the fully
|
||||
interacting regime with a dynamical background field.</p>
|
||||
<p>The FK model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to 1D by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above. Our hybrid MCMC approach elucidates a disorder-free localization mechanism within our translationally invariant system. Further, we demonstrate a significant speedup over the naive method. We show that our long-range FK in 1D retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at nonzero temperature albeit only for very large system sizes of several thousand.</p>
|
||||
<p>Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems <span class="citation" data-cites="hartLogarithmicEntanglementGrowth2020"> [<a href="#ref-hartLogarithmicEntanglementGrowth2020" role="doc-biblioref">10</a>]</span>. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling <span class="citation" data-cites="gruberGroundStatesSpinless1990"> [<a href="#ref-gruberGroundStatesSpinless1990" role="doc-biblioref">11</a>]</span> such as the devil’s staircase <span class="citation" data-cites="michelettiCompleteDevilTextquotesingles1997"> [<a href="#ref-michelettiCompleteDevilTextquotesingles1997" role="doc-biblioref">12</a>]</span> could be stabilised at finite temperature. In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged FK version of the celebrated Su-Schrieffer-Heeger model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realizable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.</p>
|
||||
<p><strong>UNCORRELATED DISORDER MODEL</strong></p>
|
||||
<p>The disorder model referred to in the main text is defined by
|
||||
replacing the spin degree of freedom in the FK model <span
|
||||
class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder
|
||||
potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span>
|
||||
controlled by a defect density <span class="math inline">\(\rho\)</span>
|
||||
such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with
|
||||
probability <span class="math inline">\(\rho/2\)</span> and <span
|
||||
class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span
|
||||
class="math inline">\(\rho/2\)</span> is used rather than <span
|
||||
class="math inline">\(\rho\)</span> so that the disorder potential takes
|
||||
on the zero temperature CDW ground state at <span
|
||||
class="math inline">\(\rho = 0\)</span> and becomes a random choice over
|
||||
spin states at <span class="math inline">\(\rho = 1\)</span> i.e the
|
||||
infinite temperature limit. <span
|
||||
class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
|
||||
\tfrac{1}{2}) \\
|
||||
& -\;t \sum_{i} c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i}
|
||||
\nonumber\end{aligned}\]</span></p>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Chapter: <a
|
||||
href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous
|
||||
Kitaev Model</a></p>
|
||||
<p>The disorder model referred to in the main text is defined by replacing the spin degree of freedom in the FK model <span class="math inline">\(S_i = \pm \tfrac{1}{2}\)</span> with a disorder potential <span class="math inline">\(d_i = \pm \tfrac{1}{2}\)</span> controlled by a defect density <span class="math inline">\(\rho\)</span> such that <span class="math inline">\(d_i = -\tfrac{1}{2}\)</span> with probability <span class="math inline">\(\rho/2\)</span> and <span class="math inline">\(d_i = \tfrac{1}{2}\)</span> otherwise. <span class="math inline">\(\rho/2\)</span> is used rather than <span class="math inline">\(\rho\)</span> so that the disorder potential takes on the zero temperature CDW ground state at <span class="math inline">\(\rho = 0\)</span> and becomes a random choice over spin states at <span class="math inline">\(\rho = 1\)</span> i.e the infinite temperature limit. <span class="math display">\[\begin{aligned}
|
||||
H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} - \tfrac{1}{2}) \\
|
||||
& -\;t \sum_{i} c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i} \nonumber\end{aligned}\]</span></p>
|
||||
<p>Could look at doping the mott insulating phase, see results like <span class="citation" data-cites="caiVisualizingEvolutionMott2016"> [<a href="#ref-caiVisualizingEvolutionMott2016" role="doc-biblioref">13</a>]</span></p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Chapter: <a href="../4_Amorphous_Kitaev_Model/4.1.2_AMK_Model.html">4 The Amorphous Kitaev Model</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K.
|
||||
Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size
|
||||
Scaling Analysis of Ising Model Block Distribution Functions</a></em>,
|
||||
Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">K. Binder, <em><a href="https://doi.org/10.1007/BF01293604">Finite Size Scaling Analysis of Ising Model Block Distribution Functions</a></em>, Z. Physik B - Condensed Matter <strong>43</strong>, 119 (1981).</div>
|
||||
</div>
|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
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|
||||
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|
||||
</div>
|
||||
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||||
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||||
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of Localization: Absence of Quantum Diffusion in Two
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||||
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(1979).</div>
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||||
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||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, <em><a href="https://doi.org/10.1103/PhysRevLett.42.673">Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions</a></em>, Phys. Rev. Lett. <strong>42</strong>, 673 (1979).</div>
|
||||
</div>
|
||||
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||||
Kramer and A. MacKinnon, <em><a
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||||
href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory
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||||
and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469
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||||
(1993).</div>
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||||
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||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">B. Kramer and A. MacKinnon, <em><a href="https://doi.org/10.1088/0034-4885/56/12/001">Localization: Theory and Experiment</a></em>, Rep. Prog. Phys. <strong>56</strong>, 1469 (1993).</div>
|
||||
</div>
|
||||
<div id="ref-eversAndersonTransitions2008" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">F.
|
||||
Evers and A. D. Mirlin, <em><a
|
||||
href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson
|
||||
Transitions</a></em>, Rev. Mod. Phys. <strong>80</strong>, 1355
|
||||
(2008).</div>
|
||||
<div id="ref-eversAndersonTransitions2008" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">F. Evers and A. D. Mirlin, <em><a href="https://doi.org/10.1103/RevModPhys.80.1355">Anderson Transitions</a></em>, Rev. Mod. Phys. <strong>80</strong>, 1355 (2008).</div>
|
||||
</div>
|
||||
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
|
||||
Croy, P. Cain, and M. Schreiber, <em><a
|
||||
href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization
|
||||
in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B
|
||||
<strong>82</strong>, 107 (2011).</div>
|
||||
<div id="ref-croyAndersonLocalization1D2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A. Croy, P. Cain, and M. Schreiber, <em><a href="https://doi.org/10.1140/epjb/e2011-20212-1">Anderson Localization in 1d Systems with Correlated Disorder</a></em>, Eur. Phys. J. B <strong>82</strong>, 107 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">I.
|
||||
Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a
|
||||
href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the
|
||||
Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its
|
||||
Appl <strong>11</strong>, 1 (1977).</div>
|
||||
<div id="ref-goldshteinPurePointSpectrum1977" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">I. Ya. Gol’dshtein, S. A. Molchanov, and L. A. Pastur, <em><a href="https://doi.org/10.1007/BF01135526">A Pure Point Spectrum of the Stochastic One-Dimensional Schrödinger Operator</a></em>, Funct Anal Its Appl <strong>11</strong>, 1 (1977).</div>
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||||
</div>
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||||
<div id="ref-hassanSpectralPropertiesChargedensitywave2007"
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||||
class="csl-entry" role="doc-biblioentry">
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||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">S.
|
||||
R. Hassan and H. R. Krishnamurthy, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in
|
||||
the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball
|
||||
Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
|
||||
<div id="ref-zondaGaplessRegimeCharge2019" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">M. Žonda, J. Okamoto, and M. Thoss, <em><a href="https://doi.org/10.1103/PhysRevB.100.075124">Gapless Regime in the Charge Density Wave Phase of the Finite Dimensional Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>100</strong>, 075124 (2019).</div>
|
||||
</div>
|
||||
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">O.
|
||||
Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a
|
||||
href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth
|
||||
from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>,
|
||||
arXiv:2009.00618 [Cond-Mat] (2020).</div>
|
||||
<div id="ref-hassanSpectralPropertiesChargedensitywave2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">S. R. Hassan and H. R. Krishnamurthy, <em><a href="https://doi.org/10.1103/PhysRevB.76.205109">Spectral Properties in the Charge-Density-Wave Phase of the Half-Filled Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>76</strong>, 205109 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">C.
|
||||
Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the
|
||||
Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B
|
||||
<strong>41</strong>, 2198 (1990).</div>
|
||||
<div id="ref-hartLogarithmicEntanglementGrowth2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">O. Hart, S. Gopalakrishnan, and C. Castelnovo, <em><a href="http://arxiv.org/abs/2009.00618">Logarithmic Entanglement Growth from Disorder-Free Localisation in the Two-Leg Compass Ladder</a></em>, arXiv:2009.00618 [Cond-Mat] (2020).</div>
|
||||
</div>
|
||||
<div id="ref-michelettiCompleteDevilTextquotesingles1997"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">C.
|
||||
Micheletti, A. B. Harris, and J. M. Yeomans, <em><a
|
||||
href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete
|
||||
Devil\textquotesingles Staircase in the Falicov - Kimball
|
||||
Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711
|
||||
(1997).</div>
|
||||
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">C. Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, <em><a href="https://doi.org/10.1103/PhysRevB.41.2198">Ground States of the Spinless Falicov-Kimball Model</a></em>, Phys. Rev. B <strong>41</strong>, 2198 (1990).</div>
|
||||
</div>
|
||||
<div id="ref-michelettiCompleteDevilTextquotesingles1997" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C. Micheletti, A. B. Harris, and J. M. Yeomans, <em><a href="https://doi.org/10.1088/0305-4470/30/21/002">A Complete Devil\textquotesingles Staircase in the Falicov - Kimball Model</a></em>, J. Phys. A: Math. Gen. <strong>30</strong>, L711 (1997).</div>
|
||||
</div>
|
||||
<div id="ref-caiVisualizingEvolutionMott2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">P. Cai, W. Ruan, Y. Peng, C. Ye, X. Li, Z. Hao, X. Zhou, D.-H. Lee, and Y. Wang, <em><a href="https://doi.org/10.1038/nphys3840">Visualizing the Evolution from the Mott Insulator to a Charge-Ordered Insulator in Lightly Doped Cuprates</a></em>, Nature Phys <strong>12</strong>, 11 (2016).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
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Load Diff
@ -30,32 +30,7 @@ image:
|
||||
<ul>
|
||||
<li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
|
||||
<ul>
|
||||
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
|
||||
Systems</a></li>
|
||||
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
|
||||
<li><a href="#the-kitaev-model" id="toc-the-kitaev-model">The Kitaev
|
||||
Model</a>
|
||||
<ul>
|
||||
<li><a href="#commutation-relations"
|
||||
id="toc-commutation-relations">Commutation relations</a></li>
|
||||
<li><a href="#the-hamiltonian" id="toc-the-hamiltonian">The
|
||||
Hamiltonian</a></li>
|
||||
<li><a href="#from-spins-to-majorana-operators"
|
||||
id="toc-from-spins-to-majorana-operators">From Spins to Majorana
|
||||
operators</a></li>
|
||||
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors"
|
||||
id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning
|
||||
the Hilbert Space into Bond sectors</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#the-majorana-hamiltonian"
|
||||
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a>
|
||||
<ul>
|
||||
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
|
||||
id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
|
||||
back from Bond Sectors to the Physical Subspace</a></li>
|
||||
<li><a href="#open-boundary-conditions"
|
||||
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous Systems</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -72,32 +47,7 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
<ul>
|
||||
<li><a href="#amk-Model" id="toc-amk-Model">The Model</a>
|
||||
<ul>
|
||||
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous
|
||||
Systems</a></li>
|
||||
<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
|
||||
<li><a href="#the-kitaev-model" id="toc-the-kitaev-model">The Kitaev
|
||||
Model</a>
|
||||
<ul>
|
||||
<li><a href="#commutation-relations"
|
||||
id="toc-commutation-relations">Commutation relations</a></li>
|
||||
<li><a href="#the-hamiltonian" id="toc-the-hamiltonian">The
|
||||
Hamiltonian</a></li>
|
||||
<li><a href="#from-spins-to-majorana-operators"
|
||||
id="toc-from-spins-to-majorana-operators">From Spins to Majorana
|
||||
operators</a></li>
|
||||
<li><a href="#partitioning-the-hilbert-space-into-bond-sectors"
|
||||
id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning
|
||||
the Hilbert Space into Bond sectors</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#the-majorana-hamiltonian"
|
||||
id="toc-the-majorana-hamiltonian">The Majorana Hamiltonian</a>
|
||||
<ul>
|
||||
<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace"
|
||||
id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping
|
||||
back from Bond Sectors to the Physical Subspace</a></li>
|
||||
<li><a href="#open-boundary-conditions"
|
||||
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#amorphous-systems" id="toc-amorphous-systems">Amorphous Systems</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -111,727 +61,40 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</div>
|
||||
<p><strong>Contributions</strong></p>
|
||||
<p>The material in this chapter expands on work presented in</p>
|
||||
<p> <span class="citation"
|
||||
data-cites="cassellaExactChiralAmorphous2022"> [<a
|
||||
href="#ref-cassellaExactChiralAmorphous2022"
|
||||
role="doc-biblioref">1</a>]</span> Cassella, G., D’Ornellas, P., Hodson,
|
||||
T., Natori, W. M., & Knolle, J. (2022). An exact chiral amorphous
|
||||
spin liquid. <em>arXiv preprint arXiv:2208.08246.</em></p>
|
||||
<p>the code is available at <span class="citation"
|
||||
data-cites="hodsonKoalaKitaevAmorphous2022"> [<a
|
||||
href="#ref-hodsonKoalaKitaevAmorphous2022"
|
||||
role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>This was a joint project of Gino, Peru and myself with advice and
|
||||
guidance from Willian and Johannes. The project grew out of an interest
|
||||
the three of us had in studying amorphous systems, coupled with
|
||||
Johannes’ expertise on the Kitaev model. The idea to use voronoi
|
||||
partitions came from <span class="citation"
|
||||
data-cites="marsalTopologicalWeaireThorpe2020"> [<a
|
||||
href="#ref-marsalTopologicalWeaireThorpe2020"
|
||||
role="doc-biblioref">3</a>]</span> and Gino did the implementation of
|
||||
this. The idea and implementation of the edge colouring using SAT
|
||||
solvers, the mapping from flux sector to bond sector using A* search
|
||||
were both entirely my work. Peru found the ground state and implemented
|
||||
the local markers. Gino and I did much of the rest of the programming
|
||||
for Koala while pair programming and ’whiteboard’ing, this included the
|
||||
phase diagram, edge mode and finite temperature analyses as well as the
|
||||
derivation of the projector in the amorphous case.</p>
|
||||
<p> <span class="citation" data-cites="cassellaExactChiralAmorphous2022"> [<a href="#ref-cassellaExactChiralAmorphous2022" role="doc-biblioref">1</a>]</span> Cassella, G., D’Ornellas, P., Hodson, T., Natori, W. M., & Knolle, J. (2022). An exact chiral amorphous spin liquid. <em>arXiv preprint arXiv:2208.08246.</em></p>
|
||||
<p>the code is available at <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>This was a joint project of Gino, Peru and myself with advice and guidance from Willian and Johannes. The project grew out of an interest the three of us had in studying amorphous systems, coupled with Johannes’ expertise on the Kitaev model. The idea to use voronoi partitions came from <span class="citation" data-cites="marsalTopologicalWeaireThorpe2020"> [<a href="#ref-marsalTopologicalWeaireThorpe2020" role="doc-biblioref">3</a>]</span> and Gino did the implementation of this. The idea and implementation of the edge colouring using SAT solvers, the mapping from flux sector to bond sector using A* search were both entirely my work. Peru found the ground state and implemented the local markers. Gino and I did much of the rest of the programming for Koala while pair programming and ’whiteboard’ing, this included the phase diagram, edge mode and finite temperature analyses as well as the derivation of the projector in the amorphous case.</p>
|
||||
<section id="amk-Model" class="level1">
|
||||
<h1>The Model</h1>
|
||||
<div id="fig:intro_figure_by_hand" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg"
|
||||
data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%"
|
||||
alt="Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span>
|
||||
<strong>(a)</strong> The standard Kitaev model is defined on a honeycomb
|
||||
lattice. The special feature of the honeycomb lattice that makes the
|
||||
model solvable is that each vertex is joined by exactly three bonds,
|
||||
i.e. the lattice is trivalent. One of three labels is assigned to each
|
||||
<strong>(b)</strong>. We represent the antisymmetric gauge degree of
|
||||
freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> with arrows
|
||||
that point in the direction <span class="math inline">\(u_{jk} =
|
||||
+1\)</span> <strong>(c)</strong>. The Majorana transformation can be
|
||||
visualised as breaking each spin into four Majoranas which then pair
|
||||
along the bonds. The pairs of x,y and z Majoranas become part of the
|
||||
classical <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field
|
||||
<span class="math inline">\(u_{ij}\)</span>. This leavies a single
|
||||
Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig:intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site." />
|
||||
<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each <strong>(b)</strong>. We represent the antisymmetric gauge degree of freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> with arrows that point in the direction <span class="math inline">\(u_{jk} = +1\)</span> <strong>(c)</strong>. The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field <span class="math inline">\(u_{ij}\)</span>. This leavies a single Majorana <span class="math inline">\(c_i\)</span> per site.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>The Kitaev Honeycomb model <span class="math display">\[H = -
|
||||
\sum_{\langle j,k\rangle_\alpha}
|
||||
J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\]</span> is remarkable
|
||||
because it combines three key properties.</p>
|
||||
<p>First, the form of the Hamiltonian plausibly be realised by a real
|
||||
material. Candidate materials, such as <span
|
||||
class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have
|
||||
sufficiently strong spin-orbit coupling and the correct lattice
|
||||
structure to behave according to the Kitaev Honeycomb model with small
|
||||
corrections <span class="citation"
|
||||
data-cites="banerjeeProximateKitaevQuantum2016 trebstKitaevMaterials2022"> [<a
|
||||
href="#ref-banerjeeProximateKitaevQuantum2016"
|
||||
role="doc-biblioref">4</a>,<a href="#ref-trebstKitaevMaterials2022"
|
||||
role="doc-biblioref"><strong>trebstKitaevMaterials2022?</strong></a>]</span>.</p>
|
||||
<p><strong>expand later: Why do we need spin orbit coupling and what
|
||||
will the corrections be?</strong></p>
|
||||
<p>Second, its ground state is the canonical example of the long sought
|
||||
after quantum spin liquid state. Its excitations are anyons, particles
|
||||
that can only exist in two dimensions that break the normal
|
||||
fermion/boson dichotomy. Anyons have been the subject of much attention
|
||||
because, among other reasons, they can be braided through spacetime to
|
||||
achieve noise tolerant quantum computations <span class="citation"
|
||||
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
|
||||
href="#ref-freedmanTopologicalQuantumComputation2003"
|
||||
role="doc-biblioref">5</a>]</span>.</p>
|
||||
<p>Third, and perhaps most importantly, this model is a rare many body
|
||||
interacting quantum system that can be treated analytically. It is
|
||||
exactly solvable. We can explicitly write down its many body ground
|
||||
states in terms of single particle states <span class="citation"
|
||||
data-cites="kitaevAnyonsExactlySolved2006"> [<a
|
||||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||||
role="doc-biblioref">6</a>]</span>. The solubility of the Kitaev
|
||||
Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes
|
||||
about because the model has extensively many conserved degrees of
|
||||
freedom. These conserved quantities can be factored out as classical
|
||||
degrees of freedom, leaving behind a non-interacting quantum model that
|
||||
is easy to solve.</p>
|
||||
<section id="amorphous-systems" class="level2">
|
||||
<h2>Amorphous Systems</h2>
|
||||
<p><strong>Insert discussion of why a generalisation to the amorphous
|
||||
case is interesting</strong></p>
|
||||
<p>This chapter details the physics of the Kitaev model on amorphous
|
||||
lattices.</p>
|
||||
<p>It starts by expanding on the physics of the Kitaev model. It will
|
||||
look at the gauge symmetries of the model as well as its solution via a
|
||||
transformation to a Majorana hamiltonian. This discussion shows that,
|
||||
for the the model to be solvable, it needs only be defined on a
|
||||
trivalent, tri-edge-colourable lattice <span class="citation"
|
||||
data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009"
|
||||
role="doc-biblioref">7</a>]</span>.</p>
|
||||
<p>The methods section discusses how to generate such lattices and
|
||||
colour them. It also explain how to map back and forth between
|
||||
configurations of the gauge field and configurations of the gauge
|
||||
invariant quantities.</p>
|
||||
<p>The results section begins by looking at the zero temperature
|
||||
physics. It presents numerical evidence that the ground state of the
|
||||
Kitaev model is given by a simple rule depending only on the number of
|
||||
sides of each plaquette. It assesses the gapless, Abelian and
|
||||
non-Abelian, phases that are present, characterising them by the
|
||||
presence of a gap and using local Chern markers. Next it looks at
|
||||
spontaneous chiral symmetry breaking and topological edge states. It
|
||||
also compares the zero temperature phase diagram to that of the Kitaev
|
||||
Honeycomb Model. Finally, we introduce flux disorder and demonstrate
|
||||
that there is a phase transition to a thermal metal state.</p>
|
||||
<p>The discussion considers possible physical realisations of this model
|
||||
and the motivations for doing so. It also discusses how a well known
|
||||
quantum error correcting code defined on the Kitaev Honeycomb model
|
||||
could be generalised to the amorphous case.</p>
|
||||
</section>
|
||||
<section id="glossary" class="level2">
|
||||
<h2>Glossary</h2>
|
||||
<ul>
|
||||
<li><p>Lattice: The underlying graph on which the models are defined.
