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---
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title: 1_Intro
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excerpt:
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title: Introduction
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excerpt: Why do we do Condensed Matter theory at all?
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layout: none
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image:
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@ -11,7 +11,8 @@ image:
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<meta charset="utf-8" />
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<meta name="generator" content="pandoc" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
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<title>1_Intro</title>
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<meta name="description" content="Why do we do Condensed Matter theory at all?" />
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<title>Introduction</title>
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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@ -34,7 +35,6 @@ Body Systems</a></li>
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Insulators</a></li>
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<li><a href="#quantum-spin-liquids"
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id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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</nav>
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@ -55,7 +55,6 @@ Body Systems</a></li>
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Insulators</a></li>
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<li><a href="#quantum-spin-liquids"
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id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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</nav>
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@ -355,43 +354,61 @@ href="#ref-antipovCriticalExponentsStrongly2014"
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role="doc-biblioref">38</a>–<a
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||||
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
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role="doc-biblioref">41</a>]</span>.</p>
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<p>In Chapter 3 I will introduce a generalized FK model in one
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dimension. With the addition of long-range interactions in the
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background field, the model shows a similarly rich phase diagram. I use
|
||||
an exact Markov chain Monte Carlo method to map the phase diagram and
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||||
compute the energy-resolved localization properties of the fermions. I
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||||
then compare the behaviour of this transitionally invariant model to an
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Anderson model of uncorrelated binary disorder about a background charge
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density wave field which confirms that the fermionic sector only fully
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localizes for very large system sizes.</p>
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<p>In Chapter 3 I will introduce a generalized Falikov-Kimball model in
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one dimension I call the Long-Range Falikov-Kimball model. With the
|
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addition of long-range interactions in the background field, the model
|
||||
shows a similarly rich phase diagram its higher dimensional cousins. I
|
||||
use an exact Markov chain Monte Carlo method to map the phase diagram
|
||||
and compute the energy-resolved localization properties of the fermions.
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||||
I then compare the behaviour of this transitionally invariant model to
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an Anderson model of uncorrelated binary disorder about a background
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charge density wave field which confirms that the fermionic sector only
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fully localizes for very large system sizes.</p>
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</section>
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<section id="quantum-spin-liquids" class="level1">
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<h1>Quantum Spin Liquids</h1>
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<p>To turn to the other key topic of this thesis, we have discussed the
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question of the magnetic ordering of local moments in the Mott
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insulating state. The local moments may form an AFM ground state.
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Alternatively they may fail to order even at zero temperature <span
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class="citation"
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data-cites="law1TTaS2QuantumSpin2017 ribakGaplessExcitationsGround2017"> [<a
|
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href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">28</a>,<a
|
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href="#ref-ribakGaplessExcitationsGround2017"
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role="doc-biblioref">29</a>]</span>, giving rise to what is known as a
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quantum spin liquid (QSL) state.</p>
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<p>Landau-Ginzburg-Wilson theory characterises phases of matter as
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<p>To turn to the other key topic of this thesis, we have already
|
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discussed the AFM ordering of local moments in the Mott insulating
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state. Landau-Ginzburg-Wilson theory characterises phases of matter as
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inextricably linked to the emergence of long range order via a
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spontaneously broken symmetry. The fractional quantum Hall (FQH) state,
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discovered in the 1980s is an explicit example of an electronic system
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that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems
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exhibit fractionalised excitations linked to their ground state having
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long range entanglement and non-trivial topological properties <span
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class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
|
||||
href="#ref-broholmQuantumSpinLiquids2020"
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role="doc-biblioref">42</a>]</span>. Quantum spin liquids are the
|
||||
analogous phase of matter for spin systems. Remarkably the existence of
|
||||
QSLs was first suggested by Anderson in 1973 <span class="citation"
|
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data-cites="andersonResonatingValenceBonds1973"> [<a
|
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spontaneously broken symmetry. So within this paradigm we would not
|
||||
expect any interesting phases of matter not associated with AFM or other
|
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long-range order. However, Anderson first proposed in 1973 <span
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class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a
|
||||
href="#ref-andersonResonatingValenceBonds1973"
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||||
role="doc-biblioref">43</a>]</span>.</p>
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role="doc-biblioref">42</a>]</span> that if long range order is
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suppressed by some mechanism, it might lead to a liquid-like state even
|
||||
at zero temperature, the Quantum Spin Liquid (QSL).</p>
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<p>This QSL state would exist at zero or very low temperatures, so we
|
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would expect quantum effects to be very strong, which will turn out to
|
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have far reaching consequences. It was the discovery of a different
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phase, however that really kickstarted interest in the topic. The
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fractional quantum Hall (FQH) state, discovered in the 1980s is an
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explicit example of an interacting electron system that falls outside of
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||||
the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological
|
||||
properties with the QSL state. They both exhibit fractionalised
|
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excitations, braiding statistics and non-trivial topological
|
||||
properties <span class="citation"
|
||||
data-cites="broholmQuantumSpinLiquids2020"> [<a
|
||||
href="#ref-broholmQuantumSpinLiquids2020"
|
||||
role="doc-biblioref">43</a>]</span>. The many-body ground state of such
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systems acts as a complex and highly entangled vacuum. This vacuum can
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support quasiparticle excitations with properties unbound from that of
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the Dirac fermions of the standard model.</p>
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<p>How do we actually make a QSL? Frustration is one mechanism that we
|
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can use to suppress magnetic order in spin models <span class="citation"
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||||
data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022"
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||||
role="doc-biblioref">44</a>]</span>. Frustration can be geometric,
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triangular lattices for instance cannot support AFM order. It can also
|
||||
come about as a result of spin-orbit coupling or other physics. There
|
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are also other routes to QSLs besides frustrated spin systems that we
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||||
will not discuss here <span class="citation"
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||||
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
|
||||
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a
|
||||
href="#ref-linExactSymmetryWeaklyinteracting1998"
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||||
role="doc-biblioref">47</a>]</span>.</p>
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||||
<!-- Experimentally, Mott insulating systems without magnetic order have been proposed as QSL systems\ [@law1TTaS2QuantumSpin2017; @ribakGaplessExcitationsGround2017]. -->
|
||||
<!-- Other exampels: Quantum spin liquids are the analogous phase of matter for spin systems. Spin ice support deconfined magnetic monopoles. -->
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||||
<div id="fig:correlation_spin_orbit_PT" class="fignos">
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<figure>
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<img src="/assets/thesis/intro_chapter/correlation_spin_orbit_PT.png"
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@ -403,97 +420,176 @@ href="#ref-TrebstPhysRep2022"
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role="doc-biblioref">44</a>]</span>.</figcaption>
|
||||
</figure>
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||||
</div>
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||||
<p>The main route to QSLs, though there are others <span
|
||||
class="citation"
|
||||
