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1109 lines
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---
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title: The Amorphous Kitaev Model - Results
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excerpt: The Amorphous Kitaev model is a chiral spin liquid!
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{% include header.html %}
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<main>
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<nav id="TOC" role="doc-toc">
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<ul>
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<li><a href="#results" id="toc-results">Results</a>
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<ul>
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<li><a href="#the-ground-state-flux-sector"
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id="toc-the-ground-state-flux-sector">The Ground State Flux
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Sector</a></li>
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<li><a href="#spontaneous-chiral-symmetry-breaking"
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id="toc-spontaneous-chiral-symmetry-breaking">Spontaneous Chiral
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Symmetry Breaking</a></li>
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<li><a href="#ground-state-phase-diagram"
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id="toc-ground-state-phase-diagram">Ground State Phase Diagram</a>
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<ul>
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<li><a href="#is-it-abelian-or-non-abelian"
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id="toc-is-it-abelian-or-non-abelian">Is it Abelian or
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non-Abelian?</a></li>
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<li><a href="#chern-number-and-edge-modes"
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id="toc-chern-number-and-edge-modes">Chern Number and Edge
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Modes</a></li>
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</ul></li>
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<li><a href="#anderson-transition-to-a-thermal-metal"
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id="toc-anderson-transition-to-a-thermal-metal">Anderson Transition to a
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Thermal Metal</a></li>
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</ul></li>
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<li><a href="#conclusion" id="toc-conclusion">Conclusion</a></li>
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<li><a href="#discussion" id="toc-discussion">Discussion</a>
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<ul>
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<li><a href="#failure-of-the-ground-state-conjecture"
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id="toc-failure-of-the-ground-state-conjecture">Failure of the ground
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state conjecture</a></li>
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<li><a href="#full-monte-carlo" id="toc-full-monte-carlo">Full Monte
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Carlo</a></li>
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</ul></li>
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<li><a href="#outlook" id="toc-outlook">Outlook</a>
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<ul>
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<li><a href="#experimental-realisations-and-signatures"
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id="toc-experimental-realisations-and-signatures">Experimental
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Realisations and Signatures</a></li>
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<li><a href="#generalisations"
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id="toc-generalisations">Generalisations</a></li>
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</ul></li>
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</ul>
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</nav>
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<h1 id="results">Results</h1>
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<h2 id="the-ground-state-flux-sector">The Ground State Flux Sector</h2>
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<p>Here I will discuss the numerical evidence that our guess for the
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ground state flux sector is correct, it relies on three key numerical
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observations arguments:</p>
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<p>First we fully eumerate the flux sectors of ~25,000 periodic systems
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with a disordered unit cell of up to 16 plaquettes (<span
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class="math inline">\(2^{16-1}\)</span> sectors). Going to larger system
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sizes in impractical because of the exponential sclaling. However, as
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discussed earlier, finite size effects play a large role for small
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systems<span class="citation"
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data-cites="kitaevAnyonsExactlySolved2006"><sup><a
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href="#ref-kitaevAnyonsExactlySolved2006"
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role="doc-biblioref">1</a></sup></span>. To get around this we look at