|
||||
Composed of sites (vertices), bonds (edges) and plaquettes
|
||||
(faces).</p></li>
|
||||
<li><p>The model : Used when I refer to properties of the the Kitaev
|
||||
model that do not depend on the particular lattice.</p></li>
|
||||
<li><p>The Honeycomb model : The Kitaev Model defined on the honeycomb
|
||||
lattice.</p></li>
|
||||
<li><p>The Amorphous model : The Kitaev Model defined on the amorphous
|
||||
lattices described here.</p></li>
|
||||
</ul>
|
||||
<p><strong>The Spin Hamiltonian</strong></p>
|
||||
<ul>
|
||||
<li>Spin Bond Operators: <span class="math inline">\(\hat{k}_{ij} =
|
||||
\sigma_i^\alpha \sigma_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} =
|
||||
\prod_{<i,j>} k_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
</ul>
|
||||
<p><strong>The Majorana Model</strong></p>
|
||||
<ul>
|
||||
<li>Majorana Operators on site <span class="math inline">\(i\)</span>:
|
||||
<span class="math inline">\(\hat{b}^x_i, \hat{b}^y_i, \hat{b}^z_i,
|
||||
\hat{c}_i\)</span></li>
|
||||
<li>Majorana Bond Operators: <span class="math inline">\(\hat{u}_{ij} =
|
||||
i \sigma_i^\alpha \sigma_j^\alpha\)</span></li>
|
||||
<li>Loop Operators: <span class="math inline">\(\hat{W_p} =
|
||||
\prod_{<i,j>} u_{ij}\)</span></li>
|
||||
<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
|
||||
<li>Gauge Operators: <span class="math inline">\(D_i = \hat{b}^x_i
|
||||
\hat{b}^y_i \hat{b}^z_i \hat{c}_i\)</span></li>
|
||||
<li>The Extended Hilbert space: The larger Hilbert space spanned by the
|
||||
Majorana operators.</li>
|
||||
<li>The physical subspace: The subspace of the extended Hilbert space
|
||||
that we identify with the Hilbert space of the original spin model.</li>
|
||||
<li>The Projector <span class="math inline">\(\hat{P}\)</span>: The
|
||||
projector onto the physical subspace.</li>
|
||||
</ul>
|
||||
<p><strong>Flux Sectors</strong></p>
|
||||
<ul>
|
||||
<li><p>Odd/Even Plaquettes: Plaquettes with an odd/even number of
|
||||
sides.</p></li>
|
||||
<li><p>Fluxes <span class="math inline">\(\phi_i\)</span>: The
|
||||
expectation values of the plaquette operators <span
|
||||
class="math inline">\(\pm 1\)</span> for even and <span
|
||||
class="math inline">\(\pm i\)</span> for odd plaquettes.</p></li>
|
||||
<li><p>Flux Sector: A subspace of Hilbert space in which the fluxes take
|
||||
particular values.</p></li>
|
||||
<li><p>Ground state flux sector: The Flux Sector containing the lowest
|
||||
energy many body state.</p></li>
|
||||
<li><p>Vortices: Flux excitations away from the ground state flux
|
||||
sector.</p></li>
|
||||
<li><p>Dual Loops: A set of <span class="math inline">\(u_{jk}\)</span>
|
||||
that correspond to loops on the dual lattice.</p></li>
|
||||
<li><p>non-contractible loops or dual loops: The two loops topologically
|
||||
distinct loops on the torus that cannot be smoothly deformed to a
|
||||
point.</p></li>
|
||||
<li><p>Topological Fluxes <span class="math inline">\(\Phi_{x},
|
||||
\Phi_{y}\)</span>: The two fluxes associated with the two
|
||||
non-contractible loops.</p></li>
|
||||
<li><p>Topological Transport Operators: <span
|
||||
class="math inline">\(\mathcal{T}_{x}, \mathcal{T}_{y}\)</span>: The two
|
||||
vortex-pair operations associated with the non-contractible
|
||||
<em>dual</em> loops.</p></li>
|
||||
</ul>
|
||||
<p><strong>Phases</strong></p>
|
||||
<ul>
|
||||
<li>The A phase: The three anisotropic regions of the phase diagram
|
||||
<span class="math inline">\(A_x, A_y, A_z\)</span> where <span
|
||||
class="math inline">\(A_\alpha\)</span> means <span
|
||||
class="math inline">\(J_\alpha >> J_\beta, J_\gamma\)</span>.</li>
|
||||
<li>The B phase: The roughly isotropic region of the phase diagram.</li>
|
||||
</ul>
|
||||
</section>
|
||||
<section id="the-kitaev-model" class="level2">
|
||||
<h2>The Kitaev Model</h2>
|
||||
<section id="commutation-relations" class="level3">
|
||||
<h3>Commutation relations</h3>
|
||||
<p>Before diving into the Hamiltonian of the Kitaev model, the following
|
||||
describes the key commutation relations of spins, fermions and
|
||||
Majoranas.</p>
|
||||
<section id="spins" class="level4">
|
||||
<h4>Spins</h4>
|
||||
<p>Skip this is you are familiar with the algebra of the Pauli matrices.
|
||||
Scalars like <span class="math inline">\(\delta_{ij}\)</span> should be
|
||||
understood to be multiplied by an implicit identity <span
|
||||
class="math inline">\(\mathbb{1}\)</span> where necessary.</p>
|
||||
<p>We can represent a single spin<span
|
||||
class="math inline">\(-1/2\)</span> particle using the Pauli matrices
|
||||
<span class="math inline">\((\sigma^x, \sigma^y, \sigma^z) =
|
||||
\vec{\sigma}\)</span>, these matrices all square to the identity <span
|
||||
class="math inline">\(\sigma^\alpha \sigma^\alpha = \mathbb{1}\)</span>
|
||||
and obey nice commutation and exchange rules: <span
|
||||
class="math display">\[\sigma^\alpha \sigma^\beta = \delta^{\alpha
|
||||
\beta} + i \epsilon^{\alpha \beta \gamma} \sigma^\gamma\]</span> <span
|
||||
class="math display">\[[\sigma^\alpha, \sigma^\beta] = 2 i
|
||||
\epsilon^{\alpha \beta \gamma} \sigma^\gamma\]</span></p>
|
||||
<p>Adding site indices, spins at different spatial sites always commute
|
||||
<span class="math inline">\([\vec{\sigma}_i, \vec{\sigma}_j] =
|
||||
0\)</span> so when <span class="math inline">\(i \neq j\)</span> <span
|
||||
class="math display">\[\sigma_i^\alpha \sigma_j^\beta = \sigma_j^\alpha
|
||||
\sigma_i^\beta\]</span> <span class="math display">\[[\sigma_i^\alpha,
|
||||
\sigma_j^\beta] = 0\]</span> while the previous equations hold for <span
|
||||
class="math inline">\(i = j\)</span>.</p>
|
||||
<p>Two extra relations useful for the Kitaev model are the value of
|
||||
<span class="math inline">\(\sigma^\alpha \sigma^\beta
|
||||
\sigma^\gamma\)</span> and <span class="math inline">\([\sigma^\alpha
|
||||
\sigma^\beta, \sigma^\gamma]\)</span> when <span
|
||||
class="math inline">\(\alpha \neq \beta \neq \gamma\)</span> these can
|
||||
be computed relatively easily by applying the above relations yielding:
|
||||
<span class="math display">\[\sigma^\alpha \sigma^\beta \sigma^\gamma =
|
||||
i \epsilon^{\alpha\beta\gamma}\]</span> and <span
|
||||
class="math display">\[[\sigma^\alpha \sigma^\beta, \sigma^\gamma] =
|
||||
0\]</span></p>
|
||||
</section>
|
||||
<section id="fermions-and-majoranas" class="level4">
|
||||
<h4>Fermions and Majoranas</h4>
|
||||
<p>The fermionic creation and anhilation operators are defined by the
|
||||
canonical anticommutation relations <span
|
||||
class="math display">\[\begin{aligned}
|
||||
\{f_i, f_j\} &= \{f^\dagger_i, f^\dagger_j\} = 0\\
|
||||
\{f_i, f^\dagger_j\} &= \delta_{ij}
|
||||
\end{aligned}\]</span> which give us the exchange statistics and Pauli
|
||||
exclusion principle.</p>
|
||||
<p>From fermionic operators, we can construct Majorana operators: <span
|
||||
class="math display">\[\begin{aligned}
|
||||
f_i &= 1/2 (a_i + ib_i)\\
|
||||
f^\dagger_i &= 1/2(a_i - ib_i)\\
|
||||
a_i &= f_i + f^\dagger_i = 2\Re f\\
|
||||
b_i &= 1/i(f_i - f^\dagger_i) = 2\Im f
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Majorana operators are the real and imaginary parts of the fermionic
|
||||
operators. Physically, they correspond to the orthogonal superpositions
|
||||
of the presence and absence of the fermion and are, thus, a kind of
|
||||
quasiparticle.</p>
|
||||
<p>Once we involve multiple fermions, there is some freedom in how we
|
||||
can perform the transformation from <span
|
||||
class="math inline">\(n\)</span> fermions <span
|
||||
class="math inline">\(f_i\)</span> to <span
|
||||
class="math inline">\(2n\)</span> Majoranas <span
|
||||
class="math inline">\(c_i\)</span>. The property that must be preserved,
|
||||
however, is that the Majoranas still anticommute:</p>
|
||||
<p><span class="math display">\[ \{c_i, c_j\} =
|
||||
2\delta_{ij}\]</span></p>
|
||||
<div id="fig:visual_kitaev_1" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg"
|
||||
data-short-caption="A Visual Intro to the Kitaev Model"
|
||||
style="width:100.0%"
|
||||
alt="Figure 2: A visual introduction to the Kitaev Model." />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> A visual
|
||||
introduction to the Kitaev Model.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
</section>
|
||||
</section>
|
||||
<section id="the-hamiltonian" class="level3">
|
||||
<h3>The Hamiltonian</h3>
|
||||
<p>To start from the fundamentals, the Kitaev Honeycomb model is a model
|
||||
of interacting spin<span class="math inline">\(-1/2\)</span>s on the
|
||||
vertices of a honeycomb lattice. Each bond in the lattice is assigned a
|
||||
label <span class="math inline">\(\alpha \in \{ x, y, z\}\)</span> and
|
||||
that bond couples its two spin neighbours along the <span
|
||||
class="math inline">\(\alpha\)</span> axis. See fig. <a
|
||||
href="#fig:visual_kitaev_1">2</a> for a diagram.</p>
|
||||
<p>This gives us the Hamiltonian <span class="math display">\[H = -
|
||||
\sum_{\langle j,k\rangle_\alpha}
|
||||
J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\]</span> where <span
|
||||
class="math inline">\(\sigma^\alpha_j\)</span> is a Pauli matrix acting
|
||||
on site <span class="math inline">\(j\)</span> and <span
|
||||
class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of
|
||||
nearest-neighbour indices connected by an <span
|
||||
class="math inline">\(\alpha\)</span>-bond with exchange coupling <span
|
||||
class="math inline">\(J^\alpha\)</span> <span class="citation"
|
||||
data-cites="kitaevAnyonsExactlySolved2006"> [<a
|
||||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||||
role="doc-biblioref">6</a>]</span>. For notational brevity, it is useful
|
||||
to introduce the bond operators <span class="math inline">\(K_{ij} =
|
||||
\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span
|
||||
class="math inline">\(\alpha\)</span> is a function of <span
|
||||
class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
|
||||
<p>This Kitaev model has a set of conserved quantities that, in the spin
|
||||
language, take the form of Wilson loop operators <span
|
||||
class="math inline">\(W_p\)</span> winding around a closed path on the
|
||||
lattice. The direction does not matter, but we will keep to clockwise
|
||||
here. We will use the term plaquette and the symbol <span
|
||||
class="math inline">\(\phi\)</span> to refer to a Wilson loop operator
|
||||
that does not enclose any other sites, such as a single hexagon in a
|
||||
honeycomb lattice.</p>
|
||||
<p><span class="math display">\[W_p = \prod_{\mathrm{i,j}\; \in\; p}
|
||||
K_{ij} = \sigma_1^z \sigma_2^x \sigma_2^y \sigma_3^y .. \sigma_n^y
|
||||
\sigma_n^y \sigma_1^z\]</span></p>
|
||||
<p><strong>add a diagram of a single plaquette with labelled site and
|
||||
bond types</strong></p>
|
||||
<p>In closed loops, each site appears twice in the product with two of
|
||||
the three bond types. Applying <span class="math inline">\(\sigma^\alpha
|
||||
\sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq
|
||||
\beta\)</span> then gives us a product containing a single Pauli matrix
|
||||
associated with each site in the loop with the type of the
|
||||
<em>outward</em> pointing bond. This shows that the <span
|
||||
class="math inline">\(W_p\)</span> associated with hexagons or shapes
|
||||
with an even number of sides all square to 1 and, hence, have
|
||||
eigenvalues <span class="math inline">\(\pm 1\)</span>.</p>
|
||||
<p>A bipartite lattice is composed of A and B sublattices with no
|
||||
intra-sublattice edges, i.e. no A-A or B-B edges. Any closed loop must
|
||||
begin and end at the same site. If we start at an A site, the loop must
|
||||
go A-B-A-B… until it returns to the original site. It must, therefore,
|
||||
contain an even number of edges to end on the same sublattice that it
|
||||
started on.</p>
|
||||
<p>As the honeycomb lattice is bipartite, there are no closed loops that
|
||||
contain an even number of edges. Therefore, all the <span
|
||||
class="math inline">\(W_p\)</span> have eigenvalues <span
|
||||
class="math inline">\(\pm 1\)</span> on bipartite lattices. Later, we
|
||||
will show that plaquettes with an odd number of sides (odd plaquettes
|
||||
for short) have eigenvalues <span class="math inline">\(\pm
|
||||
i\)</span>.</p>
|
||||
<div id="fig:regular_plaquettes" class="fignos">
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg"
|
||||
data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%"
|
||||
alt="Figure 3: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> The eigenvalues of
|
||||
a loop or plaquette operators depend on the number of bonds in its
|
||||
enclosing path.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>Remarkably, all of the spin bond operators <span
|
||||
class="math inline">\(K_{ij}\)</span> commute with all the Wilson loop
|
||||
operators <span class="math inline">\(W_p\)</span>. <span
|
||||
class="math display">\[[W_p, K_{ij}] = 0\]</span> We can prove this by
|
||||
considering three cases: 1. neither <span
|
||||
class="math inline">\(i\)</span> nor <span
|
||||
class="math inline">\(j\)</span> is part of the loop 2. one of <span
|
||||
class="math inline">\(i\)</span> or <span
|
||||
class="math inline">\(j\)</span> are part of the loop 3. both are part
|
||||
of the loop</p>
|
||||
<p>The first case is trivial. The other two require some algebra,
|
||||
outlined in fig. <a href="#fig:visual_kitaev_2">4</a>.</p>
|
||||
<div id="fig:visual_kitaev_2" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/visual_kitaev_2.svg"
|
||||
data-short-caption="Plaquette Operators are Conserved"
|
||||
style="width:100.0%"
|
||||
alt="Figure 4: Plaquette operators are conserved." />
|
||||
<figcaption aria-hidden="true"><span>Figure 4:</span> Plaquette
|
||||
operators are conserved.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>Since the Hamiltonian is a linear combination of bond operators, it
|
||||
commutes with the plaquette operators. This is helpful because it leads
|
||||
to a simultaneous eigenbasis for the Hamiltonian and the plaquette
|
||||
operators. We can, thus, work in <em>or “on”???</em> a basis in which
|
||||
the eigenvalues of the plaquette operators take on a definite value and,
|
||||
for all intents and purposes, act like classical degrees of freedom.