data-cites="balentsNodalLiquidTheory1998 balentsDualOrderParameter1999 linExactSymmetryWeaklyinteracting1998"> [<a
|
||||
href="#ref-balentsNodalLiquidTheory1998" role="doc-biblioref">45</a>–<a
|
||||
href="#ref-linExactSymmetryWeaklyinteracting1998"
|
||||
role="doc-biblioref">47</a>]</span>, is via frustration of spin models
|
||||
that would otherwise order have AFM order. This frustration can come
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geometrically, triangular lattices for instance cannot support AFM
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||||
order. It can also come about as a result of spin-orbit coupling.</p>
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||||
<p>Electron spin naturally couples to magnetic fields. Spin-orbit
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||||
coupling is a relativistic effect, that very roughly corresponds to the
|
||||
fact that in the frame of reference of a moving electron, the electric
|
||||
field of nearby nuclei look like magnetic field to which the electron
|
||||
spin couples. In certain transition metal based compounds, such as those
|
||||
based on Iridium and Rutheniun, crystal field effects, strong spin-orbit
|
||||
coupling and narrow bandwidths lead to effective spin-<span
|
||||
<p>Spin-orbit coupling is a relativistic effect, that very roughly
|
||||
corresponds to the fact that in the frame of reference of a moving
|
||||
electron, the electric field of nearby nuclei look like magnetic fields
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to which the electron spin couples. This effectively couples the spatial
|
||||
and spin parts of the electron wavefunction, meaning that the lattice
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||||
structure can influence the form of the spin-spin interactions leading
|
||||
to spatial anisotropy. This anisotropy will be how we frustrate the Mott
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||||
insulators <span class="citation"
|
||||
data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a
|
||||
href="#ref-jackeliMottInsulatorsStrong2009"
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role="doc-biblioref">48</a>,<a
|
||||
href="#ref-khaliullinOrbitalOrderFluctuations2005"
|
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role="doc-biblioref">49</a>]</span>. As we saw with the Hubbard model,
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interaction effects are only strong or weak in comparison to the
|
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bandwidth or hopping integral <span class="math inline">\(t\)</span> so
|
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what we need to see strong frustration is a material with strong
|
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spin-orbit coupling <span class="math inline">\(\lambda\)</span>
|
||||
relative to its bandwidth <span class="math inline">\(t\)</span>.</p>
|
||||
<p>In certain transition metal based compounds, such as those based on
|
||||
Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling
|
||||
and narrow bandwidths lead to effective spin-<span
|
||||
class="math inline">\(\tfrac{1}{2}\)</span> Mott insulating states with
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strongly anisotropic spin-spin couplings known as Kitaev Materials <span
|
||||
class="citation"
|
||||
strongly anisotropic spin-spin couplings. These transition metal
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||||
compounds, known Kitaev Materials, draw their name from the celebrated
|
||||
Kitaev Honeycomb Model which is expected to model their low temperature
|
||||
behaviour <span class="citation"
|
||||
data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a
|
||||
href="#ref-TrebstPhysRep2022" role="doc-biblioref">44</a>,<a
|
||||
href="#ref-Jackeli2009" role="doc-biblioref">48</a>–<a
|
||||
href="#ref-Takagi2019" role="doc-biblioref">51</a>]</span>. Kitaev
|
||||
materials draw their name from the celebrated Kitaev Honeycomb Model as
|
||||
it is believed they will realise the QSL state via the mechanisms of the
|
||||
Kitaev Model.</p>
|
||||
href="#ref-Jackeli2009" role="doc-biblioref">50</a>–<a
|
||||
href="#ref-Takagi2019" role="doc-biblioref">53</a>]</span>.</p>
|
||||
<p>At this point we can sketch out a phase diagram like that of fig. <a
|
||||
href="#fig:correlation_spin_orbit_PT">3</a>. When both electron-electron
|
||||
interactions <span class="math inline">\(U\)</span> and spin-orbit
|
||||
couplings <span class="math inline">\(\lambda\)</span> are small
|
||||
relative to the bandwidth <span class="math inline">\(t\)</span> we
|
||||
recover standard band theory of band insulators and metals. In the upper
|
||||
left we have the simple Mott insulating state as described by the
|
||||
Hubbard model. In the lower right, strong spin-orbit coupling gives rise
|
||||
to Topological insulators (TIs) characterised by symmetry protected edge
|
||||
modes and non-zero Chern number. Kitaev materials occur in the region
|
||||
where strong electron-electron interaction and spin-orbit coupling
|
||||
interact. See <span class="citation"
|
||||
data-cites="witczak-krempaCorrelatedQuantumPhenomena2014"> [<a
|
||||
href="#ref-witczak-krempaCorrelatedQuantumPhenomena2014"
|
||||
role="doc-biblioref">54</a>]</span> for a much more expansive version of
|
||||
this diagram.</p>
|
||||
<p>The Kitaev Honeycomb model <span class="citation"
|
||||
data-cites="kitaevAnyonsExactlySolved2006"> [<a
|
||||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||||
role="doc-biblioref">52</a>]</span> was the first concrete model with a
|
||||
QSL ground state. It is defined on the honeycomb lattice and provides an
|
||||
exactly solvable model whose ground state is a QSL characterized by a
|
||||
static <span class="math inline">\(\mathbb Z_2\)</span> gauge field and
|
||||
Majorana fermion excitations. It can be reduced to a free fermion
|
||||
problem via a mapping to Majorana fermions which yields an extensive
|
||||
number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes
|
||||
tied to an emergent gauge field. The model is remarkable not only for
|
||||
its QSL ground state, it supports a rich phase diagram hosting gapless,
|
||||
Abelian and non-Abelian phases <span class="citation"
|
||||
role="doc-biblioref">55</a>]</span> was the first concrete spin model
|
||||
with a QSL ground state. It is defined on the two dimensional honeycomb
|
||||
lattice and provides an exactly solvable model that can be reduced to a
|
||||
free fermion problem via a mapping to Majorana fermions. This yields an
|
||||
extensive number of static <span class="math inline">\(\mathbb
|
||||
Z_2\)</span> fluxes tied to an emergent gauge field. The model is
|
||||
remarkable not only for its QSL ground state but also for its
|
||||
fractionalised excitations with non-trivial braiding statistics. It has
|
||||
a rich phase diagram hosting gapless, Abelian and non-Abelian
|
||||
phases <span class="citation"
|
||||
data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
|
||||
href="#ref-knolleDynamicsFractionalizationQuantum2015"
|
||||
role="doc-biblioref">53</a>]</span> and a finite temperature phase
|
||||
role="doc-biblioref">56</a>]</span> and a finite temperature phase
|
||||
transition to a thermal metal state <span class="citation"
|
||||
data-cites="selfThermallyInducedMetallic2019"> [<a
|
||||
href="#ref-selfThermallyInducedMetallic2019"
|
||||
role="doc-biblioref">54</a>]</span>. It been proposed that its
|
||||
role="doc-biblioref">57</a>]</span>. It been proposed that its
|
||||
non-Abelian excitations could be used to support robust topological
|
||||
quantum computing [<span class="citation"
|
||||
data-cites="kitaev_fault-tolerant_2003"> [<a
|
||||
href="#ref-kitaev_fault-tolerant_2003"
|
||||
role="doc-biblioref">55</a>]</span>; <span class="citation"
|
||||
data-cites="freedmanTopologicalQuantumComputation2003"> [<a
|
||||
href="#ref-freedmanTopologicalQuantumComputation2003"
|
||||
role="doc-biblioref">56</a>]</span>;
|
||||
nayakNonAbelianAnyonsTopological2008].</p>
|
||||
<p>It is by now understood that the Kitaev model on any tri-coordinated
|
||||
<span class="math inline">\(z=3\)</span> graph has conserved plaquette
|
||||
operators and local symmetries <span class="citation"
|
||||
data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
|
||||
role="doc-biblioref">57</a>,<a href="#ref-Baskaran2008"
|
||||
role="doc-biblioref">58</a>]</span> which allow a mapping onto effective
|
||||
free Majorana fermion problems in a background of static <span
|
||||
class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
|
||||
data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
|
||||
href="#ref-Nussinov2009" role="doc-biblioref">59</a>–<a
|
||||
href="#ref-hermanns2015weyl" role="doc-biblioref">62</a>]</span>.
|
||||
However, depending on lattice symmetries, finding the ground state flux
|
||||
sector and understanding the QSL properties can still be
|
||||
quantum computing <span class="citation"
|
||||
data-cites="kitaev_fault-tolerant_2003 freedmanTopologicalQuantumComputation2003 nayakNonAbelianAnyonsTopological2008"> [<a
|
||||
href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">58</a>–<a
|
||||
href="#ref-nayakNonAbelianAnyonsTopological2008"
|
||||
role="doc-biblioref">60</a>]</span>.</p>
|
||||
<p>As Kitaev points out in his original paper, the model remains
|
||||
solvable on any tri-coordinated <span class="math inline">\(z=3\)</span>
|
||||
graph which can be 3-edge-coloured. Indeed many generalisations of the
|
||||
model to <span class="citation"
|
||||
data-cites="Baskaran2007 Baskaran2008 Nussinov2009 OBrienPRB2016 hermanns2015weyl"> [<a
|
||||
href="#ref-Baskaran2007" role="doc-biblioref">61</a>–<a
|
||||
href="#ref-hermanns2015weyl" role="doc-biblioref">65</a>]</span>.
|
||||
Notably, the Yao-Kivelson model <span class="citation"
|
||||
data-cites="yaoExactChiralSpin2007"> [<a
|
||||
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>
|
||||
introduces triangular plaquettes to the honeycomb lattice leading to
|
||||
spontaneous chiral symmetry breaking. These extensions all retain
|
||||
translation symmetry, likely because edge-colouring and finding the
|
||||
ground state become much harder without it. Finding the ground state
|
||||
flux sector and understanding the QSL properties can still be
|
||||
challenging <span class="citation"
|
||||
data-cites="eschmann2019thermodynamics Peri2020"> [<a
|
||||
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">63</a>,<a
|
||||
href="#ref-Peri2020" role="doc-biblioref">64</a>]</span>.</p>
|
||||
<p><strong>paragraph about amorphous lattices</strong></p>
|
||||
<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
|
||||
liquid by extending the Kitaev honeycomb model to random lattices with
|
||||
fixed coordination number three. The model retains its exact solubility
|
||||
but the presence of plaquettes with an odd number of sides leads to a
|
||||
spontaneous breaking of time reversal symmetry. I unearth a rich phase
|
||||
diagram displaying Abelian as well as a non-Abelian quantum spin liquid
|
||||
phases with a remarkably simple ground state flux pattern. Furthermore,
|
||||
I show that the system undergoes a finite-temperature phase transition
|
||||
to a conducting thermal metal state and discuss possible experimental
|
||||
realisations.</p>
|
||||
</section>
|
||||
<section id="outline" class="level1">
|
||||
<h1>Outline</h1>
|
||||
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">67</a>,<a
|
||||
href="#ref-Peri2020" role="doc-biblioref">68</a>]</span>. Undeterred,
|
||||
this gap lead us to wonder what might happen if we remove translation
|
||||
symmetry from the Kitaev Model. This might would be a model of a
|
||||
tri-coordinated, highly bond anisotropic but otherwise amorphous
|
||||
material.</p>
|
||||
<p>Amorphous materials do no have long-range lattice regularities but
|
||||
covalent compounds can induce short-range regularities in the lattice
|
||||
structure such as fixed coordination number <span
|
||||
class="math inline">\(z\)</span>. The best examples being amorphous
|
||||
Silicon and Germanium with <span class="math inline">\(z=4\)</span>
|
||||
which are used to make thin-film solar cells <span class="citation"
|
||||
data-cites="Weaire1971 betteridge1973possible"> [<a
|
||||
href="#ref-Weaire1971" role="doc-biblioref">69</a>,<a
|
||||
href="#ref-betteridge1973possible" role="doc-biblioref">70</a>]</span>.
