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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="#edge-modes" id="toc-edge-modes">Edge 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="#limits-of-the-ground-state-conjecture"
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id="toc-limits-of-the-ground-state-conjecture">Limits of the ground
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state conjecture</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. We will do this by enumerating all
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the flux sectors of many separate system realisations. However there are
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some issues we will need to address to make this argument work.</p>
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<p>We have two seemingly irreconcilable problems. Finite size effects
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have a large energetic contribution for small systems<span
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class="citation" data-cites="kitaevAnyonsExactlySolved2006"><sup><a
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href="#ref-kitaevAnyonsExactlySolved2006"
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role="doc-biblioref">1</a></sup></span> so we would like to perform our
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analysis for very large lattices. However for an amorphous system with
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<span class="math inline">\(N\)</span> plaquettes, <span
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class="math inline">\(2N\)</span> edges and <span
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class="math inline">\(3N\)</span> vertices we have <span
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class="math inline">\(2^{N-1}\)</span> flux sectors to check and
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diagonalisation scales with <span
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class="math inline">\(\mathcal{0}(N^3)\)</span>. That exponential
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scaling makes it infeasible to work with lattices much larger than <span
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class="math inline">\(16\)</span> plaquettes.</p>
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<p>To get around this we instead look at periodic systems with amorphous
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unit cells. For a similarly sized periodic system with <span
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class="math inline">\(A\)</span> unit cells and <span
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class="math inline">\(B\)</span> plaquettes in each unit cell where
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<span class="math inline">\(N \sim AB\)</span> things get much better.
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We can use Bloch’s theorem to diagonalise this system in about <span
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class="math inline">\(\mathcal{0}(A B^3)\)</span> operations, and more
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importantly there are only <span class="math inline">\(2^{B-1}\)</span>
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flux sectors to check.</p>
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<p>We fully enumerated the flux sectors of ~25,000 periodic systems with
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disordered unit cells of up to <span class="math inline">\(B =
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16\)</span> plaquettes and <span class="math inline">\(A = 100\)</span>
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unit cells.</p>
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<p>However, showing that our guess is correct for periodic systems with
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disordered unit cells is not quite convincing on its own. We have
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effectively removed longer-range disorder from our lattices.</p>
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<p>The second part of the argument is to show that the energetic effect
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of introducing periodicity scales away as we go to larger system sizes
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and has already diminished to a small enough value at 16 plaquettes,
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which is indeed what we find.</p>
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<p>From this we argue that the results for small periodic systems
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generalise to large amorphous systems. We perform this analysis for both
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the isotropic point (<span class="math inline">\(J^\alpha = 1\)</span>),
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as well as in the toric code phase (<span class="math inline">\(J^x =
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J^y = 0.25, J^z = 1\)</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 reversal
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symmetry (TRS). However, the flux <span
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class="math inline">\(\phi_p\)</span> through any plaquette with an odd
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number of sides has imaginary eigenvalues <span
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class="math inline">\(\pm i\)</span>. The ground state sector induces a
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relatively regular pattern for the imaginary fluxes with only a global
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two-fold chiral 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>So 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 discussed, the standard Honeycomb model has a Abelian,
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gapped phase in the anisotropic region (the A phase) and is gapless in
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the isotropic region. The introduction of a magnetic field breaks the
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chiral symmetry, leading to the isotropic region becoming a gapped,
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non-Abelian phase, the B phase.</p>
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<p>We set the energy scale by requiring that <span
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class="math inline">\(J_x + J_y + J_z = 1\)</span>, this restricts the
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3D phase space down to an equilateral triangle that is convenient for
|
||
diagrams. Imagine the cube defined by <span
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class="math inline">\(J_\alpha \in [0,1]\)</span> being cut by the plane
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<span class="math inline">\(J_x + J_y + J_z = 1\)</span>, we plot the
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projection of that plane in diagrams like fig. <a
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href="#fig:phase_diagram">1</a>.</p>
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<p>Similar to the Kitaev Honeycomb model with a magnetic field, we find
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that the amorphous model is only gapless along critical lines, see
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fig. <a href="#fig:phase_diagram">1</a> (Left).</p>
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<p>Interestingly, the gap closing exists in only one of the four
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topological sectors, though this is certainly a finite size effect as
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the sectors must become degenerate in the thermodynamic limit.