|
||||
These are the extensively many conserved quantities that make the model
|
||||
tractable.</p>
|
||||
<p>Plaquette operators measure flux. We will find that the ground state
|
||||
of the model corresponds to some particular choice of flux through each
|
||||
plaquette. We will refer to excitations which flip the expectation value
|
||||
of a plaquette operator away from the ground state as
|
||||
<strong>vortices</strong>.</p>
|
||||
<p>Thus, fixing a configuration of the vortices partitions the many-body
|
||||
Hilbert space into a set of ‘vortex sectors’ labelled by that particular
|
||||
flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm
|
||||
i\)</span>.</p>
|
||||
</section>
|
||||
<section id="from-spins-to-majorana-operators" class="level3">
|
||||
<h3>From Spins to Majorana operators</h3>
|
||||
<section id="for-a-single-spin" class="level4">
|
||||
<h4>For a single spin</h4>
|
||||
<p>Let us start by considering only one site and its <span
|
||||
class="math inline">\(\sigma^x, \sigma^y\)</span> and <span
|
||||
class="math inline">\(\sigma^z\)</span> operators which live in a two
|
||||
dimensional Hilbert space <span
|
||||
class="math inline">\(\mathcal{L}\)</span>.</p>
|
||||
<p>We will introduce two fermionic modes <span
|
||||
class="math inline">\(f\)</span> and <span
|
||||
class="math inline">\(g\)</span> that satisfy the canonical
|
||||
anticommutation relations along with their number operators <span
|
||||
class="math inline">\(n_f = f^\dagger f, n_g = g^\dagger g\)</span> and
|
||||
the total fermionic parity operator <span class="math inline">\(F_p =
|
||||
(2n_f - 1)(2n_g - 1)\)</span> which can be used to divide their Fock
|
||||
space up into even and odd parity subspaces. These subspaces are
|
||||
separated by the addition or removal of one fermion.</p>
|
||||
<p>From these two fermionic modes, we can build four Majorana operators:
|
||||
<span class="math display">\[\begin{aligned}
|
||||
b^x &= f + f^\dagger\\
|
||||
b^y &= -i(f - f^\dagger)\\
|
||||
b^z &= g + g^\dagger\\
|
||||
c &= -i(g - g^\dagger)
|
||||
\end{aligned}\]</span></p>
|
||||
<p>The Majoranas obey the usual commutation relations, squaring to one
|
||||
and anticommuting with each other. The fermions and Majorana live in a
|
||||
four dimensional Fock space <span
|
||||
class="math inline">\(\mathcal{\tilde{L}}\)</span>. We can therefore
|
||||
identify the two dimensional space <span
|
||||
class="math inline">\(\mathcal{M}\)</span> with one of the parity
|
||||
subspaces of <span class="math inline">\(\mathcal{\tilde{L}}\)</span>
|
||||
which will be called the <em>physical subspace</em> <span
|
||||
class="math inline">\(\mathcal{\tilde{L}}_p\)</span>. Kitaev defines the
|
||||
operator <span class="math display">\[D = b^xb^yb^zc\]</span> which can
|
||||
be expanded to <span class="math display">\[D = -(2n_f - 1)(2n_g - 1) =
|
||||
-F_p\]</span> and labels the physical subspace as the space spanned by
|
||||
states for which <span class="math display">\[ D|\phi\rangle =
|
||||
|\phi\rangle\]</span></p>
|
||||
<p>We can also think of the physical subspace as whatever is left after
|
||||
applying the projector <span class="math display">\[P = \frac{1 -
|
||||
D}{2}\]</span> This formulation will be useful for taking states that
|
||||
span the extended space <span
|
||||
class="math inline">\(\mathcal{\tilde{M}}\)</span> and projecting them
|
||||
into the physical subspace.</p>
|
||||
<p>So now, with the caveat that we are working in the physical subspace,
|
||||
we can define new Pauli operators:</p>
|
||||
<p><span class="math display">\[\tilde{\sigma}^x = i b^x c,\;
|
||||
\tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]</span></p>
|
||||
<p>These extended space Pauli operators satisfy all the usual
|
||||
commutation relations. The only difference is that if we evaluate <span
|
||||
class="math inline">\(\sigma^x \sigma^y \sigma^z = i\)</span>, we
|
||||
instead get <span class="math display">\[
|
||||
\tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]</span></p>
|
||||
<p>This makes sense if we promise to confine ourselves to the physical
|
||||
subspace <span class="math inline">\(D = 1\)</span>.</p>
|
||||
</section>
|
||||
<section id="for-multiple-spins" class="level4">
|
||||
<h4>For multiple spins</h4>
|
||||
<p>This construction easily generalises to the case of multiple spins.
|
||||
We get a set of 4 Majoranas <span class="math inline">\(b^x_j,\;
|
||||
b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j =
|
||||
b^x_jb^y_jb^z_jc_j\)</span> operator for every spin. For a state to be
|
||||
physical, we require that <span class="math inline">\(D_j |\psi\rangle =
|
||||
|\psi\rangle\)</span> for all <span
|
||||
class="math inline">\(j\)</span>.</p>
|
||||
<p>From these each Pauli operator can be constructed: <span
|
||||
class="math display">\[\tilde{\sigma}^\alpha_j = i b^\alpha_j
|
||||
c_j\]</span></p>
|
||||
<p>This is where the magic happens. We can promote the spin hamiltonian
|
||||
from <span class="math inline">\(\mathcal{L}\)</span> into the extended
|
||||
space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>, safe in
|
||||
the knowledge that nothing changes so long as we only actually work with
|
||||
physical states. The Hamiltonian <span
|
||||
class="math display">\[\begin{aligned}
|
||||
\tilde{H} &= - \sum_{\langle j,k\rangle_\alpha}
|
||||
J^{\alpha}\tilde{\sigma}_j^{\alpha}\tilde{\sigma}_k^{\alpha}\\
|
||||
&= \frac{i}{4} \sum_{\langle j,k\rangle_\alpha}
|
||||
2J^{\alpha} (ib^\alpha_i b^\alpha_j) c_i c_j\\
|
||||
&= \frac{i}{4} \sum_{\langle i,j\rangle_\alpha}
|
||||
2J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j
|
||||
\end{aligned}\]</span></p>
|
||||
<p>We can factor out the Majorana bond operators <span
|
||||
class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span>.
|
||||
Note that these bond operators are not equal to the spin bond operators
|
||||
<span class="math inline">\(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = -
|
||||
\hat{u}_{ij} c_i c_j\)</span>. In what follows, we will work much more
|
||||
frequently with the Majorana bond operators. Therefore, when we refer to
|
||||
bond operators without qualification, we are referring to the Majorana
|
||||
variety.</p>
|
||||
<p>Similarly to the argument with the spin bond operators <span
|
||||
class="math inline">\(K_{ij}\)</span>, we can quickly verify by
|
||||
considering three cases that the Majorana bond operators <span
|
||||
class="math inline">\(u_{ij}\)</span> all commute with one another. They
|
||||
square to one, so have eigenvalues <span class="math inline">\(\pm
|
||||
1\)</span>. They also commute with the <span
|
||||
class="math inline">\(c_i\)</span> operators.</p>
|
||||
<p>Importantly, the operators <span class="math inline">\(D_i = b^x_i
|
||||
b^y_i b^z_i c_i\)</span> commute with <span
|
||||
class="math inline">\(K_{ij}\)</span> and, therefore, with <span
|
||||
class="math inline">\(\tilde{H}\)</span>. We will show later that the
|
||||
action of <span class="math inline">\(D_i\)</span> on a state is to flip
|
||||
the values of the three <span class="math inline">\(u_{ij}\)</span>
|
||||
bonds that connect to site <span class="math inline">\(i\)</span>.
|
||||
Physically, this indicates that <span
|
||||
class="math inline">\(u_{ij}\)</span> is a gauge field with a high
|
||||
degree of degeneracy.</p>
|
||||
<p>In summary, Majorana bond operators <span
|
||||
class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span
|
||||
class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="partitioning-the-hilbert-space-into-bond-sectors"
|
||||
class="level3">
|
||||
<h3>Partitioning the Hilbert Space into Bond sectors</h3>
|
||||
<p>Similarly to the story with the plaquette operators from the spin
|
||||
language, we can divide the Hilbert space <span
|
||||
class="math inline">\(\mathcal{L}\)</span> into sectors labelled by a
|
||||
set of choices <span class="math inline">\(\{\pm 1\}\)</span> for the
|
||||
value of each <span class="math inline">\(u_{ij}\)</span> operator which
|
||||
we denote by <span class="math inline">\(\mathcal{L}_u\)</span>. Since
|
||||
<span class="math inline">\(u_{ij} = -u_{ji}\)</span>, we can represent
|
||||
the <span class="math inline">\(u_{ij}\)</span> graphically with an
|
||||
arrow that points along each bond in the direction in which <span
|
||||
class="math inline">\(u_{ij} = 1\)</span>.</p>
|
||||
<p>Once confined to a particular <span
|
||||
class="math inline">\(\mathcal{L}_u\)</span>, we can ‘remove the hats’
|
||||
from the <span class="math inline">\(\hat{u}_{ij}\)</span>. The
|
||||
hamiltonian becomes a quadratic, free fermion problem <span
|
||||
class="math display">\[\tilde{H_u} = \frac{i}{4} \sum_{\langle
|
||||
i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> The ground state,
|
||||
<span class="math inline">\(|\psi_u\rangle\)</span> can be found easily
|
||||
as will be explained in the next part. At this point, we may wonder
|
||||
whether the <span class="math inline">\(\mathcal{L}_u\)</span> are
|
||||
confined entirely within the physical subspace <span
|
||||
class="math inline">\(\mathcal{L}_p\)</span> and, indeed, we will see
|
||||
that they are not. However, it will be helpful to first develop the
|
||||
theory of the Majorana Hamiltonian further.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="the-majorana-hamiltonian" class="level2">
|
||||
<h2>The Majorana Hamiltonian</h2>
|
||||
<p>We now have a quadratic Hamiltonian <span class="math display">\[
|
||||
\tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha}
|
||||
u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom
|
||||
have paired along bonds to become a classical gauge field <span
|
||||
class="math inline">\(u_{ij}\)</span>. What follows is relatively
|
||||
standard theory for quadratic Majorana Hamiltonians <span
|
||||
class="citation" data-cites="BlaizotRipka1986"> [<a
|
||||
href="#ref-BlaizotRipka1986" role="doc-biblioref">8</a>]</span>.</p>
|
||||
<p>Because of the antisymmetry of the matrix with entries <span
|
||||
class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the
|
||||
Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in
|
||||
pairs <span class="math inline">\(\pm \epsilon_m\)</span>. This
|
||||
redundant information is a consequence of the doubling of the Hilbert
|
||||
space which occurred when we transformed to the Majorana
|
||||
representation.</p>
|
||||
<p>If we organise the eigenmodes of <span
|
||||
class="math inline">\(H\)</span> into pairs, such that <span
|
||||
class="math inline">\(b_m\)</span> and <span
|
||||
class="math inline">\(b_m'\)</span> have energies <span
|
||||
class="math inline">\(\epsilon_m\)</span> and <span
|
||||
class="math inline">\(-\epsilon_m\)</span>, we can construct the
|
||||
transformation <span class="math inline">\(Q\)</span> <span
|
||||
class="math display">\[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2,
|
||||
b_2' ... b_{N}, b_{N}')\]</span> and put the Hamiltonian into
|
||||
the form <span class="math display">\[\tilde{H}_u = \frac{i}{2} \sum_m
|
||||
\epsilon_m b_m b_m'\]</span></p>
|
||||
<p>The determinant of <span class="math inline">\(Q\)</span> will be
|
||||
useful later when we consider the projector from <span
|
||||
class="math inline">\(\mathcal{\tilde{L}}\)</span> to <span
|
||||
class="math inline">\(\mathcal{L}\)</span>. Otherwise, the <span
|
||||
class="math inline">\(b_m\)</span> are merely an intermediate step. From
|
||||
them, we form fermionic operators <span class="math display">\[ f_i =
|
||||
\tfrac{1}{2} (b_m + ib_m')\]</span> with their associated number
|
||||
operators <span class="math inline">\(n_i = f^\dagger_i f_i\)</span>.
|
||||
These let us write the Hamiltonian neatly as</p>
|
||||
<p><span class="math display">\[ \tilde{H}_u = \sum_m \epsilon_m (n_m -
|
||||
\tfrac{1}{2}).\]</span></p>
|
||||
<p>The ground state <span class="math inline">\(|n_m = 0\rangle\)</span>
|
||||
of the many body system at fixed <span class="math inline">\(u\)</span>
|
||||
is then <span class="math display">\[E_{u,0} = -\frac{1}{2}\sum_m
|
||||
\epsilon_m \]</span> We can construct any state from a particular choice
|
||||
of <span class="math inline">\(n_m = 0,1\)</span>.</p>
|
||||
<p>If we only care about the value of <span
|
||||
class="math inline">\(E_{u,0}\)</span>, it is possible to skip forming
|
||||
the fermionic operators. The eigenvalues obtained directly from
|
||||
diagonalising <span class="math inline">\(J^{\alpha} u_{ij}\)</span>
|
||||
come in <span class="math inline">\(\pm \epsilon_m\)</span> pairs. We
|
||||
can take half the absolute value of the whole set to recover <span
|
||||
class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
|
||||
<p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond
|
||||
Sector.</p>
|
||||
<section id="mapping-back-from-bond-sectors-to-the-physical-subspace"
|
||||
class="level3">
|
||||
<h3>Mapping back from Bond Sectors to the Physical Subspace</h3>
|
||||
<p>At this point, given a particular bond configuration <span
|
||||
class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a
|
||||
quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span>
|
||||
in the extended space and diagonalise it to find its ground state <span
|
||||
class="math inline">\(|\vec{u}, \vec{n} = 0\rangle\)</span>. This is not
|
||||
necessarily the ground state of the system as a whole, it is just the
|
||||
lowest energy state within the subspace <span
|
||||
class="math inline">\(\mathcal{L}_u\)</span></p>
|
||||
<p><strong>However, <span class="math inline">\(|u, n_m =
|
||||
0\rangle\)</span> does not lie in the physical subspace</strong>. As an
|
||||
example, consider the lowest energy state associated with <span
|
||||
class="math inline">\(u_{ij} = +1\)</span>. This state satisfies <span
|
||||
class="math display">\[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle =
|
||||
|\vec{u}=1, \vec{n} = 0\rangle\]</span> for all bonds <span
|
||||
class="math inline">\(i,j\)</span>.</p>
|
||||
<p>If we act on it, this state with one of the gauge operators <span
|
||||
class="math inline">\(D_j = b_j^x b_j^y b_j^z c_j\)</span>, we see that
|
||||
<span class="math inline">\(D_j\)</span> flips the value of the three
|
||||
bonds <span class="math inline">\(u_{ij}\)</span> that surround site
|
||||
<span class="math inline">\(k\)</span>:</p>
|
||||
<p><span class="math display">\[ |u'\rangle = D_j |u=1, n_m =
|
||||
0\rangle\]</span></p>
|
||||
<p><span class="math display">\[ \begin{aligned}
|
||||
\langle u'|u_{ij}|u'\rangle &= \langle u| b_j^x b_j^y b_j^z
|
||||
c_j \;ib^x_i b^x_j\; b_j^x b_j^y b_j^z c_j|u\rangle\\
|
||||
&= -1
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Since <span class="math inline">\(D_j\)</span> commutes with the
|
||||
Hamiltonian in the extended space <span
|
||||
class="math inline">\(\tilde{H}\)</span>, the fact that <span
|
||||
class="math inline">\(D_j\)</span> flips the value of bond operators
|
||||
indicates that there is a gauge degeneracy between the ground state of
|
||||
<span class="math inline">\(\tilde{H}_u\)</span> and the set of <span
|
||||
class="math inline">\(\tilde{H}_{u'}\)</span> related to it by gauge
|
||||
transformations <span class="math inline">\(D_j\)</span>. Thus, we can
|
||||
flip any three bonds around a vertex and the physics will stay the
|
||||
same.</p>
|
||||
<p>We can turn this into a symmetrisation procedure by taking a
|
||||
superposition of every possible gauge transformation. Every possible
|
||||
gauge transformation is just every possible subset of <span
|
||||
class="math inline">\({D_0, D_1 ... D_n}\)</span> which can be neatly
|
||||
expressed as <span class="math display">\[|\phi_w\rangle = \prod_i
|
||||
\left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\]</span> This is
|
||||
convenient because the quantity <span class="math inline">\(\frac{1 +
|
||||
D_i}{2}\)</span> is also the local projector onto the physical subspace.