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periodic systems with amorphous unit cells. This reduces the finite size
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effects but we can use Bloch’s theorem to diagonalise periodic systems
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with only a linear penalty in system area.</p>
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<p>Looking at periodic systems comes at the expense of removing
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longer-range disorder from our lattices so we bolster this by comparing
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the behaviour of periodic lattice with amorphous to unit cells to fully
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amorphous lattice as we scale the size of the unit cell. We show that
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the energetic effect of introducing perodicity scales away as we go to
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larger system sizes.</p>
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<p>From these two observations we argue that the results for small
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periodic systems generalise to large amorphous systems. We perform this
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analysis for both the isotropic point (<span
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class="math inline">\(J^\alpha = 1\)</span>), as well as in the toric
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code phase (<span class="math inline">\(J^x = J^y = 0.25, J^z =
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1\)</span>).</p>
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<p>In the isotropic case (<span class="math inline">\(J^\alpha =
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1\)</span>), our conjecture correctly predicted the ground state flux
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sector for all of the lattices we tested.</p>
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<p>For the toric code phase (<span class="math inline">\(J^x, J^y =
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0.25, J^z = 1\)</span>) all but around (<span class="math inline">\(\sim
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0.5 \%\)</span>) lattices had ground states conforming to our
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conjecture. In these cases, the energy difference between the true
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ground state and our prediction was on the order of <span
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class="math inline">\(10^{-6} J\)</span>. It is unclear whether this is
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a finite size effect or something else.</p>
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<h2 id="spontaneous-chiral-symmetry-breaking">Spontaneous Chiral
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Symmetry Breaking</h2>
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<p>The spin Kitaev Hamiltonian is real and therefore has time reveral
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symmetry. However, the flux <span class="math inline">\(\phi_p\)</span>
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through any plaquette with an odd number of sides has imaginary
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eigenvalues <span class="math inline">\(\pm i\)</span>. Further we have
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shown that the ground state sector induces a relatively regular pattern
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for the imaginary fluxes with only a global two-fold degeneracy.</p>
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<p>Thus, states with a fixed flux sector spontaneously break time
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reversal symmetry. This was first described by Yao and Kivelson for a
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translation invariant Kitaev model with odd sided plaquettes<span
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class="citation" data-cites="Yao2011"><sup><a href="#ref-Yao2011"
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role="doc-biblioref">2</a></sup></span>.</p>
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<p>Thus we have flux sectors that come in degenerate pairs, where time
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reversal is equivalent to inverting the flux through every odd
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plaquette, a general feature for lattices with odd plaquettes <span
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class="citation" data-cites="yaoExactChiralSpin2007 Peri2020"><sup><a
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href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">3</a>,<a
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href="#ref-Peri2020" role="doc-biblioref">4</a></sup></span>. This
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spontaneously broken symmetry avoids the need to explicitly break TRS
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with a magnetic field term as is done in the original honeycomb
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model.</p>
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<h2 id="ground-state-phase-diagram">Ground State Phase Diagram</h2>
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<p>As previously discusssed, the standard Honeycomb model has a Abelian,
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gapped phase in the anisotropic region and is gapless in the isotropic
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region. The introduction of a magnetic field breaks the chiral symmetry,
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leading to the isotropic region becoming a gapped, non-Abelian