|
||||
Recently is has been shown that topological insulating (TI) phases can
|
||||
exist in amorphous systems. Amorphous TIs are characterized by similar
|
||||
protected edge states to their translation invariant cousins and
|
||||
generalised topological bulk invariants <span class="citation"
|
||||
data-cites="mitchellAmorphousTopologicalInsulators2018 agarwala2019topological marsalTopologicalWeaireThorpeModels2020 costa2019toward agarwala2020higher spring2021amorphous corbae2019evidence"> [<a
|
||||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||||
role="doc-biblioref">71</a>–<a href="#ref-corbae2019evidence"
|
||||
role="doc-biblioref">77</a>]</span>. However, research on amorphous
|
||||
electronic systems has been mostly focused on non-interacting systems
|
||||
with a few exceptions, for example, to account for the observation of
|
||||
superconductivity <span class="citation"
|
||||
data-cites="buckel1954einfluss mcmillan1981electron meisel1981eliashberg bergmann1976amorphous mannaNoncrystallineTopologicalSuperconductors2022"> [<a
|
||||
href="#ref-buckel1954einfluss" role="doc-biblioref">78</a>–<a
|
||||
href="#ref-mannaNoncrystallineTopologicalSuperconductors2022"
|
||||
role="doc-biblioref">82</a>]</span> in amorphous materials or very
|
||||
recently to understand the effect of strong electron repulsion in
|
||||
TIs <span class="citation" data-cites="kim2022fractionalization"> [<a
|
||||
href="#ref-kim2022fractionalization"
|
||||
role="doc-biblioref">83</a>]</span>.</p>
|
||||
<p>Amorphous <em>magnetic</em> systems has been investigated since the
|
||||
1960s, mostly through the adaptation of theoretical tools developed for
|
||||
disordered systems <span class="citation"
|
||||
data-cites="aharony1975critical Petrakovski1981 kaneyoshi1992introduction Kaneyoshi2018"> [<a
|
||||
href="#ref-aharony1975critical" role="doc-biblioref">84</a>–<a
|
||||
href="#ref-Kaneyoshi2018" role="doc-biblioref">87</a>]</span> and with
|
||||
numerical methods <span class="citation"
|
||||
data-cites="fahnle1984monte plascak2000ising"> [<a
|
||||
href="#ref-fahnle1984monte" role="doc-biblioref">88</a>,<a
|
||||
href="#ref-plascak2000ising" role="doc-biblioref">89</a>]</span>.
|
||||
Research on classical Heisenberg and Ising models has been shown to
|
||||
account for observed behaviour of ferromagnetism, disordered
|
||||
antiferromagnetism and widely observed spin glass behaviour <span
|
||||
class="citation" data-cites="coey1978amorphous"> [<a
|
||||
href="#ref-coey1978amorphous" role="doc-biblioref">90</a>]</span>.
|
||||
However, the role of spin-anisotropic interactions and quantum effects
|
||||
in amorphous magnets has not been addressed. It is an open question
|
||||
whether frustrated magnetic interactions on amorphous lattices can give
|
||||
rise genuine quantum phases, i.e. to long-range entangled quantum spin
|
||||
liquids (QSL) <span class="citation"
|
||||
data-cites="Anderson1973 Knolle2019 Savary2016 Lacroix2011"> [<a
|
||||
href="#ref-Anderson1973" role="doc-biblioref">91</a>–<a
|
||||
href="#ref-Lacroix2011" role="doc-biblioref">94</a>]</span>.</p>
|
||||
<p>In Chapter 4 I will introduce the Amorphous Kitaev model, a
|
||||
generalisation of the Kitaev honeycomb model to random lattices with
|
||||
fixed coordination number three. We will show that this model is a
|
||||
soluble chiral amorphous quantum spin liquid. The model retains its
|
||||
exact solubility but, as with the Yao-Kivelson model <span
|
||||
class="citation" data-cites="yaoExactChiralSpin2007"> [<a
|
||||
href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">66</a>]</span>,
|
||||
the presence of plaquettes with an odd number of sides leads to a
|
||||
spontaneous breaking of time reversal symmetry. We will confirm prior
|
||||
observations that the form of the ground state can be written in terms
|
||||
of the number of sides of elementary plaquettes of the model <span
|
||||
class="citation"
|
||||
data-cites="OBrienPRB2016 eschmannThermodynamicClassificationThreedimensional2020"> [<a
|
||||
href="#ref-OBrienPRB2016" role="doc-biblioref">64</a>,<a
|
||||
href="#ref-eschmannThermodynamicClassificationThreedimensional2020"
|
||||
role="doc-biblioref">95</a>]</span>. We unearth a rich phase diagram
|
||||
displaying Abelian as well as a non-Abelian chiral spin liquid phases.
|
||||
Furthermore, I show that the system undergoes a finite-temperature phase
|
||||
transition to a conducting thermal metal state and discuss possible
|
||||
experimental realisations.</p>
|
||||
<p>The next chapter, Chapter 2, will introduce some necessary background
|
||||
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
|
||||
localisation.</p>
|
||||
<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
|
||||
greater detail. I will present results that. Chapter 4 focusses on the
|
||||
localisation. Then Chapter 3 introduces and studies the Long Range
|
||||
Falikov-Kimball Model in one dimension while Chapter 4 focusses on the
|
||||
Amorphous Kitaev Model.</p>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
@ -820,22 +916,22 @@ href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
|
||||
Dynamical Cluster Approximation Study of the Falicov-Kimball
|
||||
Model</a></em>, Phys. Rev. B <strong>94</strong>, 245114 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-broholmQuantumSpinLiquids2020" class="csl-entry"
|
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role="doc-biblioentry">
|
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|
||||
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|
||||
Senthil, <em><a href="https://doi.org/10.1126/science.aay0668">Quantum
|
||||
Spin Liquids</a></em>, Science <strong>367</strong>, eaay0668
|
||||
(2020).</div>
|
||||
</div>
|
||||
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|
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|
||||
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>
|
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|
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Senthil, <em><a href="https://doi.org/10.1126/science.aay0668">Quantum
|
||||
Spin Liquids</a></em>, Science <strong>367</strong>, eaay0668
|
||||
(2020).</div>
|
||||
</div>
|
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|
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|
||||
the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
|
||||
Compass and Kitaev Models</a></em>, Phys. Rev. Lett.
|
||||
<strong>102</strong>, 017205 (2009).</div>
|
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|
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Theoretical Physics Supplement <strong>160</strong>, 155 (2005).</div>
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the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum
|
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Compass and Kitaev Models</a></em>, Physical Review Letters
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<strong>102</strong>, 017205 (2009).</div>
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href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics
|
||||
of the Kitaev Model: Fractionalization, Dynamic Correlations, and
|
||||
@ -886,28 +998,36 @@ Material Connections</a></em>, Annual Review of Condensed Matter Physics
|
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<strong>9</strong>, 17 (2018).</div>
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|
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Kitaev Magnetism</em>, Journal of Physics: Condensed Matter
|
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<strong>29</strong>, 493002 (2017).</div>
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|
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Reviews Physics <strong>1</strong>, 264 (2019).</div>
|
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</div>
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|
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Quantum Phenomena in the Strong Spin-Orbit Regime</a></em>, Annual
|
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Review of Condensed Matter Physics <strong>5</strong>, 57 (2014).</div>
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||||
</div>
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|
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in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
|
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<strong>321</strong>, 2 (2006).</div>
|
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</div>
|
||||
<div id="ref-knolleDynamicsFractionalizationQuantum2015"
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class="csl-entry" role="doc-biblioentry">
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Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, <em><a
|
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href="https://doi.org/10.1103/PhysRevB.92.115127">Dynamics of
|
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Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
|
||||
@ -915,7 +1035,7 @@ Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
|
||||
</div>
|
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|
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href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
|
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Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
|
||||
@ -924,7 +1044,7 @@ the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
|
||||
</div>
|
||||
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
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|
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href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
|
||||
Quantum Computation by Anyons</a></em>, Annals of Physics
|
||||
@ -932,70 +1052,259 @@ Quantum Computation by Anyons</a></em>, Annals of Physics
|
||||
</div>
|
||||
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|
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href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
|
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Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
|
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(2003).</div>
|
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</div>
|
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|
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Topological Quantum Computation</a></em>, Rev. Mod. Phys.
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<strong>80</strong>, 1083 (2008).</div>
|
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</div>
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|
||||
href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for
|
||||
Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys.
|
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Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
|
||||
</div>
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|
||||
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
|
||||
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin- S Kitaev Model:
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Classical Ground States, Order from Disorder, and Exact Correlation
|
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|
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(2008).</div>
|
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</div>
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|
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|
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Physical Review B <strong>79</strong>, 214440 (2009).</div>
|
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</div>
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|
||||
href="https://doi.org/10.1103/PhysRevB.93.085101">Classification of
|
||||
Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models</a></em>,
|
||||
Phys. Rev. B <strong>93</strong>, 085101 (2016).</div>
|
||||
</div>
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|
||||
</div>
|
||||
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|
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href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
|
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|
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<strong>99</strong>, 247203 (2007).</div>
|
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</div>
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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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|
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</div>
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|
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</div>
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|
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P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine,
|
||||
<em><a href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous
|
||||
topological insulators constructed from random point sets</a></em>,
|
||||
Nature Phys <strong>14</strong>, 380 (2018).</div>
|
||||
</div>
|
||||
<div id="ref-agarwala2019topological" class="csl-entry"
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||||
Agarwala, <em>Topological Insulators in Amorphous Systems</em>, in
|
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<em>Excursions in Ill-Condensed Quantum Matter</em> (Springer, 2019),
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||||
pp. 61–79.</div>
|
||||
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|
||||
<div id="ref-marsalTopologicalWeaireThorpeModels2020" class="csl-entry"
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||||
Marsal, D. Varjas, and A. G. Grushin, <em><a
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href="https://doi.org/10.1073/pnas.2007384117">Topological Weaire-Thorpe
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Models of Amorphous Matter</a></em>, Proc. Natl. Acad. Sci. U.S.A.