|
||||
Here <span class="math inline">\(|\phi_w\rangle\)</span> is a gauge
|
||||
invariant state that lives in <span
|
||||
class="math inline">\(\mathcal{L}_p\)</span> which has been constructed
|
||||
from a set of states in different <span
|
||||
class="math inline">\(\mathcal{L}_u\)</span>.</p>
|
||||
<p>This gauge degeneracy leads us to the next topic of discussion,
|
||||
namely how to construct a set of gauge invariant quantities out of the
|
||||
<span class="math inline">\(u_{ij}\)</span>, these will turn out to just
|
||||
be the plaquette operators.</p>
|
||||
<p>Takeaway: The Bond Sectors overlap with the physical subspace but are
|
||||
not contained within it.</p>
|
||||
</section>
|
||||
<section id="open-boundary-conditions" class="level3">
|
||||
<h3>Open boundary conditions</h3>
|
||||
<p>Care must be taken when defining open boundary conditions. Simply
|
||||
removing bonds from the lattice leaves behind unpaired <span
|
||||
class="math inline">\(b^\alpha\)</span> operators that must be paired in
|
||||
some way to arrive at fermionic modes. To fix a pairing, we always start
|
||||
from a lattice defined on the torus and generate a lattice with open
|
||||
boundary conditions by defining the bond coupling <span
|
||||
class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by
|
||||
bonds <span class="math inline">\((i,j)\)</span> that we want to remove.
|
||||
This creates fermionic zero modes <span
|
||||
class="math inline">\(u_{ij}\)</span> associated with these cut bonds
|
||||
which we set to 1 when calculating the projector.</p>
|
||||
<p>Alternatively, since all the fermionic zero modes are degenerate
|
||||
anyway, an arbitrary pairing of the unpaired <span
|
||||
class="math inline">\(b^\alpha\)</span> operators could be performed.
|
||||
</i,j></i,j></p>
|
||||
<p>Next Section: <a
|
||||
href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html">Methods</a></p>
|
||||
</section>
|
||||
<p><strong>Insert discussion of why a generalisation to the amorphous case is interesting</strong></p>
|
||||
<p>This chapter details the physics of the Kitaev model on amorphous lattices.</p>
|
||||
<p>It starts by expanding on the physics of the Kitaev model. It will look at the gauge symmetries of the model as well as its solution via a transformation to a Majorana hamiltonian. This discussion shows that, for the the model to be solvable, it needs only be defined on a trivalent, tri-edge-colourable lattice <span class="citation" data-cites="Nussinov2009"> [<a href="#ref-Nussinov2009" role="doc-biblioref">4</a>]</span>.</p>
|
||||
<p>The methods section discusses how to generate such lattices and colour them. It also explain how to map back and forth between configurations of the gauge field and configurations of the gauge invariant quantities.</p>
|
||||
<p>The results section begins by looking at the zero temperature physics. It presents numerical evidence that the ground state of the Kitaev model is given by a simple rule depending only on the number of sides of each plaquette. It assesses the gapless, Abelian and non-Abelian, phases that are present, characterising them by the presence of a gap and using local Chern markers. Next it looks at spontaneous chiral symmetry breaking and topological edge states. It also compares the zero temperature phase diagram to that of the Kitaev Honeycomb Model. Finally, we introduce flux disorder and demonstrate that there is a phase transition to a thermal metal state.</p>
|
||||
<p>The discussion considers possible physical realisations of this model and the motivations for doing so. It also discusses how a well known quantum error correcting code defined on the Kitaev Honeycomb model could be generalised to the amorphous case.</p>
|
||||
<p>Next Section: <a href="../4_Amorphous_Kitaev_Model/4.2_AMK_Methods.html">Methods</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-cassellaExactChiralAmorphous2022" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">G.
|
||||
Cassella, P. D’Ornellas, T. Hodson, W. M. H. Natori, and J. Knolle,
|
||||
<em><a href="https://doi.org/10.48550/arXiv.2208.08246">An Exact Chiral
|
||||
Amorphous Spin Liquid</a></em>, arXiv:2208.08246.</div>
|
||||
<div id="ref-cassellaExactChiralAmorphous2022" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">G. Cassella, P. D’Ornellas, T. Hodson, W. M. H. Natori, and J. Knolle, <em><a href="https://doi.org/10.48550/arXiv.2208.08246">An Exact Chiral Amorphous Spin Liquid</a></em>, arXiv:2208.08246.</div>
|
||||
</div>
|
||||
<div id="ref-hodsonKoalaKitaevAmorphous2022" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T.
|
||||
Hodson, P. D’Ornellas, and G. Cassella, <em><a
|
||||
href="https://doi.org/10.5281/zenodo.6303275">Koala: Kitaev on Amorphous
|
||||
Lattices</a></em>, (2022).</div>
|
||||
<div id="ref-hodsonKoalaKitaevAmorphous2022" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">T. Hodson, P. D’Ornellas, and G. Cassella, <em><a href="https://doi.org/10.5281/zenodo.6303275">Koala: Kitaev on Amorphous Lattices</a></em>, (2022).</div>
|
||||
</div>
|
||||
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">Q.
|
||||
Marsal, D. Varjas, and A. G. Grushin, <em><a
|
||||
href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe
|
||||
Models of Amorphous Matter</a></em>, Proceedings of the National Academy
|
||||
of Sciences <strong>117</strong>, 30260 (2020).</div>
|
||||
</div>
|
||||
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">A.
|
||||
Banerjee et al., <em><a
|
||||
href="https://doi.org/10.1038/nmat4604">Proximate Kitaev Quantum Spin
|
||||
Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
|
||||
<strong>15</strong>, 733 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-freedmanTopologicalQuantumComputation2003"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">M.
|
||||
Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
|
||||
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
|
||||
Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
|
||||
(2003).</div>
|
||||
</div>
|
||||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">A.
|
||||
Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
|
||||
in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
|
||||
<strong>321</strong>, 2 (2006).</div>
|
||||
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">Q. Marsal, D. Varjas, and A. G. Grushin, <em><a href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire–Thorpe Models of Amorphous Matter</a></em>, Proceedings of the National Academy of Sciences <strong>117</strong>, 30260 (2020).</div>
|
||||
</div>
|
||||
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">Z.
|
||||
Nussinov and G. Ortiz, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
|
||||
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
|
||||
Physical Review B <strong>79</strong>, 214440 (2009).</div>
|
||||
</div>
|
||||
<div id="ref-BlaizotRipka1986" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div
|
||||
class="csl-right-inline">J.-P. Blaizot and G. Ripka, <em>Quantum Theory
|
||||
of Finite Systems</em> (The MIT Press, 1986).</div>
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">Z. Nussinov and G. Ortiz, <em><a href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>, Physical Review B <strong>79</strong>, 214440 (2009).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -31,26 +31,15 @@ image:
|
||||
<li><a href="#amk-methods" id="toc-amk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
|
||||
<li><a href="#graph-representation" id="toc-graph-representation">Graph
|
||||
Representation</a></li>
|
||||
<li><a href="#colouring-the-bonds"
|
||||
id="toc-colouring-the-bonds">Colouring the Bonds</a>
|
||||
<li><a href="#graph-representation" id="toc-graph-representation">Graph Representation</a></li>
|
||||
<li><a href="#colouring-the-bonds" id="toc-colouring-the-bonds">Colouring the Bonds</a>
|
||||
<ul>
|
||||
<li><a href="#four-colourings-and-three-colourings"
|
||||
id="toc-four-colourings-and-three-colourings">Four-colourings and
|
||||
three-colourings</a></li>
|
||||
<li><a href="#finding-lattice-colourings-with-minisat"
|
||||
id="toc-finding-lattice-colourings-with-minisat">Finding Lattice
|
||||
colourings with miniSAT</a></li>
|
||||
<li><a href="#does-it-matter-which-colouring-we-choose"
|
||||
id="toc-does-it-matter-which-colouring-we-choose">Does it matter which
|
||||
colouring we choose?</a></li>
|
||||
<li><a href="#four-colourings-and-three-colourings" id="toc-four-colourings-and-three-colourings">Four-colourings and three-colourings</a></li>
|
||||
<li><a href="#finding-lattice-colourings-with-minisat" id="toc-finding-lattice-colourings-with-minisat">Finding Lattice colourings with miniSAT</a></li>
|
||||
<li><a href="#does-it-matter-which-colouring-we-choose" id="toc-does-it-matter-which-colouring-we-choose">Does it matter which colouring we choose?</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#mapping-between-flux-sectors-and-bond-sectors"
|
||||
id="toc-mapping-between-flux-sectors-and-bond-sectors">Mapping between
|
||||
flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern
|
||||
Markers</a></li>
|
||||
<li><a href="#mapping-between-flux-sectors-and-bond-sectors" id="toc-mapping-between-flux-sectors-and-bond-sectors">Mapping between flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern Markers</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -68,26 +57,15 @@ Markers</a></li>
|
||||
<li><a href="#amk-methods" id="toc-amk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#voronisation" id="toc-voronisation">Voronisation</a></li>
|
||||
<li><a href="#graph-representation" id="toc-graph-representation">Graph
|
||||
Representation</a></li>
|
||||
<li><a href="#colouring-the-bonds"
|
||||
id="toc-colouring-the-bonds">Colouring the Bonds</a>
|
||||
<li><a href="#graph-representation" id="toc-graph-representation">Graph Representation</a></li>
|
||||
<li><a href="#colouring-the-bonds" id="toc-colouring-the-bonds">Colouring the Bonds</a>
|
||||
<ul>
|
||||
<li><a href="#four-colourings-and-three-colourings"
|
||||
id="toc-four-colourings-and-three-colourings">Four-colourings and
|
||||
three-colourings</a></li>
|
||||
<li><a href="#finding-lattice-colourings-with-minisat"
|
||||
id="toc-finding-lattice-colourings-with-minisat">Finding Lattice
|
||||
colourings with miniSAT</a></li>
|
||||
<li><a href="#does-it-matter-which-colouring-we-choose"
|
||||
id="toc-does-it-matter-which-colouring-we-choose">Does it matter which
|
||||
colouring we choose?</a></li>
|
||||
<li><a href="#four-colourings-and-three-colourings" id="toc-four-colourings-and-three-colourings">Four-colourings and three-colourings</a></li>
|
||||
<li><a href="#finding-lattice-colourings-with-minisat" id="toc-finding-lattice-colourings-with-minisat">Finding Lattice colourings with miniSAT</a></li>
|
||||
<li><a href="#does-it-matter-which-colouring-we-choose" id="toc-does-it-matter-which-colouring-we-choose">Does it matter which colouring we choose?</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#mapping-between-flux-sectors-and-bond-sectors"
|
||||
id="toc-mapping-between-flux-sectors-and-bond-sectors">Mapping between
|
||||
flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern
|
||||
Markers</a></li>
|
||||
<li><a href="#mapping-between-flux-sectors-and-bond-sectors" id="toc-mapping-between-flux-sectors-and-bond-sectors">Mapping between flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern Markers</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -99,215 +77,50 @@ Markers</a></li>
|
||||
<p>4 The Amorphous Kitaev Model</p>
|
||||
<hr />
|
||||
</div>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<section id="amk-methods" class="level1">
|
||||
<h1>Methods</h1>
|
||||
<p>The practical implementation of what is described in this section is
|
||||
available as a Python package called Koala (Kitaev On Amorphous
|
||||
LAttices) <span class="citation"
|
||||
data-cites="tomImperialCMTHKoalaFirst2022"> [<a
|
||||
href="#ref-tomImperialCMTHKoalaFirst2022"
|
||||
role="doc-biblioref"><strong>tomImperialCMTHKoalaFirst2022?</strong></a>]</span>.
|
||||
All results and figures were generated with Koala.</p>
|
||||
<p>The practical implementation of what is described in this section is available as a Python package called Koala (Kitaev On Amorphous LAttices) <span class="citation" data-cites="hodsonKoalaKitaevAmorphous2022"> [<a href="#ref-hodsonKoalaKitaevAmorphous2022" role="doc-biblioref">1</a>]</span>. All results and figures were generated with Koala.</p>
|
||||
<section id="voronisation" class="level2">
|
||||
<h2>Voronisation</h2>
|
||||
<p>To study the properties of the amorphous Kitaev model, we need to
|
||||
sample from the space of possible trivalent graphs.</p>
|
||||
<p>A simple method is to use a Voronoi partition of the torus <span
|
||||
class="citation"
|
||||
data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a
|
||||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||||
role="doc-biblioref">1</a>–<a href="#ref-florescu_designer_2009"
|
||||
role="doc-biblioref">3</a>]</span>. We start by sampling <em>seed
|
||||
points</em> uniformly (or otherwise) on the torus. Then, we compute the
|
||||
partition of the torus into regions closest (with a Euclidean metric) to
|
||||
each seed point. The straight lines (if the torus is flattened out) at
|
||||
the borders of these regions become the edges of the new lattice. The
|
||||
points where they intersect become the vertices.</p>
|
||||
<p>The graph generated by a Voronoi partition of a two dimensional
|
||||
surface is always planar. This means that no edges cross each other when
|
||||
the graph is embedded into the plane. It is also trivalent in that every
|
||||
vertex is connected to exactly three edges <strong>cite</strong>.</p>
|
||||
<p>Ideally, we would sample uniformly from the space of possible
|
||||
trivalent graphs. Indeed, there has been some work on how to do this
|
||||
using a Markov Chain Monte Carlo approach <span class="citation"
|
||||
data-cites="alyamiUniformSamplingDirected2016"> [<a
|
||||
href="#ref-alyamiUniformSamplingDirected2016"
|
||||
role="doc-biblioref">4</a>]</span>. However, it does not guarantee that
|
||||
the resulting graph is planar, which we must ensure so that the edges
|
||||
can be 3-coloured.</p>
|
||||
<p>In practice, we use a standard algorithm <span class="citation"
|
||||
data-cites="barberQuickhullAlgorithmConvex1996"> [<a
|
||||
href="#ref-barberQuickhullAlgorithmConvex1996"
|
||||
role="doc-biblioref">5</a>]</span> from Scipy <span class="citation"
|
||||
data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a
|
||||
href="#ref-virtanenSciPyFundamentalAlgorithms2020"
|
||||
role="doc-biblioref">6</a>]</span> which computes the Voronoi partition
|
||||
of the plane. To compute the Voronoi partition of the torus, we take the
|
||||
seed points and replicate them into a repeating grid. This will be
|
||||
either 3x3 or, for very small numbers of seed points, 5x5. Then, we
|
||||
identify edges in the output to construct a lattice on the torus.</p>
|
||||
<div id="fig:lattice_construction_animated" class="fignos">
|
||||
<p>To study the properties of the amorphous Kitaev model, we need to sample from the space of possible trivalent graphs.</p>
|
||||
<p>A simple method is to use a Voronoi partition of the torus <span class="citation" data-cites="mitchellAmorphousTopologicalInsulators2018 marsalTopologicalWeaireThorpeModels2020 florescu_designer_2009"> [<a href="#ref-mitchellAmorphousTopologicalInsulators2018" role="doc-biblioref">2</a>–<a href="#ref-florescu_designer_2009" role="doc-biblioref">4</a>]</span>. We start by sampling <em>seed points</em> uniformly (or otherwise) on the torus. Then, we compute the partition of the torus into regions closest (with a Euclidean metric) to each seed point. The straight lines (if the torus is flattened out) at the borders of these regions become the edges of the new lattice. The points where they intersect become the vertices.</p>
|
||||
<p>The graph generated by a Voronoi partition of a two dimensional surface is always planar. This means that no edges cross each other when the graph is embedded into the plane. It is also trivalent in that every vertex is connected to exactly three edges <strong>cite</strong>.</p>
|
||||
<p>Ideally, we would sample uniformly from the space of possible trivalent graphs. Indeed, there has been some work on how to do this using a Markov Chain Monte Carlo approach <span class="citation" data-cites="alyamiUniformSamplingDirected2016"> [<a href="#ref-alyamiUniformSamplingDirected2016" role="doc-biblioref">5</a>]</span>. However, it does not guarantee that the resulting graph is planar, which we must ensure so that the edges can be 3-coloured.</p>
|
||||
<p>In practice, we use a standard algorithm <span class="citation" data-cites="barberQuickhullAlgorithmConvex1996"> [<a href="#ref-barberQuickhullAlgorithmConvex1996" role="doc-biblioref">6</a>]</span> from Scipy <span class="citation" data-cites="virtanenSciPyFundamentalAlgorithms2020"> [<a href="#ref-virtanenSciPyFundamentalAlgorithms2020" role="doc-biblioref">7</a>]</span> which computes the Voronoi partition of the plane. To compute the Voronoi partition of the torus, we take the seed points and replicate them into a repeating grid. This will be either 3x3 or, for very small numbers of seed points, 5x5. Then, we identify edges in the output to construct a lattice on the torus.</p>
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif"
|
||||
data-short-caption="Lattice Construction" style="width:100.0%"
|
||||
alt="Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges." />
|
||||
<figcaption aria-hidden="true"><span>Figure 1:</span> (Left) Lattice
|
||||
construction begins with the Voronoi partition of the plane with respect
|
||||
to a set of seed points (black points) sampled uniformly from <span
|
||||
class="math inline">\(\mathbb{R}^2\)</span>. (Center) However, we want
|
||||
the Voronoi partition of the torus, so we tile the seed points into a
|
||||
three by three grid. The boundaries of each tile are shown in light
|
||||
grey. (Right) Finally, we identify edges corresponding to each other
|
||||
across the boundaries to produce a graph on the torus. An edge colouring
|
||||
is shown here to help the reader identify corresponding
|
||||
edges.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/lattice_construction_animated/lattice_construction_animated.gif" id="fig:lattice_construction_animated" data-short-caption="Lattice Construction" style="width:100.0%" alt="Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from \mathbb{R}^2. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges." />
|
||||
<figcaption aria-hidden="true">Figure 1: (Left) Lattice construction begins with the Voronoi partition of the plane with respect to a set of seed points (black points) sampled uniformly from <span class="math inline">\(\mathbb{R}^2\)</span>. (Center) However, we want the Voronoi partition of the torus, so we tile the seed points into a three by three grid. The boundaries of each tile are shown in light grey. (Right) Finally, we identify edges corresponding to each other across the boundaries to produce a graph on the torus. An edge colouring is shown here to help the reader identify corresponding edges.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
</section>
|
||||
<section id="graph-representation" class="level2">
|
||||
<h2>Graph Representation</h2>
|
||||
<p>Three keys pieces of information allow us to represent amorphous
|
||||
lattices.</p>
|
||||
<p>Most of the graph connectivity is encoded by an ordered list of edges
|
||||
<span class="math inline">\((i,j)\)</span>. These are ordered to
|
||||
represent both directed and undirected graphs. This is useful for
|
||||
defining the sign of bond operators <span class="math inline">\(u_{ij} =
|
||||
- u_{ji}\)</span>.</p>
|
||||
<p>Information about the embedding of the lattice onto the torus is
|
||||
encoded into a point on the unit square associated with each vertex. The
|
||||
torus is unwrapped onto the square by defining an arbitrary pair of cuts
|
||||
along the major and minor axes. For simplicity, we take these axes to be
|
||||
the lines <span class="math inline">\(x = 0\)</span> and <span
|
||||
class="math inline">\(y = 0\)</span>. We can wrap the unit square back
|
||||
up into a torus by identifying the lines <span class="math inline">\(x =
|
||||
0\)</span> with <span class="math inline">\(x = 1\)</span> and <span
|
||||
class="math inline">\(y = 0\)</span> with <span class="math inline">\(y
|
||||
= 1\)</span>.</p>
|
||||
<p>Finally, we need to encode the topology of the graph. This is
|
||||
necessary because, if we are simply given an edge <span
|
||||
class="math inline">\((i, j)\)</span> we do not know how the edge gets
|
||||
from vertex i to vertex j. One method would be taking the shortest path,
|
||||
but it could also ‘go the long way around’ by crossing one of the cuts.