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phase.</p>
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<p>Similar to the Kitaev Honeycomb model with a magnetic field, we find
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that this model is only gapless along critical lines, see ~<a
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href="#fig:phase_diagram">1</a> (Left). Interestingly, the gap closing
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exists in only one of the four topological sectors, though this is
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certainly a finite size effect as the sectors must become degenerate in
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the thermodynamic limit.</p>
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<p>In the honeycomb model, the phase boundaries are located on the
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straight line <span class="math inline">\(|J^x| = |J^y| + |J^x|\)</span>
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and permutations of <span class="math inline">\(x,y,z\)</span>, shown as
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dotted line on ~<a href="#fig:phase_diagram">1</a> (Right). We find that
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on the amorphous lattice these boundaries exhibit an inward curvature,
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similar to honeycomb Kitaev models with flux<span class="citation"
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data-cites="Nasu_Thermal_2015"><sup><a href="#ref-Nasu_Thermal_2015"
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role="doc-biblioref">5</a></sup></span> or bond<span class="citation"
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||
data-cites="knolle_dynamics_2016"><sup><a
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href="#ref-knolle_dynamics_2016" role="doc-biblioref">6</a></sup></span>
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disorder.</p>
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<div id="fig:phase_diagram" class="fignos">
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<figure>
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<img
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src="/assets/thesis/figure_code/amk_chapter/results/phase_diagram/phase_diagram.svg"
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style="width:100.0%"
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alt="Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting J_x + J_y + J_z = 1. This intersects a plane with the unit cube spanned by J_\alpha \in [0,1], giving a triangle with corners (1,0,0), (0,1,0), (0,0,1). To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the chern number \nu changes from 0 t0 \pm 1 from the A phases to the B phase." />
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<figcaption aria-hidden="true"><span>Figure 1:</span> (Center) We choose
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an energy scale for the Hamiltonian by setting <span
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||
class="math inline">\(J_x + J_y + J_z = 1\)</span>. This intersects a
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||
plane with the unit cube spanned by <span class="math inline">\(J_\alpha
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||
\in [0,1]\)</span>, giving a triangle with corners <span
|
||
class="math inline">\((1,0,0), (0,1,0), (0,0,1)\)</span>. To compute
|
||
critical lines efficiently in this space we evaluate the order parameter
|
||
of interest along rays shooting from the corners. The ray highlighted in
|
||
red defines the values of J used for the left figure. (Left) The fermion
|
||
gap as a function of J for an amorphous system with 20 plaquettes, where
|
||
the x axis is the position on the red line in the central figure from 0
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||
to 1. For finite size systems the four topological sectors are not
|
||
degenerate and only one of them has a true gap closing. (Right) The
|
||
Abelian <span class="math inline">\(A_\alpha\)</span> phases of the
|
||
model and the non-Abelian B phase separated by critical lines where the
|
||
fermion gap closes. Later we will show that the chern number <span
|
||
class="math inline">\(\nu\)</span> changes from <span
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||
class="math inline">\(0\)</span> t0 <span class="math inline">\(\pm
|
||
1\)</span> from the A phases to the B phase.</figcaption>
|
||
</figure>
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||
</div>
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<h3 id="is-it-abelian-or-non-abelian">Is it Abelian or non-Abelian?</h3>
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||
<p>The two phases of the amorphous model are clearly gapped, though see
|
||
later for a finite size scaling check on this.</p>
|
||
<p>The next question is: do these phases support Abelian or non-Abelian
|
||
statistics? To answer that we turn to Chern numbers and markers. As
|
||
discussed earlier the Chern number is a quantity intimately linked to
|
||
both the topological properties and the anyonic statistics of a model.
|
||
The Abelian/non-Abelian character of a model is linked to its Chern
|
||
number <strong>citation</strong>. However the Chern number is only
|
||
defined for the translation invariant case.</p>
|
||
<p>A family of generalisations to amorphous systems exist<span
|
||
class="citation"
|
||
data-cites="mitchellAmorphousTopologicalInsulators2018"><sup><a
|
||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||
role="doc-biblioref">7</a></sup></span> called local topological
|
||
markers. We use the crosshair marker<span class="citation"
|
||
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint"