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||||
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Letters <strong>19</strong>, 8941 (2019).</div>
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|
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Agarwala, V. Juričić, and B. Roy, <em>Higher-Order Topological
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<strong>2</strong>, 012067 (2020).</div>
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Spring, A. Akhmerov, and D. Varjas, <em>Amorphous Topological Phases
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Corbae et al., <em>Evidence for Topological Surface States in Amorphous
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Buckel and R. Hilsch, <em>Einfluß Der Kondensation Bei Tiefen
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Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für
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Verschiedene Metalle</em>, Zeitschrift Für Physik <strong>138</strong>,
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109 (1954).</div>
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McMillan and J. Mochel, <em>Electron Tunneling Experiments on Amorphous
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Ge 1- x Au x</em>, Physical Review Letters <strong>46</strong>, 556
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(1981).</div>
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</div>
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<div class="csl-left-margin">[80] </div><div class="csl-right-inline">L.
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V. Meisel and P. J. Cote, <em><a
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Amorphous Metals</a></em>, Phys. Rev. B <strong>23</strong>, 5834
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(1981).</div>
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Bergmann, <em>Amorphous Metals and Their Superconductivity</em>, Physics
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Reports <strong>27</strong>, 159 (1976).</div>
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</div>
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<div id="ref-mannaNoncrystallineTopologicalSuperconductors2022"
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Manna, S. K. Das, and B. Roy, <em><a
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href="http://arxiv.org/abs/2207.02203">Noncrystalline Topological
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Superconductors</a></em>, arXiv:2207.02203.</div>
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</div>
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Kim, A. Agarwala, and D. Chowdhury, <em>Fractionalization and Topology
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in Amorphous Electronic Solids</em>, arXiv Preprint arXiv:2205.11523
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(2022).</div>
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</div>
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<div class="csl-left-margin">[84] </div><div class="csl-right-inline">A.
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Aharony, <em>Critical Behavior of Amorphous Magnets</em>, Physical
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Review B <strong>12</strong>, 1038 (1975).</div>
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<div class="csl-left-margin">[85] </div><div class="csl-right-inline">G.
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||||
A. Petrakovskii, <em><a
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||||
href="https://doi.org/10.1070/pu1981v024n06abeh004850">Amorphous
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Magnetic Materials</a></em>, Soviet Physics Uspekhi <strong>24</strong>,
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511 (1981).</div>
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</div>
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<div class="csl-left-margin">[86] </div><div class="csl-right-inline">T.
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Kaneyoshi, <em>Introduction to Amorphous Magnets</em> (World Scientific
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||||
Publishing Company, 1992).</div>
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</div>
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Kaneyoshi, editor, <em>Amorphous Magnetism</em> (CRC Press, Boca Raton,
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2018).</div>
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||||
Fähnle, <em>Monte Carlo Study of Phase Transitions in Bond-and
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Site-Disordered Ising and Classical Heisenberg Ferromagnets</em>,
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||||
Journal of Magnetism and Magnetic Materials <strong>45</strong>, 279
|
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(1984).</div>
|
||||
</div>
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<div id="ref-plascak2000ising" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[89] </div><div class="csl-right-inline">J.
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Plascak, L. E. Zamora, and G. P. Alcazar, <em>Ising Model for Disordered
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Ferromagnetic Fe- Al Alloys</em>, Physical Review B <strong>61</strong>,
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||||
3188 (2000).</div>
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||||
</div>
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<div id="ref-coey1978amorphous" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">[90] </div><div class="csl-right-inline">J.
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||||
Coey, <em>Amorphous Magnetic Order</em>, Journal of Applied Physics
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||||
<strong>49</strong>, 1646 (1978).</div>
|
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</div>
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<div id="ref-Anderson1973" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[91] </div><div class="csl-right-inline">P.
|
||||
W. Anderson, <em>Resonating Valence Bonds: A New Kind of
|
||||
Insulator?</em>, Mater. Res. Bull. <strong>8</strong>, 153 (1973).</div>
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</div>
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<div id="ref-Knolle2019" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[92] </div><div class="csl-right-inline">J.
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||||
Knolle and R. Moessner, <em><a
|
||||
href="https://doi.org/10.1146/annurev-conmatphys-031218-013401">A Field
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||||
Guide to Spin Liquids</a></em>, Annual Review of Condensed Matter
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||||
Physics <strong>10</strong>, 451 (2019).</div>
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</div>
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<div class="csl-left-margin">[93] </div><div class="csl-right-inline">L.
|
||||
Savary and L. Balents, <em>Quantum Spin Liquids: A Review</em>, Reports
|
||||
on Progress in Physics <strong>80</strong>, 016502 (2017).</div>
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||||
</div>
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||||
<div id="ref-Lacroix2011" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[94] </div><div class="csl-right-inline">C.
|
||||
Lacroix, P. Mendels, and F. Mila, editors, <em>Introduction to
|
||||
Frustrated Magnetism</em>, Vol. 164 (Springer-Verlag, Berlin Heidelberg,
|
||||
2011).</div>
|
||||
</div>
|
||||
<div id="ref-eschmannThermodynamicClassificationThreedimensional2020"
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||||
class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[95] </div><div class="csl-right-inline">T.
|
||||
Eschmann, P. A. Mishchenko, K. O’Brien, T. A. Bojesen, Y. Kato, M.
|
||||
Hermanns, Y. Motome, and S. Trebst, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.102.075125">Thermodynamic
|
||||
Classification of Three-Dimensional Kitaev Spin Liquids</a></em>, Phys.
|
||||
Rev. B <strong>102</strong>, 075125 (2020).</div>
|
||||
</div>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: 2.1_FK_Model
|
||||
title: Background - The Falikov Kimball Model
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>2.1_FK_Model</title>
|
||||
<title>Background - The Falikov Kimball Model</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
@ -87,13 +87,15 @@ 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}.\\
|
||||
\end{aligned}\]</span></p>
|
||||
<p>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
|
||||
<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"
|
||||
<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
|
||||
@ -108,15 +110,17 @@ 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>. Doping the
|
||||
model away from the half-filled point leads to rich physics including
|
||||
superconductivity <span class="citation"
|
||||
data-cites="jedrzejewskiFalicovKimballModels2001"> [<a
|
||||
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>]</span>.</p>
|
||||
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 symmetric. That is, the Hamiltonian anticommutes with the
|
||||
particle hole operator <span
|
||||
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
|
||||
@ -127,9 +131,11 @@ 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">8</a>]</span>. The absence of a hopping term for
|
||||
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>.</p>
|
||||
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>
|
||||
<div id="fig:simple_DOS" class="fignos">
|
||||
<figure>
|
||||
<img src="/assets/thesis/background_chapter/simple_DOS.svg"
|
||||
@ -147,23 +153,21 @@ wave background with 2% binary disorder.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>We will later add a long range interaction between the localised
|
||||
electrons so 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>.</p>
|
||||
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">\[\begin{aligned}
|
||||
H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} -
|
||||
\tfrac{1}{2}) -\;t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}.\\
|
||||
\end{aligned}\]</span></p>
|
||||
<p>The FK model can be solved exaclty with dynamic mean field theory in
|
||||
<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">9</a>–<a
|
||||
role="doc-biblioref">10</a>–<a
|
||||
href="#ref-herrmannNonequilibriumDynamicalCluster2016"
|
||||
role="doc-biblioref">12</a>]</span>.</p>
|
||||
<ul>
|
||||
<li>displays disorder free localisation</li>
|
||||
</ul>
|
||||
role="doc-biblioref">13</a>]</span>.</p>
|
||||
</section>
|
||||
<section id="phase-diagrams" class="level2">
|
||||
<h2>Phase Diagrams</h2>
|
||||
@ -172,161 +176,245 @@ role="doc-biblioref">12</a>]</span>.</p>
|
||||
<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 diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right) showing temperature (T) and repulsive interaction strength (U). Hubbard model diagram adapted from [13], Falikov-Kimball model from [14,15]" />
|
||||
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
|
||||
diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right)
|
||||
showing temperature (T) and repulsive interaction strength (U). Hubbard
|
||||
model diagram adapted from <span class="citation"
|
||||
data-cites="micnasSuperconductivityNarrowbandSystems1990"> [<a
|
||||
href="#ref-micnasSuperconductivityNarrowbandSystems1990"
|
||||
role="doc-biblioref">13</a>]</span>, Falikov-Kimball model from <span
|
||||
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 antipovCriticalExponentsStrongly2014a"> [<a