|
||||
To encode this information, we store an additional vector <span
|
||||
class="math inline">\(\vec{r}\)</span> associated with each edge. <span
|
||||
class="math inline">\(r_i^x = 0\)</span> means that edge i does not
|
||||
cross the x. <span class="math inline">\(r_i^x = +1\)</span> (<span
|
||||
class="math inline">\(-1\)</span>) means it crossed the cut in a
|
||||
positive (negative) sense.</p>
|
||||
<p>This description of the lattice has a very nice relationship to
|
||||
Bloch’s theorem. Applying Bloch’s theorem to a periodic lattice
|
||||
essentially means wrappping the unit cell onto a torus. Variations that
|
||||
happen at longer length scales than the size of the unit cell are
|
||||
captured by the crystal momentum. The crystal momentum inserts a phase
|
||||
factor <span class="math inline">\(e^{i \vec{q}\cdot\vec{r}}\)</span>
|
||||
onto bonds that cross to adjacent unit cells. The vector <span
|
||||
class="math inline">\(\vec{r}\)</span> is exactly what we use to encode
|
||||
the topology of our lattices.</p>
|
||||
<div id="fig:bloch" class="fignos">
|
||||
<p>Three keys pieces of information allow us to represent amorphous lattices.</p>
|
||||
<p>Most of the graph connectivity is encoded by an ordered list of edges <span class="math inline">\((i,j)\)</span>. These are ordered to represent both directed and undirected graphs. This is useful for defining the sign of bond operators <span class="math inline">\(u_{ij} = - u_{ji}\)</span>.</p>
|
||||
<p>Information about the embedding of the lattice onto the torus is encoded into a point on the unit square associated with each vertex. The torus is unwrapped onto the square by defining an arbitrary pair of cuts along the major and minor axes. For simplicity, we take these axes to be the lines <span class="math inline">\(x = 0\)</span> and <span class="math inline">\(y = 0\)</span>. We can wrap the unit square back up into a torus by identifying the lines <span class="math inline">\(x = 0\)</span> with <span class="math inline">\(x = 1\)</span> and <span class="math inline">\(y = 0\)</span> with <span class="math inline">\(y = 1\)</span>.</p>
|
||||
<p>Finally, we need to encode the topology of the graph. This is necessary because, if we are simply given an edge <span class="math inline">\((i, j)\)</span> we do not know how the edge gets from vertex i to vertex j. One method would be taking the shortest path, but it could also ‘go the long way around’ by crossing one of the cuts. To encode this information, we store an additional vector <span class="math inline">\(\vec{r}\)</span> associated with each edge. <span class="math inline">\(r_i^x = 0\)</span> means that edge i does not cross the x. <span class="math inline">\(r_i^x = +1\)</span> (<span class="math inline">\(-1\)</span>) means it crossed the cut in a positive (negative) sense.</p>
|
||||
<p>This description of the lattice has a very nice relationship to Bloch’s theorem. Applying Bloch’s theorem to a periodic lattice essentially means wrappping the unit cell onto a torus. Variations that happen at longer length scales than the size of the unit cell are captured by the crystal momentum. The crystal momentum inserts a phase factor <span class="math inline">\(e^{i \vec{q}\cdot\vec{r}}\)</span> onto bonds that cross to adjacent unit cells. The vector <span class="math inline">\(\vec{r}\)</span> is exactly what we use to encode the topology of our lattices.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/methods/bloch.png"
|
||||
data-short-caption="Bloch's Theorem and the Torus"
|
||||
style="width:100.0%"
|
||||
alt="Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors e^{i\vec{k}\cdot\vec{r}} associated with bonds that cross unit cells that depend on the sense in which they do so \vec{r} = (\pm1, \pm1). Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries." />
|
||||
<figcaption aria-hidden="true"><span>Figure 2:</span> Bloch’s theorem
|
||||
can be thought of as transforming from a periodic Hamiltonian on the
|
||||
place to the unit cell defined an torus. In addition we get some phase
|
||||
factors <span class="math inline">\(e^{i\vec{k}\cdot\vec{r}}\)</span>
|
||||
associated with bonds that cross unit cells that depend on the sense in
|
||||
which they do so <span class="math inline">\(\vec{r} = (\pm1,
|
||||
\pm1)\)</span>. Representing graphs on the torus turns out to require a
|
||||
similar idea, we unwrap the torus to one unit cell and keep track of
|
||||
which bonds cross the cell boundaries.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/methods/bloch.png" id="fig:bloch" data-short-caption="Bloch's Theorem and the Torus" style="width:100.0%" alt="Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors e^{i\vec{k}\cdot\vec{r}} associated with bonds that cross unit cells that depend on the sense in which they do so \vec{r} = (\pm1, \pm1). Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries." />
|
||||
<figcaption aria-hidden="true">Figure 2: Bloch’s theorem can be thought of as transforming from a periodic Hamiltonian on the place to the unit cell defined an torus. In addition we get some phase factors <span class="math inline">\(e^{i\vec{k}\cdot\vec{r}}\)</span> associated with bonds that cross unit cells that depend on the sense in which they do so <span class="math inline">\(\vec{r} = (\pm1, \pm1)\)</span>. Representing graphs on the torus turns out to require a similar idea, we unwrap the torus to one unit cell and keep track of which bonds cross the cell boundaries.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
</section>
|
||||
<section id="colouring-the-bonds" class="level2">
|
||||
<h2>Colouring the Bonds</h2>
|
||||
<p>The Kitaev Model requires that each edge in the lattice be assigned a
|
||||
label <span class="math inline">\(x\)</span>, <span
|
||||
class="math inline">\(y\)</span> or <span
|
||||
class="math inline">\(z\)</span>, such that each vertex has exactly one
|
||||
edge of each type connected to it. Let <span
|
||||
class="math inline">\(\Delta\)</span> be the maximum degree of a graph
|
||||
which, in our case, is 3. If <span class="math inline">\(\Delta >
|
||||
3\)</span>, it is obviously not possible to three-colour the edges.
|
||||
However, the general theory of when this is and is not possible for
|
||||
graphs with <span class="math inline">\(\Delta \leq 3\)</span> is more
|
||||
subtle.</p>
|
||||
<p>In the graph theory literature, graphs where all vertices have degree
|
||||
three are commonly called cubic graphs. There is no term for graphs with
|
||||
maximum degree three. Planar graphs are graphs which can be embedded
|
||||
onto the plane without any edges crossing. Bridgeless graphs do not
|
||||
contain any edges that, when removed, would partition the graph into
|
||||
disconnected components.</p>
|
||||
<p>This problem must be distinguished from that considered by the famous
|
||||
four-colour theorem <span class="citation"
|
||||
data-cites="appelEveryPlanarMap1989"> [<a
|
||||
href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">7</a>]</span>.
|
||||
The 4-colour theorem is concerned with assigning colours to the
|
||||
<strong>vertices</strong> of a graph, such that no vertices that share
|
||||
an edge have the same colour. Here we are concerned with an edge
|
||||
colouring.</p>
|
||||
<p>The four-colour theorem applies to planar graphs, those that can be
|
||||
embedded onto the plane without any edges crossing. Here we are
|
||||
concerned with Toroidal graphs, which can be embedded onto the torus
|
||||
without any edges crossing. In fact, toroidal graphs require up to seven
|
||||
colours <span class="citation"
|
||||
data-cites="heawoodMapColouringTheorems"> [<a
|
||||
href="#ref-heawoodMapColouringTheorems"
|
||||
role="doc-biblioref">8</a>]</span>. The complete graph <span
|
||||
class="math inline">\(K_7\)</span> is a good example of a toroidal graph
|
||||
that requires seven colours.</p>
|
||||
<p><span class="math inline">\(\Delta + 1\)</span> colours are enough to
|
||||
edge-colour any graph. An <span
|
||||
class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it
|
||||
for a graph with <span class="math inline">\(m\)</span> edges and <span
|
||||
class="math inline">\(n\)</span> vertices <span class="citation"
|
||||
data-cites="gEstimateChromaticClass1964"> [<a
|
||||
href="#ref-gEstimateChromaticClass1964"
|
||||
role="doc-biblioref">9</a>]</span>. Restricting ourselves to graphs with
|
||||
<span class="math inline">\(\Delta = 3\)</span> like ours, those can be
|
||||
four-edge-coloured in linear time <span class="citation"
|
||||
data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a
|
||||
href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002"
|
||||
role="doc-biblioref">10</a>]</span>.</p>
|
||||
<p>However, three-edge-colouring them is more difficult. Cubic, planar,
|
||||
bridgeless graphs can be three-edge-coloured if and only if they can be
|
||||
four-face-coloured <span class="citation"
|
||||
data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks"
|
||||
role="doc-biblioref">11</a>]</span>. An <span
|
||||
class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists
|
||||
here <span class="citation" data-cites="robertson1996efficiently"> [<a
|
||||
href="#ref-robertson1996efficiently"
|
||||
role="doc-biblioref">12</a>]</span>. However, it is not clear whether
|
||||
this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
|
||||
<div id="fig:multiple_colourings" class="fignos">
|
||||
<p>The Kitaev Model requires that each edge in the lattice be assigned a label <span class="math inline">\(x\)</span>, <span class="math inline">\(y\)</span> or <span class="math inline">\(z\)</span>, such that each vertex has exactly one edge of each type connected to it. Let <span class="math inline">\(\Delta\)</span> be the maximum degree of a graph which, in our case, is 3. If <span class="math inline">\(\Delta > 3\)</span>, it is obviously not possible to three-colour the edges. However, the general theory of when this is and is not possible for graphs with <span class="math inline">\(\Delta \leq 3\)</span> is more subtle.</p>
|
||||
<p>In the graph theory literature, graphs where all vertices have degree three are commonly called cubic graphs. There is no term for graphs with maximum degree three. Planar graphs are graphs which can be embedded onto the plane without any edges crossing. Bridgeless graphs do not contain any edges that, when removed, would partition the graph into disconnected components.</p>
|
||||
<p>This problem must be distinguished from that considered by the famous four-colour theorem <span class="citation" data-cites="appelEveryPlanarMap1989"> [<a href="#ref-appelEveryPlanarMap1989" role="doc-biblioref">8</a>]</span>. The 4-colour theorem is concerned with assigning colours to the <strong>vertices</strong> of a graph, such that no vertices that share an edge have the same colour. Here we are concerned with an edge colouring.</p>
|
||||
<p>The four-colour theorem applies to planar graphs, those that can be embedded onto the plane without any edges crossing. Here we are concerned with Toroidal graphs, which can be embedded onto the torus without any edges crossing. In fact, toroidal graphs require up to seven colours <span class="citation" data-cites="heawoodMapColouringTheorems"> [<a href="#ref-heawoodMapColouringTheorems" role="doc-biblioref">9</a>]</span>. The complete graph <span class="math inline">\(K_7\)</span> is a good example of a toroidal graph that requires seven colours.</p>
|
||||
<p><span class="math inline">\(\Delta + 1\)</span> colours are enough to edge-colour any graph. An <span class="math inline">\(\mathcal{O}(mn)\)</span> algorithm exists to do it for a graph with <span class="math inline">\(m\)</span> edges and <span class="math inline">\(n\)</span> vertices <span class="citation" data-cites="gEstimateChromaticClass1964"> [<a href="#ref-gEstimateChromaticClass1964" role="doc-biblioref">10</a>]</span>. Restricting ourselves to graphs with <span class="math inline">\(\Delta = 3\)</span> like ours, those can be four-edge-coloured in linear time <span class="citation" data-cites="skulrattanakulchai4edgecoloringGraphsMaximum2002"> [<a href="#ref-skulrattanakulchai4edgecoloringGraphsMaximum2002" role="doc-biblioref">11</a>]</span>.</p>
|
||||
<p>However, three-edge-colouring them is more difficult. Cubic, planar, bridgeless graphs can be three-edge-coloured if and only if they can be four-face-coloured <span class="citation" data-cites="tait1880remarks"> [<a href="#ref-tait1880remarks" role="doc-biblioref">12</a>]</span>. An <span class="math inline">\(\mathcal{O}(n^2)\)</span> algorithm exists here <span class="citation" data-cites="robertson1996efficiently"> [<a href="#ref-robertson1996efficiently" role="doc-biblioref">13</a>]</span>. However, it is not clear whether this extends to cubic, <strong>toroidal</strong> bridgeless graphs.</p>
|
||||
<figure>
|
||||
<img
|
||||
src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg"
|
||||
data-short-caption="Colourings of an Amorphous Lattice"
|
||||
style="width:100.0%"
|
||||
alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> Three different
|
||||
valid 3-edge-colourings of amorphous lattices. Colors that differ from
|
||||
the leftmost panel are highlighted.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/multiple_colourings/multiple_colourings.svg" id="fig:multiple_colourings" data-short-caption="Colourings of an Amorphous Lattice" style="width:100.0%" alt="Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted." />
|
||||
<figcaption aria-hidden="true">Figure 3: Three different valid 3-edge-colourings of amorphous lattices. Colors that differ from the leftmost panel are highlighted.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<section id="four-colourings-and-three-colourings" class="level3">
|
||||
<h3>Four-colourings and three-colourings</h3>
|
||||
<p><strong>add diagram of this</strong></p>
|
||||
<p>A four-face-colouring can be converted into a three-edge-colouring
|
||||
quite easily: 1. Assume the faces of G can be four-coloured with labels
|
||||
(0,1,2,3) 2. Label each edge of G according to <span
|
||||
class="math inline">\(i + j \;\textrm{mod}\; 3\)</span> where i and j
|
||||
are the labels of the face adjacent to that edge. For each edge label
|
||||
there are two face label pairs that do not share any face labels. i,e
|
||||
the edge label <span class="math inline">\(0\)</span> can come about
|
||||
either from faces <span class="math inline">\(0 + 3\)</span> or <span
|
||||
class="math inline">\(1 + 2\)</span>.</p>
|
||||
<p>A four-face-colouring can be converted into a three-edge-colouring quite easily: 1. Assume the faces of G can be four-coloured with labels (0,1,2,3) 2. Label each edge of G according to <span class="math inline">\(i + j \;\textrm{mod}\; 3\)</span> where i and j are the labels of the face adjacent to that edge. For each edge label there are two face label pairs that do not share any face labels. i,e the edge label <span class="math inline">\(0\)</span> can come about either from faces <span class="math inline">\(0 + 3\)</span> or <span class="math inline">\(1 + 2\)</span>.</p>
|
||||
<p>Explicitly, the mapping from face labels to edge labels is:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
0 + 3 \;\mathrm{or}\; 1 + 2 &= 0 \;\mathrm{mod}\; 3\\
|
||||
@ -316,377 +129,131 @@ class="math inline">\(1 + 2\)</span>.</p>
|
||||
\end{aligned}
|
||||
\]</span></p>
|
||||
<ol start="3" type="1">
|
||||
<li><p>In a cubic planar G, a vertex v in G is always part of three
|
||||
faces and the colours of those faces determine the colours of the edges
|
||||
that connect to v. The three faces must take three distinct colours from
|
||||
the set <span class="math inline">\(\{0,1,2,3\}\)</span>.</p></li>
|
||||
<li><p>From there, one can easily be convinced that those three distinct
|
||||
face colours can never produce repeated edge colours according to the
|
||||
<span class="math inline">\(i+j \;\mathrm{mod}\; 3\)</span>
|
||||
rule.</p></li>
|
||||
<li><p>In a cubic planar G, a vertex v in G is always part of three faces and the colours of those faces determine the colours of the edges that connect to v. The three faces must take three distinct colours from the set <span class="math inline">\(\{0,1,2,3\}\)</span>.</p></li>
|
||||
<li><p>From there, one can easily be convinced that those three distinct face colours can never produce repeated edge colours according to the <span class="math inline">\(i+j \;\mathrm{mod}\; 3\)</span> rule.</p></li>
|
||||
</ol>
|
||||
<p>This implies that all cubic planar graphs are three-edge-colourable.