|
||
role="doc-biblioref">8</a></sup></span> here to assess the
|
||
Abelian/non-Abelian character of the phases.</p>
|
||
<p>Like the honeycomb model, the amorphous model retains an Abelian
|
||
gapped phase in the anisotropic region with <span
|
||
class="math inline">\(\nu=0\)</span>. This phase is the amorphous
|
||
analogue of the abelian toric-code quantum spin liquid<span
|
||
class="citation" data-cites="kitaev_fault-tolerant_2003"><sup><a
|
||
href="#ref-kitaev_fault-tolerant_2003"
|
||
role="doc-biblioref">9</a></sup></span>.</p>
|
||
<p>The isotropic region has <span
|
||
class="math inline">\(\nu=\pm1\)</span> so is a non-Abelian chiral spin
|
||
liquid (CSL) similar to that of the Yao-Kivelson model<span
|
||
class="citation" data-cites="yaoExactChiralSpin2007"><sup><a
|
||
href="#ref-yaoExactChiralSpin2007"
|
||
role="doc-biblioref">3</a></sup></span>. Hereafter we focus our
|
||
attention on this phase.</p>
|
||
<div id="fig:phase_diagram_chern" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/results/phase_diagram_chern/phase_diagram_chern.svg"
|
||
style="width:100.0%"
|
||
alt="Figure 2: (Center) The crosshair marker8, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the center) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian." />
|
||
<figcaption aria-hidden="true"><span>Figure 2:</span> (Center) The
|
||
crosshair marker<span class="citation"
|
||
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint"
|
||
role="doc-biblioref">8</a></sup></span>, a local topological marker,
|
||
evaluated on the Amorphous Kitaev Model. The marker is defined around a
|
||
point, denoted by the dotted crosshair. Information about the local
|
||
topological properties of the system are encoded within a region around
|
||
that point. (Left) Summing these contributions up to some finite radius
|
||
(dotted line here, dotted circle in the center) gives a generalised
|
||
version of the Chern number for the system which becomes quantised in
|
||
the thermodynamic limit. The radius must be chosen large enough to
|
||
capture information about the local properties of the lattice while not
|
||
so large as to include contributions from the edge states. The isotropic
|
||
regime <span class="math inline">\(J_\alpha = 1\)</span> in red has
|
||
<span class="math inline">\(\nu = \pm 1\)</span> implying it supports
|
||
excitations with non-Abelian statistics, while the anisotropic regime in
|
||
orange has <span class="math inline">\(\nu = \pm 0\)</span> implying it
|
||
has Abelian statistics. (Right) Extending this analysis to the whole
|
||
<span class="math inline">\(J_\alpha\)</span> phase diagram with fixed
|
||
<span class="math inline">\(r = 0.3\)</span> nicely confirms that the
|
||
isotropic phase is non-Abelian.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<h3 id="chern-number-and-edge-modes">Chern Number and Edge Modes</h3>
|
||
<p>The QSLs separated by these lines are distinguished by a real-space
|
||
analogue of the Chern number<span class="citation"
|
||
data-cites="bianco_mapping_2011 Hastings_Almost_2010"><sup><a
|
||
href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>,<a
|
||
href="#ref-Hastings_Almost_2010"
|
||
role="doc-biblioref">11</a></sup></span>. A similar topological number
|
||
was discussed by Kitaev on the honeycomb lattice<span class="citation"
|
||
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
|
||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||
role="doc-biblioref">1</a></sup></span> that we shall use here with a
|
||
slight modification<span class="citation"
|
||
data-cites="peru_preprint mitchellAmorphousTopologicalInsulators2018"><sup><a
|
||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||
role="doc-biblioref">7</a>,<a href="#ref-peru_preprint"
|
||
role="doc-biblioref">8</a></sup></span>. For a choice of flux sector, we
|
||
calculate the projector <span class="math inline">\(P\)</span> onto the
|
||
negative energy eigenstates of the matrix <span
|
||
class="math inline">\(iA\)</span> defined in eqn. <a
|
||
href="#eqn:majorana_hamiltonian" data-reference-type="ref"
|
||
data-reference="eqn:majorana_hamiltonian">[eqn:majorana_hamiltonian]</a>.
|
||
The local Chern number around a point <span
|
||
class="math inline">\(\textbf{R}\)</span> in the bulk is given by <span
|
||
class="math display">\[\begin{aligned}
|
||
\nu (\textbf{R}) = 4\pi \Im \mathrm{Tr}_{\mathrm{Bulk}}
|
||
\left (
|
||
P\theta_{R_x} P \theta_{R_y} P
|
||
\right ),\end{aligned}\]</span> where <span
|
||
class="math inline">\(\theta_{R_x}\)</span> is a step function in the
|
||
<span class="math inline">\(x\)</span>-direction, with the step located
|
||
at <span class="math inline">\(x = R_x\)</span>, <span
|
||
class="math inline">\(\theta_{R_y}\)</span> is defined analogously. The
|
||
trace is taken over a region around <span
|
||
class="math inline">\(\textbf{R}\)</span> in the bulk of the material,
|
||
where care must be taken not to include any points close to the edges.
|
||
Provided that the point <span class="math inline">\(\textbf{R}\)</span>
|
||
is sufficiently far from the edges, this quantity will be very close to
|
||
quantised to the Chern number.</p>
|
||
<p>The local Chern marker distinguishes between an Abelian phase (A)
|
||
with <span class="math inline">\(\nu = 0\)</span>, and a non-Abelian (B)
|
||
phase characterized by <span class="math inline">\(\nu = \pm 1\)</span>.
|
||
The (A) phase is equivalent to the toric code on an amorphous
|
||
system<span class="citation"
|
||
data-cites="kitaev_fault-tolerant_2003"><sup><a
|
||
href="#ref-kitaev_fault-tolerant_2003"
|
||
role="doc-biblioref">9</a></sup></span>.</p>
|
||
<p>Since the (A) phase displays the "topological" degeneracy described
|
||
above, I think that "topologically trivial" is not a good term to
|
||
describe it. Another thing that I think it should be considered here.