|
||||
data-cites="antipovInteractionTunedAndersonMott2016 antipovCriticalExponentsStrongly2014"> [<a
|
||||
href="#ref-antipovCriticalExponentsStrongly2014"
|
||||
role="doc-biblioref">10</a>,<a
|
||||
href="#ref-antipovInteractionTunedAndersonMott2016"
|
||||
role="doc-biblioref">14</a>,<a
|
||||
href="#ref-antipovCriticalExponentsStrongly2014a"
|
||||
role="doc-biblioref">15</a>]</span></figcaption>
|
||||
role="doc-biblioref">14</a>]</span></figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<ul>
|
||||
<li>rich phase diagram in 2d Despite its simplicity, the FK model has a
|
||||
rich phase diagram in <span class="math inline">\(D \geq 2\)</span>
|
||||
dimensions. For example, it shows an interaction-induced gap opening
|
||||
even at high temperatures, similar to the corresponding Hubbard
|
||||
Model <span class="citation"
|
||||
data-cites="brandtThermodynamicsCorrelationFunctions1989"> [<a
|
||||
href="#ref-brandtThermodynamicsCorrelationFunctions1989"
|
||||
role="doc-biblioref">16</a>]</span>.</li>
|
||||
</ul>
|
||||
<p>At half filling and 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 charge
|
||||
density wave state in which the spins order antiferromagnetically. This
|
||||
corresponds to the heavy electrons occupying one of the two sublattices
|
||||
A and B <span class="citation"
|
||||
<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">17</a>]</span>. In the disordered region above
|
||||
<span class="math inline">\(T_c(U)\)</span> there is a transition
|
||||
between an Anderson insulator phase at weak interaction and a Mott
|
||||
insulator phase in the strongly interacting regime <span
|
||||
class="citation" data-cites="andersonAbsenceDiffusionCertain1958"> [<a
|
||||
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">18</a>]</span>.</p>
|
||||
<ul>
|
||||
<li>superconductivity when doped</li>
|
||||
</ul>
|
||||
<p>In 1D, the ground state phenomenology as the model is doped away from
|
||||
the half-filled state can be rich <span class="citation"
|
||||
data-cites="gruberGroundStatesSpinless1990"> [<a
|
||||
href="#ref-gruberGroundStatesSpinless1990"
|
||||
role="doc-biblioref">19</a>]</span> but the system is disordered for all
|
||||
<span class="math inline">\(T > 0\)</span> <span class="citation"
|
||||
data-cites="kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">20</a>]</span>.</p>
|
||||
<p>In the one dimensional FK model there is no ordered CDW phase <span
|
||||
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">21</a>]</span>. The supression of phase transition
|
||||
is a common phenomena in one dimensional systems. It can be understood
|
||||
via Peierls’ argument <span class="citation"
|
||||
data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986"> [<a
|
||||
href="#ref-kennedyItinerantElectronModel1986"
|
||||
role="doc-biblioref">20</a>,<a
|
||||
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">22</a>]</span> to be a consequence of the low
|
||||
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>
|
||||
</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. Short range interactions produce a constant
|
||||
energy penalty for such a domain wall that does not scale with system
|
||||
size. In contrast, the number of such single domain wall states scales
|
||||
linearly 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> so they can support ordered
|
||||
phases.</p>
|
||||
</section>
|
||||
<section id="long-ranged-ising-model" class="level2">
|
||||
<h2>Long Ranged Ising model</h2>
|
||||
<p>Our extension to the FK model could now be though of as spinless
|
||||
fermions coupled to a long range Ising (LRI) model. The LRI model has
|
||||
been extensively studied and its behaviour may be bear relation to the
|
||||
behaviour of our modified FK model.</p>
|
||||
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">\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|)
|
||||
\tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i
|
||||
\tau_j\]</span></p>
|
||||
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 $1 < < 2 $ <span class="citation"
|
||||
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">23</a>]</span>. Peierls’ argument can be
|
||||
role="doc-biblioref">33</a>]</span>. Peierls’ argument can be
|
||||
extended <span class="citation"
|
||||
data-cites="thoulessLongRangeOrderOneDimensional1969"> [<a
|
||||
href="#ref-thoulessLongRangeOrderOneDimensional1969"
|
||||
role="doc-biblioref">24</a>]</span> to long range interactions to
|
||||
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">\(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)</span>
|
||||
class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)</span>
|
||||
and a domain wall state <span
|
||||
class="math inline">\(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\)</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
|
||||
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. 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 in the thermodynamic limit then the free energy
|
||||
is analytic <span class="citation"
|
||||
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">25</a>]</span>. This rules out a finite order phase
|
||||
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">23</a>]</span>.</p>
|
||||
role="doc-biblioref">33</a>]</span>.</p>
|
||||
<p>With a power law form for <span class="math inline">\(J(n)\)</span>,
|
||||
there are three cases to consider:</p>
|
||||
<ol type="1">
|
||||
<li>$ = 0$ For infinite range interactions the Ising model is exactly
|
||||
solveable and mean field theory is exact <span class="citation"
|
||||
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">26</a>]</span>.</li>
|
||||
<li>$ $ For slowly decaying interactions <span
|
||||
class="math inline">\(\sum_n J(n)\)</span> does not converge so the
|
||||
Hamiltonian is non-extensive, a case which won’t be further considered
|
||||
here.</li>
|
||||
<li>$ 1 < < 2 $ A phase transition to an ordered state at a finite
|
||||
temperature.</li>
|
||||
<li>$ = 2 $ The energy of domain walls diverges logarithmically, and
|
||||
this turns out to be a Kostelitz-Thouless transition <span
|
||||
class="citation"
|
||||
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">24</a>]</span>.</li>
|
||||
<li>$ 2 < $ For quickly decaying interactions, domain walls have a
|
||||
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.</li>
|
||||
</ol>
|
||||
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">
|
||||
<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: " />
|
||||
<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
|
||||
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>
|
||||
</figure>
|
||||
</div>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliography" class="level1 unnumbered">
|
||||
@ -388,16 +476,24 @@ href="http://adsabs.harvard.edu/abs/2001AcPPB..32.3243J">Falicov--Kimball
|
||||
Models of Collective Phenomena in Solids (A Concise Guide)</a></em>,
|
||||
Acta Physica Polonica B <strong>32</strong>, 3243 (2001).</div>
|
||||
</div>
|
||||
<div id="ref-gruberFalicovKimballModel2005" class="csl-entry"
|
||||
<div id="ref-gruberGroundStatesSpinless1990" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </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>
|
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|
||||
<div id="ref-wangCommentNonextensiveHamiltonian2003" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">Q.
|
||||
A. Wang, <em><a
|
||||
href="https://hal.archives-ouvertes.fr/hal-00009462">Comment on
|
||||
<span>“Nonextensive Hamiltonian Systems Follow Boltzmann’s Principle Not
|
||||
Tsallis Statistics-Phase Transition, Second Law of
|
||||
Thermodynamics”</span> by Gross</a></em>, (2003).</div>
|
||||
</div>
|
||||
<div id="ref-fisherCriticalExponentsLongRange1972" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[39] </div><div class="csl-right-inline">M.
|
||||
E. Fisher, S. Ma, and B. G. Nickel, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.29.917">Critical Exponents for
|
||||
Long-Range Interactions</a></em>, Phys. Rev. Lett. <strong>29</strong>,
|
||||
917 (1972).</div>
|
||||
</div>
|
||||
<div id="ref-angeliniRelationsShortrangeLongrange2014" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<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>
|
||||
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: 2.2_HKM_Model
|
||||
title: Background - The Kitaev Honeycomb Model
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>2.2_HKM_Model</title>
|
||||
<title>Background - The Kitaev Honeycomb Model</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
@ -30,7 +30,7 @@ image:
|
||||
<li><a href="#the-kitaev-honeycomb-model"
|
||||
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
|
||||
<ul>
|
||||
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
|
||||
<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>
|
||||
@ -57,7 +57,7 @@ Diagram</a></li>
|
||||
<li><a href="#the-kitaev-honeycomb-model"
|
||||
id="toc-the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</a>
|
||||
<ul>
|
||||
<li><a href="#the-model" id="toc-the-model">The Model</a></li>
|
||||
<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>
|
||||
@ -88,7 +88,7 @@ with long range entanglement (not simple paramagnet)</p>
|
||||
<li>experimental probes include inelastic neutron scattering, Raman
|
||||
scattering</li>
|
||||
</ul>
|
||||
<section id="the-model" class="level2">
|
||||
<section id="bg-hkm-model" class="level2">
|
||||
<h2>The Model</h2>
|
||||
<div id="fig:intro_figure_by_hand" class="fignos">
|
||||
<figure>
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: 2.3_Disorder
|
||||
title: Background - Disorder & Localisation
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>2.3_Disorder</title>
|
||||
<title>Background - Disorder & Localisation</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
@ -27,8 +27,8 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#disorder-and-localisation"
|
||||
id="toc-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<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:
|
||||
@ -52,8 +52,8 @@ id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#disorder-and-localisation"
|
||||
id="toc-disorder-and-localisation">Disorder and Localisation</a>
|
||||
<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:
|
||||
@ -70,7 +70,7 @@ id="toc-amorphous-magnetism">Amorphous Magnetism</a></li>
|
||||
-->
|
||||
|
||||
<!-- Main Page Body -->
|
||||
<section id="disorder-and-localisation" class="level1">
|
||||
<section id="bg-disorder-and-localisation" class="level1">
|
||||
<h1>Disorder and Localisation</h1>
|
||||
<section id="localisation-anderson-many-body-and-disorder-free"
|
||||
class="level2">
|
||||
|
||||
@ -27,7 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The 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>
|
||||
</nav>
|
||||
@ -41,7 +41,7 @@ image:
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-Model" id="toc-sec:FK-Model">The 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>
|
||||
</nav>
|
||||
@ -72,13 +72,13 @@ analysis presented here.</p>
|
||||
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 id="sec:FK-Model" class="level1">
|
||||
<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 delocalization <span class="citation"
|
||||
induce delocalisation <span class="citation"
|
||||
data-cites="aubryAnalyticityBreakingAnderson1980 dassarmaLocalizationMobilityEdges1990 dunlapAbsenceLocalizationRandomdimer1990"> [<a
|
||||
href="#ref-aubryAnalyticityBreakingAnderson1980"
|
||||
role="doc-biblioref">3</a>–<a
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: 3.2_LRFK_Methods
|
||||
title: The Long Range Falikov-Kimball Model - Methods