|
||||
This does not apply to toroidal graphs. We have not yet generated a
|
||||
Voronoi lattices on the torus that is not three-edge-colourable. This
|
||||
suggests that Voronoi lattices may have additional structures that make
|
||||
them three-edge-colourable. Intuitively, it seems that the kinds of
|
||||
toroidal graphs that cannot be three-edge-coloured could never be
|
||||
generated by a Voronoi partition with more than a few seed points.</p>
|
||||
<p>This implies that all cubic planar graphs are three-edge-colourable. This does not apply to toroidal graphs. We have not yet generated a Voronoi lattices on the torus that is not three-edge-colourable. This suggests that Voronoi lattices may have additional structures that make them three-edge-colourable. Intuitively, it seems that the kinds of toroidal graphs that cannot be three-edge-coloured could never be generated by a Voronoi partition with more than a few seed points.</p>
|
||||
</section>
|
||||
<section id="finding-lattice-colourings-with-minisat" class="level3">
|
||||
<h3>Finding Lattice colourings with miniSAT</h3>
|
||||
<p>Some issues are harder in theory than in practice.
|
||||
Three-edge-colouring cubic toroidal graphs appears to be one of those
|
||||
things.</p>
|
||||
<p>To find colourings, we use a Boolean Satisfiability Solver or SAT
|
||||
solver. A SAT problem is a set of statements about some number of
|
||||
boolean variables , such as “<span class="math inline">\(x_1\)</span> or
|
||||
not <span class="math inline">\(x_3\)</span> is true”, and looks for an
|
||||
assignment <span class="math inline">\(x_i \in {0,1}\)</span> that
|
||||
satisfies all the statements <span class="citation"
|
||||
data-cites="Karp1972"> [<a href="#ref-Karp1972"
|
||||
role="doc-biblioref">13</a>]</span>.</p>
|
||||
<p>General purpose, high performance programs for solving SAT problems
|
||||
have been an area of active research for decades <span class="citation"
|
||||
data-cites="alounehComprehensiveStudyAnalysis2019"> [<a
|
||||
href="#ref-alounehComprehensiveStudyAnalysis2019"
|
||||
role="doc-biblioref">14</a>]</span>. Such programs are useful because,
|
||||
by the Cook-Levin theorem, any NP problem can be encoded in polynomial
|
||||
time as an instance of a SAT problem . This property is what makes SAT
|
||||
one of the subset of NP problems called NP-Complete <span
|
||||
class="citation"
|
||||
data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a
|
||||
href="#ref-cookComplexityTheoremprovingProcedures1971"
|
||||
role="doc-biblioref">15</a>,<a href="#ref-levin1973universal"
|
||||
role="doc-biblioref">16</a>]</span>.</p>
|
||||
<p>Thus, it is a relatively standard technique in the computer science
|
||||
community to solve NP problems by first transforming them to SAT
|
||||
instances and then using an off the shelf SAT solver. The output of this
|
||||
can then be mapped back to the original problem domain.</p>
|
||||
<p>NP problems can be loosely considered as those which do not have a
|
||||
special structure than can be exploited to compute their solution in
|
||||
polynomial time. Our three-edge-colouring problem is likely not in NP.
|
||||
However, since we do not know what special structure it might have that
|
||||
could be used to speed up its solution, using a SAT solver appears to be
|
||||
a reasonable first method to try. As will be discussed later, this
|
||||
turned out to work well enough and looking for a better solution was not
|
||||
necessary.</p>
|
||||
<p>We use a solver called <code>MiniSAT</code> <span class="citation"
|
||||
data-cites="imms-sat18"> [<a href="#ref-imms-sat18"
|
||||
role="doc-biblioref">17</a>]</span>. Like most modern SAT solvers,
|
||||
<code>MiniSAT</code> requires the input problem to be specified in
|
||||
Conjunctive Normal Form (CNF). CNF requires that the constraints be
|
||||
encoded as a set of <em>clauses</em> of the form <span
|
||||
class="math display">\[x_1 \;\textrm{or}\; -x_3 \;\textrm{or}\;
|
||||
x_5\]</span> that contain logical ORs of some subset of the variables
|
||||
where any of the variables may also be logically NOT’d, which we
|
||||
represent by negation here.</p>
|
||||
<p>A solution of the problem is one that makes all the clauses
|
||||
simultaneously true.</p>
|
||||
<p>We encode the edge colouring problem by assigning <span
|
||||
class="math inline">\(3B\)</span> boolean variables to each of the <span
|
||||
class="math inline">\(B\)</span> edges of the graph, <span
|
||||
class="math inline">\(x_{i\alpha}\)</span> where <span
|
||||
class="math inline">\(x_{i\alpha} = 1\)</span> indicates that edge <span
|
||||
class="math inline">\(i\)</span> has colour <span
|
||||
class="math inline">\(\alpha\)</span>.</p>
|
||||
<p>For edge colouring graphs we need two types of constraints: 1. Each
|
||||
edge is exactly one colour. 2. No neighbouring edges are the same
|
||||
colour.</p>
|
||||
<p>The first constraint is a product of doing this mapping to boolean
|
||||
variables. The solver does not know anything about the structure of the
|
||||
problem unless it is encoded into the variables.</p>
|
||||
<p>Let’s say we have three variables that correspond to particular edge
|
||||
being red <span class="math inline">\(r\)</span>, green <span
|
||||
class="math inline">\(g\)</span> or blue <span
|
||||
class="math inline">\(b\)</span>.</p>
|
||||
<p>To require that exactly one of the variables be true, we can enforce
|
||||
that no pair of variables be true:
|
||||
<code>-(r and b) -(r and g) -(b and g)</code></p>
|
||||
<p>However, these clauses are not in CNF form. Therefore, we also have
|
||||
to use the fact that <code>-(a and b) = (-a OR -b)</code>. To enforce
|
||||
that at least one of these is true we simply OR them all together
|
||||
<code>(r or b or g)</code></p>
|
||||
<p>To encode the fact that no adjacent edges can have the same colour,
|
||||
we emit a clause that, for each pair of adjacent edges, they cannot be
|
||||
both red, both green or both blue.</p>
|
||||
<p>We get a solution or set of solutions from the solver, which we can
|
||||
map back to a labelling of the edges. fig. <a
|
||||
href="#fig:multiple_colourings">3</a> shows some examples.</p>
|
||||
<p>The solution presented here works well enough for our purposes. It
|
||||
does not take up a substantial fraction of the overall computation time,
|
||||
see +fig:times but other approaches could likely work.</p>
|
||||
<p>When translating problems to CNF form, there is often some
|
||||
flexibility. For instance, we used three boolean variables to encode the
|
||||
colour of each edge and, then, additional constraints to require that
|
||||
only one of these variables be true. An alternative method which we did
|
||||
not try would be to encode the label of each edge using two variables,
|
||||
yielding four states per edge, and then add a constraint that one of the
|
||||
states, say (true, true) is disallowed. This would, however, have added
|
||||
some complexity to the encoding of the constraint that no adjacent edges
|
||||
can have the same colour.</p>
|
||||
<p>The popular <em>Networkx</em> Python library uses a greedy graph
|
||||
colouring algorithm. It simply iterates over the vertices/edges/faces of
|
||||
a graph and assigns them a colour that is not already disallowed. This
|
||||
does not work for our purposes because it is not designed to look for a
|
||||
particular n-colouring. However, it does include the option of using a
|
||||
heuristic function that determine the order in which vertices will be
|
||||
coloured <span class="citation"
|
||||
data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a
|
||||
href="#ref-kosowski2004classical" role="doc-biblioref">18</a>,<a
|
||||
href="#ref-matulaSmallestlastOrderingClustering1983"
|
||||
role="doc-biblioref">19</a>]</span>. Perhaps</p>
|
||||
<div id="fig:times" class="fignos">
|
||||
<p>Some issues are harder in theory than in practice. Three-edge-colouring cubic toroidal graphs appears to be one of those things.</p>
|
||||
<p>To find colourings, we use a Boolean Satisfiability Solver or SAT solver. A SAT problem is a set of statements about some number of boolean variables , such as “<span class="math inline">\(x_1\)</span> or not <span class="math inline">\(x_3\)</span> is true”, and looks for an assignment <span class="math inline">\(x_i \in {0,1}\)</span> that satisfies all the statements <span class="citation" data-cites="Karp1972"> [<a href="#ref-Karp1972" role="doc-biblioref">14</a>]</span>.</p>
|
||||
<p>General purpose, high performance programs for solving SAT problems have been an area of active research for decades <span class="citation" data-cites="alounehComprehensiveStudyAnalysis2019"> [<a href="#ref-alounehComprehensiveStudyAnalysis2019" role="doc-biblioref">15</a>]</span>. Such programs are useful because, by the Cook-Levin theorem, any NP problem can be encoded in polynomial time as an instance of a SAT problem . This property is what makes SAT one of the subset of NP problems called NP-Complete <span class="citation" data-cites="cookComplexityTheoremprovingProcedures1971 levin1973universal"> [<a href="#ref-cookComplexityTheoremprovingProcedures1971" role="doc-biblioref">16</a>,<a href="#ref-levin1973universal" role="doc-biblioref">17</a>]</span>.</p>
|
||||
<p>Thus, it is a relatively standard technique in the computer science community to solve NP problems by first transforming them to SAT instances and then using an off the shelf SAT solver. The output of this can then be mapped back to the original problem domain.</p>
|
||||
<p>NP problems can be loosely considered as those which do not have a special structure than can be exploited to compute their solution in polynomial time. Our three-edge-colouring problem is likely not in NP. However, since we do not know what special structure it might have that could be used to speed up its solution, using a SAT solver appears to be a reasonable first method to try. As will be discussed later, this turned out to work well enough and looking for a better solution was not necessary.</p>
|
||||
<p>We use a solver called <code>MiniSAT</code> <span class="citation" data-cites="imms-sat18"> [<a href="#ref-imms-sat18" role="doc-biblioref">18</a>]</span>. Like most modern SAT solvers, <code>MiniSAT</code> requires the input problem to be specified in Conjunctive Normal Form (CNF). CNF requires that the constraints be encoded as a set of <em>clauses</em> of the form <span class="math display">\[x_1 \;\textrm{or}\; -x_3 \;\textrm{or}\; x_5\]</span> that contain logical ORs of some subset of the variables where any of the variables may also be logically NOT’d, which we represent by negation here.</p>
|
||||
<p>A solution of the problem is one that makes all the clauses simultaneously true.</p>
|
||||
<p>We encode the edge colouring problem by assigning <span class="math inline">\(3B\)</span> boolean variables to each of the <span class="math inline">\(B\)</span> edges of the graph, <span class="math inline">\(x_{i\alpha}\)</span> where <span class="math inline">\(x_{i\alpha} = 1\)</span> indicates that edge <span class="math inline">\(i\)</span> has colour <span class="math inline">\(\alpha\)</span>.</p>
|
||||
<p>For edge colouring graphs we need two types of constraints: 1. Each edge is exactly one colour. 2. No neighbouring edges are the same colour.</p>
|
||||
<p>The first constraint is a product of doing this mapping to boolean variables. The solver does not know anything about the structure of the problem unless it is encoded into the variables.</p>
|
||||
<p>Let’s say we have three variables that correspond to particular edge being red <span class="math inline">\(r\)</span>, green <span class="math inline">\(g\)</span> or blue <span class="math inline">\(b\)</span>.</p>
|
||||
<p>To require that exactly one of the variables be true, we can enforce that no pair of variables be true: <code>-(r and b) -(r and g) -(b and g)</code></p>
|
||||
<p>However, these clauses are not in CNF form. Therefore, we also have to use the fact that <code>-(a and b) = (-a OR -b)</code>. To enforce that at least one of these is true we simply OR them all together <code>(r or b or g)</code></p>
|
||||
<p>To encode the fact that no adjacent edges can have the same colour, we emit a clause that, for each pair of adjacent edges, they cannot be both red, both green or both blue.</p>
|
||||
<p>We get a solution or set of solutions from the solver, which we can map back to a labelling of the edges. fig. <a href="#fig:multiple_colourings">3</a> shows some examples.</p>
|
||||
<p>The solution presented here works well enough for our purposes. It does not take up a substantial fraction of the overall computation time, see +fig:times but other approaches could likely work.</p>
|
||||
<p>When translating problems to CNF form, there is often some flexibility. For instance, we used three boolean variables to encode the colour of each edge and, then, additional constraints to require that only one of these variables be true. An alternative method which we did not try would be to encode the label of each edge using two variables, yielding four states per edge, and then add a constraint that one of the states, say (true, true) is disallowed. This would, however, have added some complexity to the encoding of the constraint that no adjacent edges can have the same colour.</p>
|
||||
<p>The popular <em>Networkx</em> Python library uses a greedy graph colouring algorithm. It simply iterates over the vertices/edges/faces of a graph and assigns them a colour that is not already disallowed. This does not work for our purposes because it is not designed to look for a particular n-colouring. However, it does include the option of using a heuristic function that determine the order in which vertices will be coloured <span class="citation" data-cites="kosowski2004classical matulaSmallestlastOrderingClustering1983"> [<a href="#ref-kosowski2004classical" role="doc-biblioref">19</a>,<a href="#ref-matulaSmallestlastOrderingClustering1983" role="doc-biblioref">20</a>]</span>. Perhaps</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/methods/times/times.svg"
|
||||
data-short-caption="Computation Time Spent on Different Procedures."