|
||
The abelian phase is expected to have 2x4 degeneracy, where the factor
|
||
of 2 comes from time-reversal. On the other hand, the non-Abelian phase
|
||
should display 2x3 degeneracy, as discussed by<span class="citation"
|
||
data-cites="yaoExactChiralSpin2007"><sup><a
|
||
href="#ref-yaoExactChiralSpin2007"
|
||
role="doc-biblioref">3</a></sup></span>. Did you get any evidence of
|
||
this?</p>
|
||
<p>By contrast, the (B) phase is a <em>chiral spin liquid</em>, the
|
||
magnetic analogue of the fractional quantum Hall state. Topologically
|
||
protected edge modes are predicted to occur in these states on periodic
|
||
boundary conditions following the bulk-boundary correspondence<span
|
||
class="citation" data-cites="qi_general_2006"><sup><a
|
||
href="#ref-qi_general_2006" role="doc-biblioref">12</a></sup></span>.
|
||
The probability density of one such edge mode is given in <a
|
||
href="#fig:edge_modes" data-reference-type="ref"
|
||
data-reference="fig:edge_modes">1</a> (a), where it is shown to be
|
||
exponentially localised to the boundary of the system. The localization
|
||
of these modes can be quantified by their inverse participation ratio
|
||
(IPR), <span class="math display">\[\mathrm{IPR} = \int
|
||
d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\]</span> where <span
|
||
class="math inline">\(L\sim\sqrt{N}\)</span> is the characteristic
|
||
linear dimension of the amorphous lattices and <span
|
||
class="math inline">\(\tau\)</span> dimensional scaling exponent of
|
||
IPR.</p>
|
||
<p>Finally, the CSL density of states in open boundary conditions
|
||
indicates the low-energy modes within the gap of Majorana bands in <a
|
||
href="#fig:edge_modes" data-reference-type="ref"
|
||
data-reference="fig:edge_modes">1</a> (b).</p>
|
||
<p>The phase diagram of the amorphous model in <a
|
||
href="#fig:example_lattice" data-reference-type="ref"
|
||
data-reference="fig:example_lattice">[fig:example_lattice]</a>(c)
|
||
displays a reduced parameter space for the non-Abelian phase when
|
||
compared to the honeycomb model. Interestingly, similar inward
|
||
deformations of the critical lines were found on the Kitaev honeycomb
|
||
model subject to disorder by proliferating flux vortices<span
|
||
class="citation" data-cites="Nasu_Thermal_2015"><sup><a
|
||
href="#ref-Nasu_Thermal_2015" role="doc-biblioref">5</a></sup></span> or
|
||
exchange disorder<span class="citation"
|
||
data-cites="knolle_dynamics_2016"><sup><a
|
||
href="#ref-knolle_dynamics_2016"
|
||
role="doc-biblioref">6</a></sup></span>.</p>
|
||
<div id="fig:figure_2_bashed" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/figure_code/amk_paper/figure_2_bashed.svg"
|
||
style="width:57.0%" alt="Figure 3: " />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> </figcaption>
|
||
</figure>
|
||
</div>
|
||
<h2 id="anderson-transition-to-a-thermal-metal">Anderson Transition to a
|
||
Thermal Metal</h2>
|
||
<div id="fig:fermion_gap_vs_L" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/results/fermion_gap_vs_L/fermion_gap_vs_L.svg"
|
||
style="width:114.0%"
|
||
alt="Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03" />
|
||
<figcaption aria-hidden="true"><span>Figure 4:</span> Within a flux
|
||
sector, the fermion gap <span class="math inline">\(\Delta_f\)</span>
|
||
measures the energy between the fermionic ground state and the first
|
||
excited state. This graph shows the fermion gap as a function of system
|
||
size for the ground state flux sector and for a configuration of random
|
||
fluxes. We see that the disorder induced by an putting the Kitaev model
|
||
on an amorphous lattice does not close the gap in the ground state. The
|
||
gap closes in the flux disordered limit is good evidence that the system
|
||
transitions to a gapless thermal metal state at high temperature. Each
|
||
point shows an average over 100 lattice realisations. System size <span
|
||
class="math inline">\(L\)</span> is defined <span
|
||
class="math inline">\(\sqrt{N}\)</span> where N is the number of
|
||
plaquettes in the system. Error bars shown are <span
|
||
class="math inline">\(3\)</span> times the standard error of the mean.