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>3.2_LRFK_Methods</title>
|
||||
<title>The Long Range Falikov-Kimball Model - Methods</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
@ -27,7 +27,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a>
|
||||
<li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#markov-chain-monte-carlo"
|
||||
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
@ -116,7 +116,7 @@ Trick</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-Methods" id="toc-sec:FK-Methods">Methods</a>
|
||||
<li><a href="#fk-methods" id="toc-fk-methods">Methods</a>
|
||||
<ul>
|
||||
<li><a href="#markov-chain-monte-carlo"
|
||||
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
@ -198,7 +198,7 @@ Trick</a></li>
|
||||
-->
|
||||
|
||||
<!-- Main Page Body -->
|
||||
<section id="sec:FK-Methods" class="level1">
|
||||
<section id="fk-methods" class="level1">
|
||||
<h1>Methods</h1>
|
||||
<section id="markov-chain-monte-carlo" class="level2">
|
||||
<h2>Markov Chain Monte Carlo</h2>
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: 3.3_LRFK_Results
|
||||
title: The Long Range Falikov-Kimball Model - Results
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>3.3_LRFK_Results</title>
|
||||
<title>The Long Range Falikov-Kimball Model - Results</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
@ -27,16 +27,15 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a>
|
||||
<li><a href="#fk-results" id="toc-fk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</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="#discussion-and-conclusion-secamk-conclusion"
|
||||
id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
|
||||
Conclusion {sec:AMK-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>
|
||||
@ -50,23 +49,22 @@ Conclusion {sec:AMK-Conclusion}</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:FK-results" id="toc-sec:FK-results">Results</a>
|
||||
<li><a href="#fk-results" id="toc-fk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</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="#discussion-and-conclusion-secamk-conclusion"
|
||||
id="toc-discussion-and-conclusion-secamk-conclusion">Discussion and
|
||||
Conclusion {sec:AMK-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>
|
||||
-->
|
||||
|
||||
<!-- Main Page Body -->
|
||||
<section id="sec:FK-results" class="level1">
|
||||
<section id="fk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<div id="fig:phase_diagram" class="fignos">
|
||||
<figure>
|
||||
@ -115,7 +113,7 @@ alt="Figure 2: Hello I am the figure caption!" />
|
||||
figure caption!</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<section id="phase-diagram" class="level2">
|
||||
<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
|
||||
@ -402,9 +400,8 @@ model is that we can explore very large system sizes for a complete
|
||||
understanding.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="discussion-and-conclusion-secamk-conclusion"
|
||||
class="level1">
|
||||
<h1>Discussion and Conclusion {sec:AMK-Conclusion}</h1>
|
||||
<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
|
||||
|
||||
@ -1048,21 +1048,21 @@ ground states in the non-Abelian phase <span class="math inline">\((+1,
|
||||
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
|
||||
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>,<a
|
||||
href="#ref-chungTopologicalQuantumPhase2010"
|
||||
role="doc-biblioref">15</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>
|
||||
role="doc-biblioref"><strong>chungTopologicalQuantumPhase2010?</strong></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="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
|
||||
href="#ref-poulinStabilizerFormalismOperator2005"
|
||||
role="doc-biblioref">16</a>,<a
|
||||
role="doc-biblioref">15</a>,<a
|
||||
href="#ref-hastingsDynamicallyGeneratedLogical2021"
|
||||
role="doc-biblioref">17</a>,<a
|
||||
role="doc-biblioref">16</a>,<a
|
||||
href="#ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
|
||||
</section>
|
||||
@ -1173,18 +1173,9 @@ 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-chungTopologicalQuantumPhase2010" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[15] </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-poulinStabilizerFormalismOperator2005" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">D.
|
||||
<div class="csl-left-margin">[15] </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.
|
||||
@ -1192,7 +1183,7 @@ Lett. <strong>95</strong>, 230504 (2005).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">M.
|
||||
<div class="csl-left-margin">[16] </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>
|
||||
|
||||
@ -28,7 +28,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
|
||||
<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>
|
||||
@ -70,7 +70,7 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Model" id="toc-sec:AMK-Model">The Model</a>
|
||||
<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>
|
||||
@ -107,24 +107,32 @@ id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
<!-- Main Page Body -->
|
||||
<p><strong>Contributions</strong></p>
|
||||
<p>The material in this chapter expands on work presented in</p>
|
||||
<p><strong>Insert citation of amorphous Kitaev paper here</strong></p>
|
||||
<p>which was a joint project of the first three authors with advice and
|
||||
<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
|
||||
Gino, Peru and I had in studying amorphous systems, coupled with
|
||||
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">1</a>]</span> and Gino did the implementation of
|
||||
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 came up with the ground state
|
||||
conjecture 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="sec:AMK-Model" class="level1">
|
||||
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>
|
||||
@ -160,7 +168,7 @@ 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">2</a>,<a href="#ref-trebstKitaevMaterials2022"
|
||||
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>
|
||||
@ -172,14 +180,14 @@ 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">3</a>]</span>.</p>
|
||||
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">4</a>]</span>. The solubility of the Kitaev
|
||||
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
|
||||
@ -197,7 +205,7 @@ 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">5</a>]</span>.</p>
|
||||
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
|
||||
@ -395,7 +403,7 @@ 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">4</a>]</span>. For notational brevity, it is useful
|
||||
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
|
||||
@ -635,7 +643,7 @@ 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">6</a>]</span>.</p>
|
||||
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
|
||||
@ -762,9 +770,23 @@ class="math inline">\(b^\alpha\)</span> operators could be performed.
|
||||
<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>
|
||||
<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">[1] </div><div class="csl-right-inline">Q.
|
||||
<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
|
||||
@ -772,7 +794,7 @@ of Sciences <strong>117</strong>, 30260 (2020).</div>
|
||||
</div>
|
||||
<div id="ref-banerjeeProximateKitaevQuantum2016" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">A.
|
||||
<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
|
||||
@ -780,7 +802,7 @@ Liquid Behaviour in {\Alpha}-RuCl$_3$</a></em>, Nature Mater
|
||||
</div>
|
||||
<div id="ref-freedmanTopologicalQuantumComputation2003"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M.
|
||||
<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
|
||||
@ -788,20 +810,20 @@ Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
|
||||
</div>
|
||||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">A.
|
||||
<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>
|
||||
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">Z.
|
||||
<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">[6] </div><div
|
||||
<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>
|
||||
|
||||
@ -28,7 +28,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
|
||||
<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
|
||||
@ -65,7 +65,7 @@ Markers</a></li>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Methods" id="toc-sec:AMK-Methods">Methods</a>
|
||||
<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
|
||||
@ -97,7 +97,7 @@ Markers</a></li>
|
||||
<!-- Main Page Body -->
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<section id="sec:AMK-Methods" class="level1">
|
||||
<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
|
||||
|
||||
@ -28,7 +28,7 @@ image:
|
||||
<br>
|
||||
<nav aria-label="Table of Contents" class="page-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
|
||||
<li><a href="#amk-results" id="toc-amk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#the-ground-state-flux-sector"
|
||||
id="toc-the-ground-state-flux-sector">The Ground State Flux
|
||||
@ -52,8 +52,9 @@ Thermal Metal</a></li>
|
||||
and Conclusion</a>
|
||||
<ul>
|
||||
<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
|
||||
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
|
||||
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
|
||||
<li><a href="#amk-discussion"
|
||||
id="toc-amk-discussion">Discussion</a></li>
|
||||
<li><a href="#amk-outlook" id="toc-amk-outlook">Outlook</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -68,7 +69,7 @@ and Conclusion</a>
|
||||
<!-- Table of Contents -->
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#sec:AMK-Results" id="toc-sec:AMK-Results">Results</a>
|
||||
<li><a href="#amk-results" id="toc-amk-results">Results</a>
|
||||
<ul>
|
||||
<li><a href="#the-ground-state-flux-sector"
|
||||
id="toc-the-ground-state-flux-sector">The Ground State Flux
|
||||
@ -92,8 +93,9 @@ Thermal Metal</a></li>
|
||||
and Conclusion</a>
|
||||
<ul>
|
||||
<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
|
||||
<li><a href="#discussion" id="toc-discussion">Discussion</a></li>
|
||||
<li><a href="#outlook" id="toc-outlook">Outlook</a></li>
|
||||
<li><a href="#amk-discussion"
|
||||
id="toc-amk-discussion">Discussion</a></li>
|
||||
<li><a href="#amk-outlook" id="toc-amk-outlook">Outlook</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
@ -101,7 +103,7 @@ and Conclusion</a>
|
||||
-->
|
||||
|
||||
<!-- Main Page Body -->
|
||||
<section id="sec:AMK-Results" class="level1">
|
||||
<section id="amk-results" class="level1">
|
||||
<h1>Results</h1>
|
||||
<section id="the-ground-state-flux-sector" class="level2">
|
||||
<h2>The Ground State Flux Sector</h2>
|
||||
@ -562,7 +564,7 @@ spin liquid phase.</p>
|
||||
Anderson transition to a thermal metal phase, driven by the
|
||||
proliferation of vortices with increasing temperature.</p>
|
||||
</section>
|
||||
<section id="discussion" class="level2">
|
||||
<section id="amk-discussion" class="level2">
|
||||
<h2>Discussion</h2>
|
||||
<p><strong>Limits of the ground state conjecture</strong></p>
|
||||
<p>We found a small number of lattices for which the ground state
|
||||
@ -581,7 +583,7 @@ code A phase. It is possible that some property of the particular
|
||||
colouring chosen is what leads to failure of the ground state conjecture
|
||||
here.</p>
|
||||
</section>
|
||||
<section id="outlook" class="level2">
|
||||
<section id="amk-outlook" class="level2">
|
||||
<h2>Outlook</h2>
|
||||
<p>This exactly solvable chiral QSL provides a first example of a
|
||||
topological quantum many-body phase in amorphous magnets, which raises a
|
||||
@ -946,7 +948,7 @@ Its Application to the Cuprous Oxides</a></em>, Rev. Mod. Phys.