|
||||
style="width:100.0%"
|
||||
alt="Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates." />
|
||||
<figcaption aria-hidden="true"><span>Figure 4:</span> The proportion of
|
||||
computation time taken up by the four longest running steps when
|
||||
generating a lattice. For larger systems, the time taken to perform the
|
||||
diagonalisation dominates.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/methods/times/times.svg" id="fig:times" data-short-caption="Computation Time Spent on Different Procedures." style="width:100.0%" alt="Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates." />
|
||||
<figcaption aria-hidden="true">Figure 4: The proportion of computation time taken up by the four longest running steps when generating a lattice. For larger systems, the time taken to perform the diagonalisation dominates.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
</section>
|
||||
<section id="does-it-matter-which-colouring-we-choose" class="level3">
|
||||
<h3>Does it matter which colouring we choose?</h3>
|
||||
<p>In the isotropic case <span class="math inline">\(J^\alpha =
|
||||
1\)</span>, it is easy to show that choosing a particular valid
|
||||
colouring cannot make a difference. As the choice of how we define the
|
||||
four Majoranas at a site is arbitrary, we can define a local operator
|
||||
that transforms the colouring of any particular site to another
|
||||
permutation. The operators commute with the Hamiltonian and, by
|
||||
composing such operators, we can transform the Hamiltonian generated by
|
||||
one colouring into that generated by another.</p>
|
||||
<p>We cannot do this in the anisotropic case. It remains an open
|
||||
question whether particular physical properties could arise by
|
||||
engineering the colouring in this phase though we expect them to exhibit
|
||||
a self averaging behaviour.</p>
|
||||
<p>In the isotropic case <span class="math inline">\(J^\alpha = 1\)</span>, it is easy to show that choosing a particular valid colouring cannot make a difference. As the choice of how we define the four Majoranas at a site is arbitrary, we can define a local operator that transforms the colouring of any particular site to another permutation. The operators commute with the Hamiltonian and, by composing such operators, we can transform the Hamiltonian generated by one colouring into that generated by another.</p>
|
||||
<p>We cannot do this in the anisotropic case. It remains an open question whether particular physical properties could arise by engineering the colouring in this phase though we expect them to exhibit a self averaging behaviour.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="mapping-between-flux-sectors-and-bond-sectors"
|
||||
class="level2">
|
||||
<section id="mapping-between-flux-sectors-and-bond-sectors" class="level2">
|
||||
<h2>Mapping between flux sectors and bond sectors</h2>
|
||||
<p>Constructing the Majorana representation of the model requires the
|
||||
particular bond configuration <span class="math inline">\(u_{jk} = \pm
|
||||
1\)</span>. However, the large number of gauge symmetries of the bond
|
||||
sector makes it unwieldy to work with. Therefore, we need a way to
|
||||
quickly map between bond sectors and flux sectors.</p>
|
||||
<p>Going from the bond sector to flux sector is easy. We can compute it
|
||||
directly by taking the product of <span class="math inline">\(i
|
||||
u_{jk}\)</span> around each plaquette <span class="math display">\[
|
||||
\phi_i = \prod_{(j,k) \; \in \; \partial \phi_i} i u_{jk}\]</span></p>
|
||||
<p>Going from flux sector to bond sector requires more thought. The
|
||||
algorithm we use is this:</p>
|
||||
<p>Constructing the Majorana representation of the model requires the particular bond configuration <span class="math inline">\(u_{jk} = \pm 1\)</span>. However, the large number of gauge symmetries of the bond sector makes it unwieldy to work with. Therefore, we need a way to quickly map between bond sectors and flux sectors.</p>
|
||||
<p>Going from the bond sector to flux sector is easy. We can compute it directly by taking the product of <span class="math inline">\(i u_{jk}\)</span> around each plaquette <span class="math display">\[ \phi_i = \prod_{(j,k) \; \in \; \partial \phi_i} i u_{jk}\]</span></p>
|
||||
<p>Going from flux sector to bond sector requires more thought. The algorithm we use is this:</p>
|
||||
<ol type="1">
|
||||
<li><p>Fix the gauge by choosing some arbitrary <span
|
||||
class="math inline">\(u_{jk}\)</span> configuration. In practice, we use
|
||||
<span class="math inline">\(u_{jk} = +1\)</span>. This chooses an
|
||||
arbitrary one of the four topological sectors.</p></li>
|
||||
<li><p>Compute the current flux configuration and how it differs from
|
||||
the target one. We refer to a plaquette that differs from the target as
|
||||
a “defect”.</p></li>
|
||||
<li><p>Find any adjacent pairs of defects and flip the <span
|
||||
class="math inline">\(u_jk\)</span> between them. This leaves a set of
|
||||
isolated defects.</p></li>
|
||||
<li><p>Fix the gauge by choosing some arbitrary <span class="math inline">\(u_{jk}\)</span> configuration. In practice, we use <span class="math inline">\(u_{jk} = +1\)</span>. This chooses an arbitrary one of the four topological sectors.</p></li>
|
||||
<li><p>Compute the current flux configuration and how it differs from the target one. We refer to a plaquette that differs from the target as a “defect”.</p></li>
|
||||
<li><p>Find any adjacent pairs of defects and flip the <span class="math inline">\(u_jk\)</span> between them. This leaves a set of isolated defects.</p></li>
|
||||
<li><p>Pair the defects up using a greedy algorithm.</p></li>
|
||||
<li><p>Compute paths along the dual lattice between each pair of
|
||||
plaquettes. Flipping the corresponding set of bonds transports one flux
|
||||
to the other and annihilates them.</p></li>
|
||||
<li><p>Compute paths along the dual lattice between each pair of plaquettes. Flipping the corresponding set of bonds transports one flux to the other and annihilates them.</p></li>
|
||||
</ol>
|
||||
<div id="fig:flux_finding" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/flux_finding/flux_finding.svg"
|
||||
data-short-caption="Finding Bond Sectors from Flux Sectors"
|
||||
style="width:100.0%"
|
||||
alt="Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which u_{jk} = +1. Plaquettes are coloured blue when \hat{\phi}_i = -1 (-i) for even/odd plaquettes and orange when \hat{\phi}_i = +1 (+i) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all +1/+i), we first flip any u_{jk} that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding u_{jk} (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes." />
|
||||
<figcaption aria-hidden="true"><span>Figure 5:</span> (Left) The ground
|
||||
state flux sector and bond sector for an amorphous lattice. Bond arrows
|
||||
indicate the direction in which <span class="math inline">\(u_{jk} =
|
||||
+1\)</span>. Plaquettes are coloured blue when <span
|
||||
class="math inline">\(\hat{\phi}_i = -1\)</span> (<span
|
||||
class="math inline">\(-i\)</span>) for even/odd plaquettes and orange
|
||||
when <span class="math inline">\(\hat{\phi}_i = +1\)</span> (<span
|
||||
class="math inline">\(+i\)</span>) for even/odd plaquettes. (Centre) To
|
||||
transform this to the target flux sector (all <span
|
||||
class="math inline">\(+1\)</span>/<span
|
||||
class="math inline">\(+i\)</span>), we first flip any <span
|
||||
class="math inline">\(u_{jk}\)</span> that are between two fluxes. This
|
||||
leaves a set of isolated fluxes that must be annihilated. Then, these
|
||||
are paired up as indicated by the black lines. (Right) A* search is used
|
||||
to find paths (coloured plaquettes) on the dual lattice between each
|
||||
pair of fluxes and the corresponding <span
|
||||
class="math inline">\(u_{jk}\)</span> (shown in black) are flipped. One
|
||||
flux will remain because the starting and target flux sectors differed
|
||||
by an odd number of fluxes.</figcaption>
|
||||
<img src="/assets/thesis/amk_chapter/flux_finding/flux_finding.svg" id="fig:flux_finding" data-short-caption="Finding Bond Sectors from Flux Sectors" style="width:100.0%" alt="Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which u_{jk} = +1. Plaquettes are coloured blue when \hat{\phi}_i = -1 (-i) for even/odd plaquettes and orange when \hat{\phi}_i = +1 (+i) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all +1/+i), we first flip any u_{jk} that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding u_{jk} (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes." />
|
||||
<figcaption aria-hidden="true">Figure 5: (Left) The ground state flux sector and bond sector for an amorphous lattice. Bond arrows indicate the direction in which <span class="math inline">\(u_{jk} = +1\)</span>. Plaquettes are coloured blue when <span class="math inline">\(\hat{\phi}_i = -1\)</span> (<span class="math inline">\(-i\)</span>) for even/odd plaquettes and orange when <span class="math inline">\(\hat{\phi}_i = +1\)</span> (<span class="math inline">\(+i\)</span>) for even/odd plaquettes. (Centre) To transform this to the target flux sector (all <span class="math inline">\(+1\)</span>/<span class="math inline">\(+i\)</span>), we first flip any <span class="math inline">\(u_{jk}\)</span> that are between two fluxes. This leaves a set of isolated fluxes that must be annihilated. Then, these are paired up as indicated by the black lines. (Right) A* search is used to find paths (coloured plaquettes) on the dual lattice between each pair of fluxes and the corresponding <span class="math inline">\(u_{jk}\)</span> (shown in black) are flipped. One flux will remain because the starting and target flux sectors differed by an odd number of fluxes.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
</section>
|
||||
<section id="chern-markers" class="level2">
|
||||
<h2>Chern Markers</h2>
|
||||
<p>We know that the standard Kitaev model supports both Abelian and
|
||||
non-Abelian phases. Therefore, how can we assess whether this is also
|
||||
the case for the amorphous Kitaev model?</p>
|
||||
<p>We have already discussed the fact that topology and anyonic
|
||||
statistics are intimately linked. This will help here. The Chern number
|
||||
is a quantity that measures the topological characteristics of a
|
||||
material.</p>
|
||||
<p>The original definition of the Chern number relies on the model
|
||||
having translation symmetry. This leads to the development of <em>local
|
||||
markers</em>. These are operators that generalise the notion of the
|
||||
Chern number to an observable over some region smaller than the entire
|
||||
system.</p>
|
||||
<p>We know that the standard Kitaev model supports both Abelian and non-Abelian phases. Therefore, how can we assess whether this is also the case for the amorphous Kitaev model?</p>
|
||||
<p>We have already discussed the fact that topology and anyonic statistics are intimately linked. This will help here. The Chern number is a quantity that measures the topological characteristics of a material.</p>
|
||||
<p>The original definition of the Chern number relies on the model having translation symmetry. This leads to the development of <em>local markers</em>. These are operators that generalise the notion of the Chern number to an observable over some region smaller than the entire system.</p>
|
||||
<p><strong>Expand on definition here</strong></p>
|
||||
<p><strong>Discuss link between Chern number and Anyonic
|
||||
Statistics</strong></p>
|
||||
<p>Next Section: <a
|
||||
href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html">Results</a></p>
|
||||
<p><strong>Discuss link between Chern number and Anyonic Statistics</strong></p>
|
||||
<p>Next Section: <a href="../4_Amorphous_Kitaev_Model/4.3_AMK_Results.html">Results</a></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
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</div>
|
||||
</section>
|
||||
|
||||
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||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#material-realisations"
|
||||
id="toc-material-realisations">Material Realisations</a>
|
||||
<li><a href="#chap:5-conclusion" id="toc-chap:5-conclusion">5 Conclusion</a></li>
|
||||
<li><a href="#material-realisations" id="toc-material-realisations">Material Realisations</a>
|
||||
<ul>
|
||||
<li><a href="#amorphous-materials"
|
||||
id="toc-amorphous-materials">Amorphous Materials</a></li>
|
||||
<li><a href="#metal-organic-frameworks"
|
||||
id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
<li><a href="#amorphous-materials" id="toc-amorphous-materials">Amorphous Materials</a></li>
|
||||
<li><a href="#metal-organic-frameworks" id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
|
||||
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
|
||||
@ -49,13 +47,11 @@ id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#material-realisations"
|
||||
id="toc-material-realisations">Material Realisations</a>
|
||||
<li><a href="#chap:5-conclusion" id="toc-chap:5-conclusion">5 Conclusion</a></li>
|
||||
<li><a href="#material-realisations" id="toc-material-realisations">Material Realisations</a>
|
||||
<ul>
|
||||
<li><a href="#amorphous-materials"
|
||||
id="toc-amorphous-materials">Amorphous Materials</a></li>
|
||||
<li><a href="#metal-organic-frameworks"
|
||||
id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
<li><a href="#amorphous-materials" id="toc-amorphous-materials">Amorphous Materials</a></li>
|
||||
<li><a href="#metal-organic-frameworks" id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
|
||||
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
|
||||
@ -68,6 +64,9 @@ id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
<p>5 Conclusion</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="chap:5-conclusion" class="level1">
|
||||
<h1>5 Conclusion</h1>
|
||||
</section>
|
||||
<section id="material-realisations" class="level1">
|
||||
<h1>Material Realisations</h1>
|
||||
<section id="amorphous-materials" class="level2">
|
||||
@ -82,8 +81,7 @@ id="toc-metal-organic-frameworks">Metal Organic Frameworks</a></li>
|
||||
</section>
|
||||
<section id="outlook" class="level1">
|
||||
<h1>Outlook</h1>
|
||||
<p>Next Chapter: <a
|
||||
href="../6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></p>
|
||||
<p>Next Chapter: <a href="../6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></p>
|
||||
</section>
|
||||
|
||||
|
||||
|
||||
@ -27,9 +27,8 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#evaluation-of-the-fermion-free-energy"
|
||||
id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion
|
||||
Free Energy</a></li>
|
||||
<li><a href="#chap:appendices" id="toc-chap:appendices">Appendices</a></li>
|
||||
<li><a href="#evaluation-of-the-fermion-free-energy" id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion Free Energy</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
{% endcapture %}
|
||||
@ -42,9 +41,8 @@ Free Energy</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#evaluation-of-the-fermion-free-energy"
|
||||
id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion
|
||||
Free Energy</a></li>
|
||||
<li><a href="#chap:appendices" id="toc-chap:appendices">Appendices</a></li>
|
||||
<li><a href="#evaluation-of-the-fermion-free-energy" id="toc-evaluation-of-the-fermion-free-energy">Evaluation of the Fermion Free Energy</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -54,54 +52,33 @@ Free Energy</a></li>
|
||||
<p>Appendices</p>
|
||||
<hr />
|
||||
</div>
|
||||
<section id="chap:appendices" class="level1">
|
||||
<h1>Appendices</h1>
|
||||
</section>
|
||||
<section id="evaluation-of-the-fermion-free-energy" class="level1">
|
||||
<h1>Evaluation of the Fermion Free Energy</h1>
|
||||
<p>There are <span class="math inline">\(2^N\)</span> possible ion
|
||||
configurations <span class="math inline">\(\{ n_i \}\)</span>, we define
|
||||
<span class="math inline">\(n^k_i\)</span> to be the occupation of the
|
||||
ith site of the kth configuration. The quantum part of the free energy
|
||||
can then be defined through the quantum partition function <span
|
||||
class="math inline">\(\mathcal{Z}^k\)</span> associated with each ionic
|
||||
state <span class="math inline">\(n^k_i\)</span>: <span
|
||||
class="math display">\[\begin{aligned}
|
||||
<p>There are <span class="math inline">\(2^N\)</span> possible ion configurations <span class="math inline">\(\{ n_i \}\)</span>, we define <span class="math inline">\(n^k_i\)</span> to be the occupation of the ith site of the kth configuration. The quantum part of the free energy can then be defined through the quantum partition function <span class="math inline">\(\mathcal{Z}^k\)</span> associated with each ionic state <span class="math inline">\(n^k_i\)</span>: <span class="math display">\[\begin{aligned}
|
||||
F^k &= -1/\beta \ln{\mathcal{Z}^k} \\
|
||||
\end{aligned}\]</span> % Such that the overall partition function is:
|
||||
<span class="math display">\[\begin{aligned}
|
||||
\end{aligned}\]</span> % Such that the overall partition function is: <span class="math display">\[\begin{aligned}
|
||||
\mathcal{Z} &= \sum_k e^{- \beta H^k} Z^k \\
|
||||
&= \sum_k e^{-\beta (H^k + F^k)} \\
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Because fermions are limited to occupation numbers of 0 or 1 <span
|
||||
class="math inline">\(Z^k\)</span> simplifies nicely. If <span
|
||||
class="math inline">\(m^j_i = \{0,1\}\)</span> is defined as the
|
||||
occupation of the level with energy <span
|
||||
class="math inline">\(\epsilon^k_i\)</span> then the partition function
|
||||
is a sum over all the occupation states labelled by j: <span
|
||||
class="math display">\[\begin{aligned}
|
||||
Z^k &= \mathrm{Tr} e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i
|
||||
\epsilon^k_i}\\
|
||||
&= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i
|
||||
\sum_j e^{- \beta m^j_i \epsilon^k_i}\\
|
||||
<p>Because fermions are limited to occupation numbers of 0 or 1 <span class="math inline">\(Z^k\)</span> simplifies nicely. If <span class="math inline">\(m^j_i = \{0,1\}\)</span> is defined as the occupation of the level with energy <span class="math inline">\(\epsilon^k_i\)</span> then the partition function is a sum over all the occupation states labelled by j: <span class="math display">\[\begin{aligned}
|
||||
Z^k &= \mathrm{Tr} e^{-\beta F^k} = \sum_j e^{-\beta \sum_i m^j_i \epsilon^k_i}\\
|
||||
&= \sum_j \prod_i e^{- \beta m^j_i \epsilon^k_i}= \prod_i \sum_j e^{- \beta m^j_i \epsilon^k_i}\\
|
||||
&= \prod_i (1 + e^{- \beta \epsilon^k_i})\\
|
||||
F^k &= -1/\beta \sum_k \ln{(1 + e^{- \beta \epsilon^k_i})}
|
||||
\end{aligned}\]</span> % Observables can then be calculated from the
|
||||
partition function, for examples the occupation numbers:</p>
|
||||
\end{aligned}\]</span> % Observables can then be calculated from the partition function, for examples the occupation numbers:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
\langle N \rangle &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial
|
||||
Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\
|
||||
&= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu}
|
||||
\sum_k e^{-\beta (H^k + F^k)}\\
|
||||
&= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}})
|
||||
e^{-\beta (H^k + F^k)}\\
|
||||
\langle N \rangle &= \frac{1}{\beta} \frac{1}{Z} \frac{\partial Z}{\partial \mu} = - \frac{\partial F}{\partial \mu}\\
|
||||
&= \frac{1}{\beta} \frac{1}{Z} \frac{\partial}{\partial \mu} \sum_k e^{-\beta (H^k + F^k)}\\
|
||||
&= 1/Z \sum_k (N^k_{\mathrm{ion}} + N^k_{\mathrm{electron}}) e^{-\beta (H^k + F^k)}\\
|
||||
\end{aligned}\]</span> % with the definitions:</p>
|
||||
<p><span class="math display">\[\begin{aligned}
|
||||
N^k_{\mathrm{ion}} &= - \frac{\partial H^k}{\partial \mu} = \sum_i
|
||||
n^k_i\\
|
||||
N^k_{\mathrm{electron}} &= - \frac{\partial F^k}{\partial \mu} =
|
||||
\sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}\\
|
||||
N^k_{\mathrm{ion}} &= - \frac{\partial H^k}{\partial \mu} = \sum_i n^k_i\\
|
||||
N^k_{\mathrm{electron}} &= - \frac{\partial F^k}{\partial \mu} = \sum_i \left(1 + e^{\beta \epsilon^k_i}\right)^{-1}\\
|
||||
\end{aligned}\]</span></p>
|
||||
<p>Next Section: <a
|
||||
href="../6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html">Particle-Hole
|
||||
Symmetry</a></p>
|
||||
<p>Next Section: <a href="../6_Appendices/A.1_Particle_Hole_Symmetry.html">Particle-Hole Symmetry</a></p>
|
||||
</section>
|
||||
|
||||
|
||||
|
||||
@ -27,8 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#particle-hole-symmetry"
|
||||
id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -42,8 +41,7 @@ id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#particle-hole-symmetry"
|
||||
id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -56,75 +54,24 @@ id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
</div>
|
||||
<section id="particle-hole-symmetry" class="level1">
|
||||
<h1>Particle-Hole Symmetry</h1>
|
||||
<p>The Hubbard and FK models on a bipartite lattice have particle-hole
|
||||
(PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H
|
||||
\mathcal{P} = - H\)</span>, accordingly they have symmetric energy
|
||||
spectra. The associated symmetry operator <span
|
||||
class="math inline">\(\mathcal{P}\)</span> exchanges creation and
|
||||
annihilation operators along with a sign change between the two
|
||||
sublattices. In the language of the Hubbard model of electrons <span
|
||||
class="math inline">\(c_{\alpha,i}\)</span> with spin <span
|
||||
class="math inline">\(\alpha\)</span> at site <span
|
||||
class="math inline">\(i\)</span> the particle hole operator corresponds
|
||||
to the substitution of new fermion operators <span
|
||||
class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators
|
||||
<span class="math inline">\(m_{\alpha,i}\)</span> where</p>
|
||||
<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i
|
||||
c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} =
|
||||
d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
|
||||
<p>the lattices must be bipartite because to make this work we set <span
|
||||
class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
|
||||
<span class="math inline">\(-1\)</span> for the even sublattice <span
|
||||
class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
|
||||
href="#ref-gruberFalicovKimballModel2005"
|
||||
role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The entirely filled state <span class="math inline">\(\ket{\Omega} =
|
||||
\sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new
|
||||
vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} =
|
||||
(-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} =
|
||||
0.\]</span></p>
|
||||
<p>The number operator <span class="math inline">\(m_{\alpha,i} =
|
||||
0,1\)</span> counts holes rather than electrons <span
|
||||
class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i}
|
||||
= 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
|
||||
<p>With the last equality following from the fermionic commutation
|
||||
relations. In the case of nearest neighbour hopping on a bipartite
|
||||
lattice this transformation also leaves the hopping term unchanged
|
||||
because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span>
|
||||
when <span class="math inline">\(i\)</span> and <span
|
||||
class="math inline">\(j\)</span> are on different sublattices: <span
|
||||
class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i
|
||||
\epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i}
|
||||
c_{\alpha,j} \]</span></p>
|
||||
<p>Defining the particle density <span
|
||||
class="math inline">\(\rho\)</span> as the number of fermions per site:
|
||||
<span class="math display">\[
|
||||
\rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow}
|
||||
\right)
|
||||
<p>The Hubbard and FK models on a bipartite lattice have particle-hole (PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H \mathcal{P} = - H\)</span>, accordingly they have symmetric energy spectra. The associated symmetry operator <span class="math inline">\(\mathcal{P}\)</span> exchanges creation and annihilation operators along with a sign change between the two sublattices. In the language of the Hubbard model of electrons <span class="math inline">\(c_{\alpha,i}\)</span> with spin <span class="math inline">\(\alpha\)</span> at site <span class="math inline">\(i\)</span> the particle hole operator corresponds to the substitution of new fermion operators <span class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators <span class="math inline">\(m_{\alpha,i}\)</span> where</p>
|
||||
<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} = d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
|
||||
<p>the lattices must be bipartite because to make this work we set <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the even sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The entirely filled state <span class="math inline">\(\ket{\Omega} = \sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} = (-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} = 0.\]</span></p>
|
||||
<p>The number operator <span class="math inline">\(m_{\alpha,i} = 0,1\)</span> counts holes rather than electrons <span class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i} = 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
|
||||
<p>With the last equality following from the fermionic commutation relations. In the case of nearest neighbour hopping on a bipartite lattice this transformation also leaves the hopping term unchanged because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span> when <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> are on different sublattices: <span class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i \epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i} c_{\alpha,j} \]</span></p>
|
||||
<p>Defining the particle density <span class="math inline">\(\rho\)</span> as the number of fermions per site: <span class="math display">\[
|
||||
\rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right)
|
||||
\]</span></p>
|
||||
<p>The PH symmetry maps the Hamiltonian to itself with the sign of the
|
||||
chemical potential reversed and the density inverted about half filling:
|
||||
<span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t,
|
||||
U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 -
|
||||
\rho \]</span></p>
|
||||
<p>The Hamiltonian is symmetric under PH at <span
|
||||
class="math inline">\(\mu = 0\)</span> and so must all the observables,
|
||||
hence half filling <span class="math inline">\(\rho = 1\)</span> occurs
|
||||
here. This symmetry and known observable acts as a useful test for the
|
||||
numerical calculations.</p>
|
||||
<p>Next Section: <a
|
||||
href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Markov Chain
|
||||
Monte Carlo</a></p>
|
||||
<p>The PH symmetry maps the Hamiltonian to itself with the sign of the chemical potential reversed and the density inverted about half filling: <span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t, U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 - \rho \]</span></p>
|
||||
<p>The Hamiltonian is symmetric under PH at <span class="math inline">\(\mu = 0\)</span> and so must all the observables, hence half filling <span class="math inline">\(\rho = 1\)</span> occurs here. This symmetry and known observable acts as a useful test for the numerical calculations.</p>
|
||||
<p>Next Section: <a href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Markov Chain Monte Carlo</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C.