|
||
The lines shown are fits of <span
|
||
class="math inline">\(\tfrac{\Delta_f}{J} = aL ^ b\)</span> with fit
|
||
parameters: Ground State: <span class="math inline">\(a = 0.138 \pm
|
||
0.002, b = -0.0972 \pm 0.004\)</span> Random Flux Sector: <span
|
||
class="math inline">\(a = 1.8 \pm 0.2, b = -2.21 \pm
|
||
0.03\)</span></figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>a thermal-induced Anderson transition to a thermal metal phase<span
|
||
class="citation" data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">13</a></sup></span>.</p>
|
||
<p>An Ising non-Abelian anyon is formed by Majorana zero-modes bound to
|
||
a topological defect<span class="citation"
|
||
data-cites="Beenakker2013"><sup><a href="#ref-Beenakker2013"
|
||
role="doc-biblioref">14</a></sup></span>. Interactions between anyons
|
||
are modeled by pairwise projectors whose strength absolute value decays
|
||
exponentially with the separation between the particles, and whose sign
|
||
oscillates in analogy to RKKY exchanges<span class="citation"
|
||
data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"><sup><a
|
||
href="#ref-Laumann2012" role="doc-biblioref">15</a>–<a
|
||
href="#ref-lahtinenTopologicalLiquidNucleation2012"
|
||
role="doc-biblioref">17</a></sup></span>. Disorder can induce a finite
|
||
density of anyons whose hybridization lead to a macroscopically
|
||
degenerate state known as <em>thermal metal</em><span class="citation"
|
||
data-cites="Laumann2012"><sup><a href="#ref-Laumann2012"
|
||
role="doc-biblioref">15</a></sup></span>. One instance of this phase can
|
||
be settled on the Kitaev CSL. In this case, the topological defects
|
||
correspond to the <span class="math inline">\(W_p \neq +1\)</span>
|
||
fluxes, which naturally emerge from thermal fluctuations at nonzero
|
||
temperature<span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">13</a></sup></span>.</p>
|
||
<p>We demonstrated that the amorphous CSL undergoes the same form of
|
||
Anderson transition by studying its properties as a function of
|
||
disorder. Unfortunately, we could not perform a complete study of its
|
||
properties as a function of the temperature as it was not feasible to
|
||
evaluate an ever-present boundary condition dependent factor<span
|
||
class="citation"
|
||
data-cites="pedrocchiPhysicalSolutionsKitaev2011 Zschocke_Physical_states2015"><sup><a
|
||
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
|
||
role="doc-biblioref">18</a>,<a href="#ref-Zschocke_Physical_states2015"
|
||
role="doc-biblioref">19</a></sup></span> for random networks. Instead,
|
||
we evaluated the fermionic density of states (DOS) and the IPR as a
|
||
function of the vortex density <span class="math inline">\(\rho\)</span>
|
||
as a proxy for temperature. This approximation is exact in the limits
|
||
<span class="math inline">\(T = 0\)</span> (corresponding to <span
|
||
class="math inline">\(\rho = 0\)</span>) and <span
|
||
class="math inline">\(T \to \infty\)</span> (corresponding to <span
|
||
class="math inline">\(\rho = 0.5\)</span>). At intermediate temperatures
|
||
the method neglects to include the influence of defect-defect
|
||
correlations.</p>
|
||
<p>However, such an approximation is enough to show the onset of
|
||
low-energy excitations for <span class="math inline">\(\rho \sim
|
||
10^{-2}-10^{-1}\)</span>, as displayed on the top graphic of <a
|
||
href="#fig:DOS_Oscillations" data-reference-type="ref"
|
||
data-reference="fig:DOS_Oscillations">[fig:DOS_Oscillations]</a>(a). We
|
||
characterized these gapless excitations using the dimensional scaling
|
||
exponential <span class="math inline">\(\tau\)</span> of the IPR on the
|
||
bottom graphic of the same figure. At small <span