|
||||
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[38] </div><div class="csl-right-inline">G.
|
||||
Baskaran, D. Sen, and R. Shankar, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
|
||||
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin- S Kitaev Model:
|
||||
Classical Ground States, Order from Disorder, and Exact Correlation
|
||||
Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
|
||||
(2008).</div>
|
||||
|
||||
@ -1,205 +0,0 @@
|
||||
---
|
||||
title: A.1_Markov_Chain_Monte_Carlo
|
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excerpt:
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layout: none
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image:
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---
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|
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||||
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
|
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-->
|
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|
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
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|
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|
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|
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<script src="/assets/js/index.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
|
||||
<!--Capture the table of contents from pandoc as a jekyll variable -->
|
||||
{% capture tableOfContents %}
|
||||
<br>
|
||||
Contents:
|
||||
<ul>
|
||||
<li><a href="#markov-chain-monte-carlo"
|
||||
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
<!-- Give the table of contents to header as a variable -->
|
||||
{% include header.html extra=tableOfContents %}
|
||||
|
||||
<main>
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#markov-chain-monte-carlo"
|
||||
id="toc-markov-chain-monte-carlo">Markov Chain Monte Carlo</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
<h1 id="markov-chain-monte-carlo">Markov Chain Monte Carlo</h1>
|
||||
</main>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: A.1_Particle_Hole_Symmetry
|
||||
title: Particle-Hole Symmetry
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.1_Particle_Hole_Symmetry</title>
|
||||
<title>Particle-Hole Symmetry</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
@ -1,270 +0,0 @@
|
||||
---
|
||||
title: A.2_Lattice_Generation
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
|
||||
---
|
||||
<!DOCTYPE html>
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
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<head>
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<meta charset="utf-8" />
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<meta name="generator" content="pandoc" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
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<title>A.2_Lattice_Generation</title>
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table {
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font-variant-numeric: lining-nums tabular-nums;
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table caption {
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tbody {
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th {
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td {
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header {
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margin-bottom: 4em;
|
||||
text-align: center;
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}
|
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#TOC li {
|
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list-style: none;
|
||||
}
|
||||
#TOC ul {
|
||||
padding-left: 1.3em;
|
||||
}
|
||||
#TOC > ul {
|
||||
padding-left: 0;
|
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}
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#TOC a:not(:hover) {
|
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text-decoration: none;
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code{white-space: pre-wrap;}
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|
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|
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|
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padding: 0 4px; width: 4em;
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|
||||
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
|
||||
div.sourceCode
|
||||
{ }
|
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@media screen {
|
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pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
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|
||||
</style> -->
|
||||
|
||||
<!-- -->
|
||||
|
||||
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
||||
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
|
||||
-->
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
|
||||
<!--[if lt IE 9]>
|
||||
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
|
||||
<![endif]-->
|
||||
<link rel="stylesheet" href="/assets/css/styles.css">
|
||||
<script src="/assets/js/index.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
|
||||
<!--Capture the table of contents from pandoc as a jekyll variable -->
|
||||
{% capture tableOfContents %}
|
||||
<br>
|
||||
Contents:
|
||||
<ul>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
|
||||
Generation</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
<!-- Give the table of contents to header as a variable -->
|
||||
{% include header.html extra=tableOfContents %}
|
||||
|
||||
<main>
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#lattice-generation" id="toc-lattice-generation">Lattice
|
||||
Generation</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
<h1 id="lattice-generation">Lattice Generation</h1>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</main>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: A.2_Markov_Chain_Monte_Carlo
|
||||
title: Markov Chain Monte Carlo
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.2_Markov_Chain_Monte_Carlo</title>
|
||||
<title>Markov Chain Monte Carlo</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
@ -1,270 +0,0 @@
|
||||
---
|
||||
title: A.3_Lattice_Colouring
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
|
||||
---
|
||||
<!DOCTYPE html>
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.3_Lattice_Colouring</title>
|
||||
<!-- <style>
|
||||
html {
|
||||
line-height: 1.5;
|
||||
font-family: Georgia, serif;
|
||||
font-size: 20px;
|
||||
color: #1a1a1a;
|
||||
background-color: #fdfdfd;
|
||||
}
|
||||
body {
|
||||
margin: 0 auto;
|
||||
max-width: 36em;
|
||||
padding-left: 50px;
|
||||
padding-right: 50px;
|
||||
padding-top: 50px;
|
||||
padding-bottom: 50px;
|
||||
hyphens: auto;
|
||||
overflow-wrap: break-word;
|
||||
text-rendering: optimizeLegibility;
|
||||
font-kerning: normal;
|
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|
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|
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font-size: 0.9em;
|
||||
padding: 1em;
|
||||
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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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|
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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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|
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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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|
||||
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|
||||
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|
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font-size: 1em;
|
||||
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|
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|
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|
||||
font-weight: normal;
|
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|
||||
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|
||||
padding-left: 1.7em;
|
||||
margin-top: 1em;
|
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|
||||
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|
||||
margin-top: 0;
|
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|
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blockquote {
|
||||
margin: 1em 0 1em 1.7em;
|
||||
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|
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|
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font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
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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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|
||||
padding: 0;
|
||||
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|
||||
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|
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|
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.sourceCode {
|
||||
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|
||||
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|
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|
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|
||||
background-color: #1a1a1a;
|
||||
border: none;
|
||||
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|
||||
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|
||||
}
|
||||
table {
|
||||
margin: 1em 0;
|
||||
border-collapse: collapse;
|
||||
width: 100%;
|
||||
overflow-x: auto;
|
||||
display: block;
|
||||
font-variant-numeric: lining-nums tabular-nums;
|
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}
|
||||
table caption {
|
||||
margin-bottom: 0.75em;
|
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}
|
||||
tbody {
|
||||
margin-top: 0.5em;
|
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border-top: 1px solid #1a1a1a;
|
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border-bottom: 1px solid #1a1a1a;
|
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}
|
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th {
|
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border-top: 1px solid #1a1a1a;
|
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|
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|
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td {
|
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padding: 0.125em 0.5em 0.25em 0.5em;
|
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}
|
||||
header {
|
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margin-bottom: 4em;
|
||||
text-align: center;
|
||||
}
|
||||
#TOC li {
|
||||
list-style: none;
|
||||
}
|
||||
#TOC ul {
|
||||
padding-left: 1.3em;
|
||||
}
|
||||
#TOC > ul {
|
||||
padding-left: 0;
|
||||
}
|
||||
#TOC a:not(:hover) {
|
||||
text-decoration: none;
|
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}
|
||||
code{white-space: pre-wrap;}
|
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span.smallcaps{font-variant: small-caps;}
|
||||
span.underline{text-decoration: underline;}
|
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div.column{display: inline-block; vertical-align: top; width: 50%;}
|
||||
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
|
||||