|
||||
Gruber and D. Ueltschi, <em><a
|
||||
href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball
|
||||
Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
|
||||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C. Gruber and D. Ueltschi, <em><a href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
@ -27,8 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#particle-hole-symmetry"
|
||||
id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -42,8 +41,7 @@ id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#particle-hole-symmetry"
|
||||
id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
@ -56,75 +54,24 @@ id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
</div>
|
||||
<section id="particle-hole-symmetry" class="level1">
|
||||
<h1>Particle-Hole Symmetry</h1>
|
||||
<p>The Hubbard and FK models on a bipartite lattice have particle-hole
|
||||
(PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H
|
||||
\mathcal{P} = - H\)</span>, accordingly they have symmetric energy
|
||||
spectra. The associated symmetry operator <span
|
||||
class="math inline">\(\mathcal{P}\)</span> exchanges creation and
|
||||
annihilation operators along with a sign change between the two
|
||||
sublattices. In the language of the Hubbard model of electrons <span
|
||||
class="math inline">\(c_{\alpha,i}\)</span> with spin <span
|
||||
class="math inline">\(\alpha\)</span> at site <span
|
||||
class="math inline">\(i\)</span> the particle hole operator corresponds
|
||||
to the substitution of new fermion operators <span
|
||||
class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators
|
||||
<span class="math inline">\(m_{\alpha,i}\)</span> where</p>
|
||||
<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i
|
||||
c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} =
|
||||
d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
|
||||
<p>the lattices must be bipartite because to make this work we set <span
|
||||
class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and
|
||||
<span class="math inline">\(-1\)</span> for the even sublattice <span
|
||||
class="citation" data-cites="gruberFalicovKimballModel2005"> [<a
|
||||
href="#ref-gruberFalicovKimballModel2005"
|
||||
role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The entirely filled state <span class="math inline">\(\ket{\Omega} =
|
||||
\sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new
|
||||
vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} =
|
||||
(-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} =
|
||||
0.\]</span></p>
|
||||
<p>The number operator <span class="math inline">\(m_{\alpha,i} =
|
||||
0,1\)</span> counts holes rather than electrons <span
|
||||
class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i}
|
||||
= 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
|
||||
<p>With the last equality following from the fermionic commutation
|
||||
relations. In the case of nearest neighbour hopping on a bipartite
|
||||
lattice this transformation also leaves the hopping term unchanged
|
||||
because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span>
|
||||
when <span class="math inline">\(i\)</span> and <span
|
||||
class="math inline">\(j\)</span> are on different sublattices: <span
|
||||
class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i
|
||||
\epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i}
|
||||
c_{\alpha,j} \]</span></p>
|
||||
<p>Defining the particle density <span
|
||||
class="math inline">\(\rho\)</span> as the number of fermions per site:
|
||||
<span class="math display">\[
|
||||
\rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow}
|
||||
\right)
|
||||
<p>The Hubbard and FK models on a bipartite lattice have particle-hole (PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H \mathcal{P} = - H\)</span>, accordingly they have symmetric energy spectra. The associated symmetry operator <span class="math inline">\(\mathcal{P}\)</span> exchanges creation and annihilation operators along with a sign change between the two sublattices. In the language of the Hubbard model of electrons <span class="math inline">\(c_{\alpha,i}\)</span> with spin <span class="math inline">\(\alpha\)</span> at site <span class="math inline">\(i\)</span> the particle hole operator corresponds to the substitution of new fermion operators <span class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators <span class="math inline">\(m_{\alpha,i}\)</span> where</p>
|
||||
<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} = d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
|
||||
<p>the lattices must be bipartite because to make this work we set <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the even sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">1</a>]</span>.</p>
|
||||
<p>The entirely filled state <span class="math inline">\(\ket{\Omega} = \sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} = (-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} = 0.\]</span></p>
|
||||
<p>The number operator <span class="math inline">\(m_{\alpha,i} = 0,1\)</span> counts holes rather than electrons <span class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i} = 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
|
||||
<p>With the last equality following from the fermionic commutation relations. In the case of nearest neighbour hopping on a bipartite lattice this transformation also leaves the hopping term unchanged because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span> when <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> are on different sublattices: <span class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i \epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i} c_{\alpha,j} \]</span></p>
|
||||
<p>Defining the particle density <span class="math inline">\(\rho\)</span> as the number of fermions per site: <span class="math display">\[
|
||||
\rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right)
|
||||
\]</span></p>
|
||||
<p>The PH symmetry maps the Hamiltonian to itself with the sign of the
|
||||
chemical potential reversed and the density inverted about half filling:
|
||||
<span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t,
|
||||
U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 -
|
||||
\rho \]</span></p>
|
||||
<p>The Hamiltonian is symmetric under PH at <span
|
||||
class="math inline">\(\mu = 0\)</span> and so must all the observables,
|
||||
hence half filling <span class="math inline">\(\rho = 1\)</span> occurs
|
||||
here. This symmetry and known observable acts as a useful test for the
|
||||
numerical calculations.</p>
|
||||
<p>Next Section: <a
|
||||
href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Applying MCMC
|
||||
to the FK model</a></p>
|
||||
<p>The PH symmetry maps the Hamiltonian to itself with the sign of the chemical potential reversed and the density inverted about half filling: <span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t, U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 - \rho \]</span></p>
|
||||
<p>The Hamiltonian is symmetric under PH at <span class="math inline">\(\mu = 0\)</span> and so must all the observables, hence half filling <span class="math inline">\(\rho = 1\)</span> occurs here. This symmetry and known observable acts as a useful test for the numerical calculations.</p>
|
||||
<p>Next Section: <a href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Markov Chain Monte Carlo</a></p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C.
|
||||
Gruber and D. Ueltschi, <em><a
|
||||
href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball
|
||||
Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
|
||||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C. Gruber and D. Ueltschi, <em><a href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
File diff suppressed because it is too large
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@ -27,8 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
|
||||
Generation</a></li>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice Generation</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
{% endcapture %}
|
||||
@ -41,8 +40,7 @@ Generation</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
|
||||
Generation</a></li>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice Generation</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -54,11 +52,8 @@ Generation</a></li>
|
||||
</div>
|
||||
<section id="lattice-generation" class="level1">
|
||||
<h1>Lattice Generation</h1>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Section: <a
|
||||
href="../6_Appendices/A.4_Lattice_Colouring.html">Lattice
|
||||
Colouring</a></p>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p>Next Section: <a href="../6_Appendices/A.4_Lattice_Colouring.html">Lattice Colouring</a></p>
|
||||
</section>
|
||||
|
||||
|
||||
|
||||
@ -27,8 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
|
||||
Colouring</a></li>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice Colouring</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
{% endcapture %}
|
||||
@ -41,8 +40,7 @@ Colouring</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
|
||||
Colouring</a></li>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice Colouring</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -54,8 +52,7 @@ Colouring</a></li>
|
||||
</div>
|
||||
<section id="lattice-colouring" class="level1">
|
||||
<h1>Lattice Colouring</h1>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<div class="sourceCode" id="cb1"><pre class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</section>
|
||||
|
||||
|
||||
|
||||
@ -27,8 +27,8 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#app-the-projector" id="toc-app-the-projector">The
|
||||
Projector</a></li>
|
||||
<li><a href="#app-the-projector" id="toc-app-the-projector">The Projector</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
{% endcapture %}
|
||||
@ -41,8 +41,8 @@ Projector</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#app-the-projector" id="toc-app-the-projector">The
|
||||
Projector</a></li>
|
||||
<li><a href="#app-the-projector" id="toc-app-the-projector">The Projector</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -54,6 +54,40 @@ Projector</a></li>
|
||||
</div>
|
||||
<section id="app-the-projector" class="level1">
|
||||
<h1>The Projector</h1>
|
||||
<p>The projection from the extended space to the physical space will not be particularly important for the results presented here. However, the theory remains useful to explain why this is.</p>
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/hilbert_spaces.svg" id="fig:hilbert_spaces" data-short-caption="How the different Hilbert Spaces relate to one another" style="width:100.0%" alt="Figure 1: The relationship between the different Hilbert spaces used in the solution. needs updating" />
|
||||
<figcaption aria-hidden="true">Figure 1: The relationship between the different Hilbert spaces used in the solution. <strong>needs updating</strong></figcaption>
|
||||
</figure>
|
||||
<p>The physical states are defined as those for which <span class="math inline">\(D_i |\phi\rangle = |\phi\rangle\)</span> for all <span class="math inline">\(D_i\)</span>. Since <span class="math inline">\(D_i\)</span> has eigenvalues <span class="math inline">\(\pm1\)</span>, the quantity <span class="math inline">\(\tfrac{(1+D_i)}{2}\)</span> has eigenvalue <span class="math inline">\(1\)</span> for physical states and <span class="math inline">\(0\)</span> for extended states so is the local projector onto the physical subspace.</p>
|
||||
<p>Therefore, the global projector is <span class="math display">\[ \mathcal{P} = \prod_{i=1}^{2N} \left( \frac{1 + D_i}{2}\right)\]</span></p>
|
||||
<p>for a toroidal trivalent lattice with <span class="math inline">\(N\)</span> plaquettes <span class="math inline">\(2N\)</span> vertices and <span class="math inline">\(3N\)</span> edges. As discussed earlier, the product over <span class="math inline">\((1 + D_j)\)</span> can also be thought of as the sum of all possible subsets <span class="math inline">\(\{i\}\)</span> of the <span class="math inline">\(D_j\)</span> operators, which is the set of all possible gauge symmetry operations.</p>
|
||||
<p><span class="math display">\[ \mathcal{P} = \frac{1}{2^{2N}} \sum_{\{i\}} \prod_{i\in\{i\}} D_i\]</span></p>
|
||||
<p>Since the gauge operators <span class="math inline">\(D_j\)</span> commute and square to one, we can define the complement operator <span class="math inline">\(C = \prod_{i=1}^{2N} D_i\)</span> and see that it takes each set of <span class="math inline">\(\prod_{i \in \{i\}} D_j\)</span> operators and gives us the complement of that set. We will shortly see why <span class="math inline">\(C\)</span> is the identity in the physical subspace, as noted earlier.</p>
|
||||
<p>We use the complement operator to rewrite the projector as a sum over half the subsets of <span class="math inline">\(\{i\}\)</span> - referred to as <span class="math inline">\(\Lambda\)</span>. The complement operator deals with the other half</p>
|
||||
<p><span class="math display">\[ \mathcal{P} = \left( \frac{1}{2^{2N-1}} \sum_{\Lambda} \prod_{i\in\{i\}} D_i\right) \left(\frac{1 + \prod_i^{2N} D_i}{2}\right) = \mathcal{S} \cdot \mathcal{P}_0\]</span></p>
|
||||
<p>To compute <span class="math inline">\(\mathcal{P}_0\)</span>, the main quantity needed is the product of the local projectors <span class="math inline">\(D_i\)</span> <span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i b^y_i b^z_i c_i \]</span> for a toroidal trivalent lattice with <span class="math inline">\(N\)</span> plaquettes <span class="math inline">\(2N\)</span> vertices and <span class="math inline">\(3N\)</span> edges.</p>
|
||||
<p>First, we reorder the operators by bond type. This does not require any information about the underlying lattice.</p>
|
||||
<p><span class="math display">\[\prod_i^{2N} D_i = \prod_i^{2N} b^x_i \prod_i^{2N} b^y_i \prod_i^{2N} b^z_i \prod_i^{2N} c_i\]</span></p>
|
||||
<p>The product over <span class="math inline">\(c_i\)</span> operators reduces to a determinant of the Q matrix and the fermion parity, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">1</a>]</span>. The only difference from the honeycomb case is that we cannot explicitly compute the factors <span class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from reordering the b operators such that pairs of vertices linked by the corresponding bonds are adjacent.</p>
|
||||
<p><span class="math display">\[\prod_i^{2N} b^\alpha_i = p_\alpha \prod_{(i,j)}b^\alpha_i b^\alpha_j\]</span></p>
|
||||
<p>However, they are simply the parity of the permutation from one ordering to the other and can be computed in linear time with a cycle decomposition <strong>cite</strong>.</p>
|
||||
<p>We find that <span class="math display">\[\mathcal{P}_0 = 1 + p_x\;p_y\;p_z\; \hat{\pi} \; \mathrm{det}(Q^u) \; \prod_{\{i,j\}} -iu_{ij}\]</span></p>
|
||||
<p>where <span class="math inline">\(p_x\;p_y\;p_z = \pm 1\)</span> are lattice structure factors and <span class="math inline">\(\mathrm{det}(Q^u)\)</span> is the determinant of the matrix mentioned earlier that maps <span class="math inline">\(c_i\)</span> operators to normal mode operators <span class="math inline">\(b'_i, b''_i\)</span>. These depend only on the lattice structure.</p>
|
||||
<p><span class="math inline">\(\hat{\pi} = \prod{i}^{N} (1 - 2\hat{n}_i)\)</span> is the parity of the particular many body state determined by fermionic occupation numbers <span class="math inline">\(n_i\)</span>. As discussed in <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">1</a>]</span>, <span class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
|
||||
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> is also a gauge invariant quantity. In translation invariant models this quantity which can be related to the parity of the number of vortex pairs in the system <span class="citation" data-cites="yaoAlgebraicSpinLiquid2009"> [<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>All these factors take values <span class="math inline">\(\pm 1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or 1 for a particular state. Since <span class="math inline">\(\mathcal{S}\)</span> corresponds to symmetrising over all the gauge configurations and cannot be 0, once we have determined the single particle eigenstates of a bond sector, the true many body ground state has the same energy as either the empty state with <span class="math inline">\(n_i = 0\)</span> or a state with a single fermion in the lowest level.</p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
<h1 class="unnumbered">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">F. L. Pedrocchi, S. Chesi, and D. Loss, <em><a href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>, 165414 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">H. Yao, S.-C. Zhang, and S. A. Kivelson, <em><a href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett. <strong>102</strong>, 217202 (2009).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
|
||||
@ -9,7 +9,7 @@
|
||||
<ul>
|
||||
<li><a href="./2_Background/2.1_FK_Model.html">The Falikov Kimball Model</a></li>
|
||||
<li><a href="./2_Background/2.2_HKM_Model.html#the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a></li>
|
||||
<li><a href="./2_Background/2.3_Disorder.html#disorder-and-localisation">Disorder and Localisation</a></li>
|
||||
<li><a href="./2_Background/2.4_Disorder.html#disorder-and-localisation">Disorder and Localisation</a></li>
|
||||
</ul>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html">3 The Long Range Falikov-Kimball Model</a></li>
|
||||
<ul>
|
||||
@ -34,10 +34,8 @@
|
||||
<li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Appendices</a></li>
|
||||
<ul>
|
||||
<li><a href="./6_Appendices/A.1.2_Fermion_Free_Energy.html">Evaluation of the Fermion Free Energy</a></li>
|
||||
<li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry-Copy1.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Particle-Hole Symmetry</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
<li><a href="./6_Appendices/A.2_Markov_Chain_Monte_Carlo.html#[\[app:balance\]]">[\[app:balance\]]</a></li>
|
||||
<li><a href="./6_Appendices/A.3_Lattice_Generation.html#lattice-generation">Lattice Generation</a></li>
|
||||
<li><a href="./6_Appendices/A.4_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</a></li>
|
||||
<li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>
|
||||
|
||||
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