|
||
class="math inline">\(\rho\)</span>, the states populating the gap
|
||
possess <span class="math inline">\(\tau\approx0\)</span>, indicating
|
||
that they are localised states pinned to the defects, and the system
|
||
remains insulating. At large <span class="math inline">\(\rho\)</span>,
|
||
the in-gap states merge with the bulk band and become extensive, closing
|
||
the gap, and the system transitions to a metallic phase.</p>
|
||
<p>The thermal metal DOS displays a logarithmic divergence at zero
|
||
energy and characteristic oscillations at small energies.<span
|
||
class="citation"
|
||
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">13</a>,<a href="#ref-bocquet_disordered_2000"
|
||
role="doc-biblioref">20</a></sup></span>. These features were indeed
|
||
observed by the averaged density of states in the <span
|
||
class="math inline">\(\rho = 0.5\)</span> case shown in <a
|
||
href="#fig:DOS_Oscillations" data-reference-type="ref"
|
||
data-reference="fig:DOS_Oscillations">[fig:DOS_Oscillations]</a>(b) for
|
||
amorphous lattice. We emphasize that the CSL studied here emerges
|
||
without an applied magnetic field as opposed to the CSL on the honeycomb
|
||
lattice studied in Ref.<span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">13</a></sup></span> I have the impression that <a
|
||
href="#fig:DOS_Oscillations" data-reference-type="ref"
|
||
data-reference="fig:DOS_Oscillations">[fig:DOS_Oscillations]</a>(b) on
|
||
the top is very similar to Fig. 3 of<span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">13</a></sup></span>. Maybe a more instructive
|
||
figure would be the DOS of the amorphous toric code at the infinite
|
||
temperature limit. In this case, the lack of non-Abelian anyons would be
|
||
reflected by a gap on the DOS, which would contrast nicely to the
|
||
thermal metal phase</p>
|
||
<div id="fig:figure_3_bashed" class="fignos">
|
||
<figure>
|
||
<img src="/assets/thesis/figure_code/amk_paper/figure_3_bashed.svg"
|
||
style="width:100.0%" alt="Figure 5: " />
|
||
<figcaption aria-hidden="true"><span>Figure 5:</span> </figcaption>
|
||
</figure>
|
||
</div>
|
||
<h1 id="conclusion">Conclusion</h1>
|
||
<p>We have studied an extension of the Kitaev honeycomb model to
|
||
amorphous lattices with coordination number <span
|
||
class="math inline">\(z= 3\)</span>. We found that it is able to support
|
||
two quantum spin liquid phases that can be distinguished using a
|
||
real-space generalisation of the Chern number. The presence of odd-sided
|
||
plaquettes on these lattices let to a spontaneous breaking of time
|
||
reversal symmetry, leading to the emergence of a chiral spin liquid
|
||
phase. Furthermore we found evidence that the amorphous system undergoes
|
||
an Anderson transition to a thermal metal phase, driven by the
|
||
proliferation of vortices with increasing temperature.</p>
|
||
<h1 id="discussion">Discussion</h1>
|
||
<h2 id="failure-of-the-ground-state-conjecture">Failure of the ground
|
||
state conjecture</h2>
|
||
<p>We did find a small number of lattices for the ground state
|
||
conjecture did not correctly predict the true ground state flux sector.
|
||
I see two possibilities for what could cause this.</p>
|
||
<p>Firstly it could be a a finite size effect that is somehow amplfied
|
||
by certain rare lattice configurations. It would be interesting to try
|
||
to elucidate what lattice features are present when the ground state
|
||
conjecture fails.</p>
|
||
<p>Alternatively, it might be telling that the ground state conjecture
|
||
failed in the toric code phase where the couplings are anisotropic.
|
||
Clearly the colouring does not matter much in the isotropic phase.