ul.task-list{list-style: none;}
|
||||
pre > code.sourceCode { white-space: pre; position: relative; }
|
||||
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
|
||||
pre > code.sourceCode > span:empty { height: 1.2em; }
|
||||
.sourceCode { overflow: visible; }
|
||||
code.sourceCode > span { color: inherit; text-decoration: inherit; }
|
||||
div.sourceCode { margin: 1em 0; }
|
||||
pre.sourceCode { margin: 0; }
|
||||
@media screen {
|
||||
div.sourceCode { overflow: auto; }
|
||||
}
|
||||
@media print {
|
||||
pre > code.sourceCode { white-space: pre-wrap; }
|
||||
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
|
||||
}
|
||||
pre.numberSource code
|
||||
{ counter-reset: source-line 0; }
|
||||
pre.numberSource code > span
|
||||
{ position: relative; left: -4em; counter-increment: source-line; }
|
||||
pre.numberSource code > span > a:first-child::before
|
||||
{ content: counter(source-line);
|
||||
position: relative; left: -1em; text-align: right; vertical-align: baseline;
|
||||
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|
||||
-webkit-touch-callout: none; -webkit-user-select: none;
|
||||
-khtml-user-select: none; -moz-user-select: none;
|
||||
-ms-user-select: none; user-select: none;
|
||||
padding: 0 4px; width: 4em;
|
||||
color: #aaaaaa;
|
||||
}
|
||||
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
|
||||
div.sourceCode
|
||||
{ }
|
||||
@media screen {
|
||||
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
|
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}
|
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code span.al { color: #ff0000; font-weight: bold; } /* Alert */
|
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code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
|
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code span.at { color: #7d9029; } /* Attribute */
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code span.bn { color: #40a070; } /* BaseN */
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|
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|
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|
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|
||||
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|
||||
|
||||
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
||||
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
|
||||
-->
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
|
||||
<!--[if lt IE 9]>
|
||||
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
|
||||
<![endif]-->
|
||||
<link rel="stylesheet" href="/assets/css/styles.css">
|
||||
<script src="/assets/js/index.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
|
||||
<!--Capture the table of contents from pandoc as a jekyll variable -->
|
||||
{% capture tableOfContents %}
|
||||
<br>
|
||||
Contents:
|
||||
<ul>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
|
||||
Colouring</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
<!-- Give the table of contents to header as a variable -->
|
||||
{% include header.html extra=tableOfContents %}
|
||||
|
||||
<main>
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#lattice-colouring" id="toc-lattice-colouring">Lattice
|
||||
Colouring</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
<h1 id="lattice-colouring">Lattice Colouring</h1>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
</main>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: A.3_Lattice_Generation
|
||||
title: Lattice Generation
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.3_Lattice_Generation</title>
|
||||
<title>Lattice Generation</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: A.4_Lattice_Colouring
|
||||
title: Lattice Colouring
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.4_Lattice_Colouring</title>
|
||||
<title>Lattice Colouring</title>
|
||||
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
@ -1,205 +0,0 @@
|
||||
---
|
||||
title: A.4_The_Projector
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
|
||||
---
|
||||
<!DOCTYPE html>
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
|
||||
<title>A.4_The_Projector</title>
|
||||
<!-- <style>
|
||||
html {
|
||||
line-height: 1.5;
|
||||
font-family: Georgia, serif;
|
||||
font-size: 20px;
|
||||
color: #1a1a1a;
|
||||
background-color: #fdfdfd;
|
||||
}
|
||||
body {
|
||||
margin: 0 auto;
|
||||
max-width: 36em;
|
||||
padding-left: 50px;
|
||||
padding-right: 50px;
|
||||
padding-top: 50px;
|
||||
padding-bottom: 50px;
|
||||
hyphens: auto;
|
||||
overflow-wrap: break-word;
|
||||
text-rendering: optimizeLegibility;
|
||||
font-kerning: normal;
|
||||
}
|
||||
@media (max-width: 600px) {
|
||||
body {
|
||||
font-size: 0.9em;
|
||||
padding: 1em;
|
||||
}
|
||||
h1 {
|
||||
font-size: 1.8em;
|
||||
}
|
||||
}
|
||||
@media print {
|
||||
body {
|
||||
background-color: transparent;
|
||||
color: black;
|
||||
font-size: 12pt;
|
||||
}
|
||||
p, h2, h3 {
|
||||
orphans: 3;
|
||||
widows: 3;
|
||||
}
|
||||
h2, h3, h4 {
|
||||
page-break-after: avoid;
|
||||
}
|
||||
}
|
||||
p {
|
||||
margin: 1em 0;
|
||||
}
|
||||
a {
|
||||
color: #1a1a1a;
|
||||
}
|
||||
a:visited {
|
||||
color: #1a1a1a;
|
||||
}
|
||||
img {
|
||||
max-width: 100%;
|
||||
}
|
||||
h1, h2, h3, h4, h5, h6 {
|
||||
margin-top: 1.4em;
|
||||
}
|
||||
h5, h6 {
|
||||
font-size: 1em;
|
||||
font-style: italic;
|
||||
}
|
||||
h6 {
|
||||
font-weight: normal;
|
||||
}
|
||||
ol, ul {
|
||||
padding-left: 1.7em;
|
||||
margin-top: 1em;
|
||||
}
|
||||
li > ol, li > ul {
|
||||
margin-top: 0;
|
||||
}
|
||||
blockquote {
|
||||
margin: 1em 0 1em 1.7em;
|
||||
padding-left: 1em;
|
||||
border-left: 2px solid #e6e6e6;
|
||||
color: #606060;
|
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}
|
||||
code {
|
||||
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
|
||||
font-size: 85%;
|
||||
margin: 0;
|
||||
}
|
||||
pre {
|
||||
margin: 1em 0;
|
||||
overflow: auto;
|
||||
}
|
||||
pre code {
|
||||
padding: 0;
|
||||
overflow: visible;
|
||||
overflow-wrap: normal;
|
||||
}
|
||||
.sourceCode {
|
||||
background-color: transparent;
|
||||
overflow: visible;
|
||||
}
|
||||
hr {
|
||||
background-color: #1a1a1a;
|
||||
border: none;
|
||||
height: 1px;
|
||||
margin: 1em 0;
|
||||
}
|
||||
table {
|
||||
margin: 1em 0;
|
||||
border-collapse: collapse;
|
||||
width: 100%;
|
||||
overflow-x: auto;
|
||||
display: block;
|
||||
font-variant-numeric: lining-nums tabular-nums;
|
||||
}
|
||||
table caption {
|
||||
margin-bottom: 0.75em;
|
||||
}
|
||||
tbody {
|
||||
margin-top: 0.5em;
|
||||
border-top: 1px solid #1a1a1a;
|
||||
border-bottom: 1px solid #1a1a1a;
|
||||
}
|
||||
th {
|
||||
border-top: 1px solid #1a1a1a;
|
||||
padding: 0.25em 0.5em 0.25em 0.5em;
|
||||
}
|
||||
td {
|
||||
padding: 0.125em 0.5em 0.25em 0.5em;
|
||||
}
|
||||
header {
|
||||
margin-bottom: 4em;
|
||||
text-align: center;
|
||||
}
|
||||
#TOC li {
|
||||
list-style: none;
|
||||
}
|
||||
#TOC ul {
|
||||
padding-left: 1.3em;
|
||||
}
|
||||
#TOC > ul {
|
||||
padding-left: 0;
|
||||
}
|
||||
#TOC a:not(:hover) {
|
||||
text-decoration: none;
|
||||
}
|
||||
code{white-space: pre-wrap;}
|
||||
span.smallcaps{font-variant: small-caps;}
|
||||
span.underline{text-decoration: underline;}
|
||||
div.column{display: inline-block; vertical-align: top; width: 50%;}
|
||||
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
|
||||
ul.task-list{list-style: none;}
|
||||
</style> -->
|
||||
|
||||
<!-- -->
|
||||
|
||||
<!-- <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
||||
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3.0.1/es5/tex-mml-chtml.js"></script>
|
||||
-->
|
||||
|
||||
<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
|
||||
|
||||
|
||||
<!--[if lt IE 9]>
|
||||
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
|
||||
<![endif]-->
|
||||
<link rel="stylesheet" href="/assets/css/styles.css">
|
||||
<script src="/assets/js/index.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
|
||||
<!--Capture the table of contents from pandoc as a jekyll variable -->
|
||||
{% capture tableOfContents %}
|
||||
<br>
|
||||
Contents:
|
||||
<ul>
|
||||
<li><a href="#the-projector" id="toc-the-projector">The
|
||||
Projector</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
<!-- Give the table of contents to header as a variable -->
|
||||
{% include header.html extra=tableOfContents %}
|
||||
|
||||
<main>
|
||||
<!-- <nav id="TOC" role="doc-toc">
|
||||
<ul>
|
||||
<li><a href="#the-projector" id="toc-the-projector">The
|
||||
Projector</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
<h1 id="the-projector">The Projector</h1>
|
||||
</main>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,5 +1,5 @@
|
||||
---
|
||||
title: A.5_The_Projector
|
||||
title: The Projector
|
||||
excerpt:
|
||||
layout: none
|
||||
image:
|
||||
@ -11,7 +11,7 @@ image:
|
||||
<meta charset="utf-8" />
|
||||
<meta name="generator" content="pandoc" />
|
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<title>The Projector</title>
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<br>
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@ -41,14 +41,14 @@ Projector</a></li>
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---
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title: Introduction
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excerpt: Why do we do Condensed Matter theory at all?
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<script src="/assets/mathjax/tex-mml-svg.js" id="MathJax-script" async></script>
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<body>
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{% include header.html %}
|
||||
|
||||
<main>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="op">%%</span>html</span></code></pre></div>
|
||||
</main>
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</html>
|
||||
@ -4,7 +4,6 @@
|
||||
<li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#mott-insulators">Mott Insulators</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#quantum-spin-liquids">Quantum Spin Liquids</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#outline">Outline</a></li>
|
||||
</ul>
|
||||
<li><a href="./2_Background/2.1_FK_Model.html#the-falikov-kimball-model">Background</a></li>
|
||||
<ul>
|
||||
@ -27,14 +26,10 @@
|
||||
<li><a href="./4_Amorphous_Kitaev_Model/4.3_AMK_Results.html#discussion-and-conclusion">Discussion and Conclusion</a></li>
|
||||
</ul>
|
||||
<li><a href="./5_Conclusion/5_Conclusion.html#discussion">Conclusion</a></li>
|
||||
<li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Appendices</a></li>
|
||||
<li><a href="./6_Appendices/A.1_Particle_Hole_Symmetry.html#particle-hole-symmetry">Appendices</a></li>
|
||||
<ul>
|
||||
<li><a href="./6_Appendices/A.1_Markov_Chain_Monte_Carlo.html#markov-chain-monte-carlo">Markov Chain Monte Carlo</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_Lattice_Generation.html#lattice-generation">Lattice Generation</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.3_Lattice_Colouring.html#lattice-colouring">Lattice Colouring</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.4_The_Projector.html#the-projector">The Projector</a></li>
|
||||
<li><a href="./6_Appendices/A.5_The_Projector.html#the-projector">The Projector</a></li>
|
||||
|
||||
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