|
||
However an avenue that I did not explore was whether the particular
|
||
choice of colouring for a lattice affects the physical properties in the
|
||
toris code phase. It is possible that some property of the particular
|
||
colouring chosen is what leads to failure of the ground state conjecture
|
||
here.</p>
|
||
<h2 id="full-monte-carlo">Full Monte Carlo</h2>
|
||
<h1 id="outlook">Outlook</h1>
|
||
<h2 id="experimental-realisations-and-signatures">Experimental
|
||
Realisations and Signatures</h2>
|
||
<p>The next step is to search for an experimental realisation in
|
||
amorphous Kitaev materials, which can be created from crystalline ones
|
||
using several methods<span class="citation"
|
||
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"><sup><a
|
||
href="#ref-Weaire1976" role="doc-biblioref">21</a>–<a
|
||
href="#ref-Kaneyoshi2018" role="doc-biblioref">23</a></sup></span>.</p>
|
||
<p>Following the evidence for an induced chiral spin liquid phase in
|
||
crystalline Kitaev materials<span class="citation"
|
||
data-cites="Kasahara2018 Yokoi2021 Yamashita2020 Bruin2022"><sup><a
|
||
href="#ref-Kasahara2018" role="doc-biblioref">24</a>–<a
|
||
href="#ref-Bruin2022" role="doc-biblioref">27</a></sup></span>, it would
|
||
be interesting to investigate if a similar state is produced on its
|
||
amorphous counterpart.</p>
|
||
<p>Probably one way to make this theory experimentally relevant is to do
|
||
experiments on amorphous phases of Kitaev materials. These phases can be
|
||
obtained by liquifying the material and cooling it fast. Apparently,
|
||
most of crystalline magnets can be transformed into amorphous ones
|
||
through this process.</p>
|
||
<p>Besides the usual half-quantized signature on thermal Hall
|
||
effect<span class="citation"
|
||
data-cites="Kasahara2018 Yokoi2021 Yamashita2020 Bruin2022"><sup><a
|
||
href="#ref-Kasahara2018" role="doc-biblioref">24</a>–<a
|
||
href="#ref-Bruin2022" role="doc-biblioref">27</a></sup></span>, such a
|
||
CSL could be also characterized using local probes such as
|
||
spin-polarized scanning-tunneling microscopy<span class="citation"
|
||
data-cites="Feldmeier2020 Konig2020 Udagawa2021"><sup><a
|
||
href="#ref-Feldmeier2020" role="doc-biblioref">28</a>–<a
|
||
href="#ref-Udagawa2021" role="doc-biblioref">30</a></sup></span>. The
|
||
same probes would also be useful to manipulate non-Abelian anyons<span
|
||
class="citation" data-cites="Pereira2020"><sup><a
|
||
href="#ref-Pereira2020" role="doc-biblioref">31</a></sup></span>,
|
||
thereby implementing elementary operations for topological quantum
|
||
computation. Finally, the thermal metal phase can be diagnosed using
|
||
bulk heat transport measurements<span class="citation"
|
||
data-cites="Beenakker2013"><sup><a href="#ref-Beenakker2013"
|
||
role="doc-biblioref">14</a></sup></span>.</p>
|
||
<h2 id="generalisations">Generalisations</h2>
|
||
<p>This work could be generalized in several ways.</p>
|
||
<p>Introduction of symmetry allowed perturbations on the model<span
|
||
class="citation"
|
||
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"><sup><a
|
||
href="#ref-Rau2014" role="doc-biblioref">32</a>–<a
|
||
href="#ref-Winter2016" role="doc-biblioref">36</a></sup></span>.</p>
|
||
<p>Generalizations to higher-spin models in random networks with
|
||
different coordination numbers<span class="citation"
|
||
data-cites="Baskaran2008 Yao2009 Nussinov2009 Yao2011 Chua2011 Natori2020 Chulliparambil2020 Chulliparambil2021 Seifert2020 WangHaoranPRB2021 Wu2009"><sup><a
|
||
href="#ref-Yao2011" role="doc-biblioref">2</a>,<a
|
||
href="#ref-Baskaran2008" role="doc-biblioref">37</a>–<a
|
||
href="#ref-Wu2009" role="doc-biblioref">46</a></sup></span></p>
|
||
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
|
||
role="doc-bibliography">
|
||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">1. </div><div
|
||
class="csl-right-inline">Kitaev, A. <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</em>
|
||
<strong>321</strong>, 2–111 (2006-01-01, 2006).</div>
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href="https://doi.org/10.1103/PhysRevB.79.214440">Bond algebras and
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exact solvability of <span>Hamiltonians</span>: Spin
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<span>S</span>=<span><span
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<div class="csl-left-margin">41. </div><div
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two-dimensional quantum spin-orbital liquid: <span>Spectroscopic</span>
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signatures of fermionic magnons</a>. <em>Phys. Rev. Lett.</em>
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<div id="ref-Chulliparambil2020" class="csl-entry"
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<div class="csl-left-margin">42. </div><div
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M., Janssen, L. & Tu, H.-H. <a
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<em>Phys. Rev. B</em> <strong>102</strong>, 201111 (2020).</div>
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<div class="csl-left-margin">43. </div><div
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<div class="csl-left-margin">44. </div><div
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<div class="csl-left-margin">45. </div><div
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<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div>
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<div class="csl-left-margin">46. </div><div class="csl-right-inline">Wu,
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||
</div>
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||
</div>
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</main>
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