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||
Nevertheless this could be a useful way to define the (0, 0) topological
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flux sector for the amorphous model.</p>
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<p>In the honeycomb model, the phase boundaries are located on the
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straight lines <span class="math inline">\(|J^x| = |J^y| +
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|J^x|\)</span> and permutations of <span
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class="math inline">\(x,y,z\)</span>, shown as dotted line on ~<a
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||
href="#fig:phase_diagram">1</a> (Right). We find that on the amorphous
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lattice these boundaries exhibit an inward curvature, similar to
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||
honeycomb Kitaev models with flux<span class="citation"
|
||
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"
|
||
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 to \pm 1 from the A phases to the B phase. Indeed the gap must close in order for the Chern number to change citation." />
|
||
<figcaption aria-hidden="true"><span>Figure 1:</span> (Center) We choose
|
||
an energy scale for the Hamiltonian by setting <span
|
||
class="math inline">\(J_x + J_y + J_z = 1\)</span>. This intersects a
|
||
plane with the unit cube spanned by <span class="math inline">\(J_\alpha
|
||
\in [0,1]\)</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
|
||
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
|
||
class="math inline">\(0\)</span> to <span class="math inline">\(\pm
|
||
1\)</span> from the A phases to the B phase. Indeed the gap
|
||
<em>must</em> close in order for the Chern number to change
|
||
<strong>citation</strong>.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<h3 id="is-it-abelian-or-non-abelian">Is it Abelian or non-Abelian?</h3>
|
||
<p>The two phases of the amorphous model are clearly gapped, though
|
||
later I’ll double check this with finite size scaling.</p>
|
||
<p>The next question is: do these phases support excitations with
|
||
Abelian or non-Abelian statistics? To answer that we turn to Chern
|
||
numbers<span class="citation"
|
||
data-cites="berryQuantalPhaseFactors1984 simonHolonomyQuantumAdiabatic1983 thoulessQuantizedHallConductance1982"><sup><a
|
||
href="#ref-berryQuantalPhaseFactors1984" role="doc-biblioref">7</a>–<a
|
||
href="#ref-thoulessQuantizedHallConductance1982"
|
||
role="doc-biblioref">9</a></sup></span>. As discussed earlier the Chern
|
||
number is a quantity intimately linked to both the topological
|
||
properties and the anyonic statistics of a model. Here we will make use
|
||
of the fact that 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 because it
|
||
relies on integrals defined in k-space.</p>
|
||
<p>A family of real space generalisations of the Chern number that work
|
||
for amorphous systems exist called local topological markers<span
|
||
class="citation"
|
||
data-cites="bianco_mapping_2011 Hastings_Almost_2010 mitchellAmorphousTopologicalInsulators2018"><sup><a
|
||
href="#ref-bianco_mapping_2011" role="doc-biblioref">10</a>–<a
|
||
href="#ref-mitchellAmorphousTopologicalInsulators2018"
|
||
role="doc-biblioref">12</a></sup></span> and indeed Kitaev defines one
|
||
in his original paper on the model<span class="citation"
|
||
data-cites="kitaevAnyonsExactlySolved2006"><sup><a
|
||
href="#ref-kitaevAnyonsExactlySolved2006"
|
||
role="doc-biblioref">1</a></sup></span>.</p>
|
||
<p>Here we use the crosshair marker of<span class="citation"
|
||
data-cites="peru_preprint"><sup><a href="#ref-peru_preprint"
|
||
role="doc-biblioref">13</a></sup></span> because it works well on
|
||
smaller systems. We calculate the projector <span
|
||
class="math inline">\(P = \sum_i |\psi_i\rangle \langle \psi_i|\)</span>
|
||
onto the occupied fermion eigenstates of the system in open boundary
|
||
conditions. The projector encodes local information about the occupied
|
||
eigenstates of the system and is typically exponentially localised
|
||
<strong>[cite]</strong>. The name <em>crosshair</em> comes from the fact
|
||
that the marker is defined with respect to a particular point <span
|
||
class="math inline">\((x_0, y_0)\)</span> by step functions in x and
|
||
y</p>
|
||
<p><span class="math display">\[\begin{aligned}
|
||
\nu (x, y) = 4\pi \; \Im\; \mathrm{Tr}_{\mathrm{B}}
|
||
\left (
|
||
\hat{P}\;\hat{\theta}(x-x_0)\;\hat{P}\;\hat{\theta}(y-y_0)\; \hat{P}
|
||
\right ),
|
||
\end{aligned}\]</span></p>
|
||
<p>when the trace is taken over a region <span
|
||
class="math inline">\(B\)</span> around <span
|
||
class="math inline">\((x_0, y_0)\)</span> that is large enough to
|
||
include local information about the system but does not come too close
|
||
to the edges. If these conditions are met then then this quantity will
|
||
be very close to quantised to the Chern number, see fig. <a
|
||
href="#fig:phase_diagram_chern">2</a>.</p>
|
||
<p>We’ll use the crosshair marker to assess the Abelian/non-Abelian
|
||
character of the phases.</p>
|
||
<p>In the A phase of the amorphous model we find that <span
|
||
class="math inline">\(\nu=0\)</span> and hence the excitations have
|
||
Abelian character, similar to the honeycomb model. This phase is thus
|
||
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">14</a></sup></span>.</p>
|
||
<p>The B phase has <span class="math inline">\(\nu=\pm1\)</span> so is a
|
||
non-Abelian <em>chiral spin liquid</em> (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>. The CSL state is the the
|
||
magnetic analogue of the fractional quantum Hall state
|
||
<strong>[cite]</strong>. 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 marker13, 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 centre) 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">13</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 centre) 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="edge-modes">Edge Modes</h3>
|
||
<p>Chiral Spin Liquids support topological protected edge modes on open
|
||
boundary conditions<span class="citation"
|
||
data-cites="qi_general_2006"><sup><a href="#ref-qi_general_2006"
|
||
role="doc-biblioref">15</a></sup></span>. fig. <a
|
||
href="#fig:edge_modes">3</a> shows the probability density of one such
|
||
edge mode. It is near zero energy and exponentially localised to the
|
||
boundary of the system. While the model is gapped in periodic boundary
|
||
conditions (i.e on the torus) these edge modes appear in the gap when
|
||
the boundary is cut.</p>
|
||
<p>The localization of the edge 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 linear dimension of
|
||
the amorphous lattices and <span class="math inline">\(\tau\)</span> the
|
||
dimensional scaling exponent of IPR. This is relevant because localised
|
||
in-gap states do not participate in transport and hence do not turn band
|
||
insulators into metals. It is only when the gap fills with extended
|
||
states that we get a metallic state.</p>
|
||
<div id="fig:edge_modes" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/results/edge_modes/edge_modes.svg"
|
||
style="width:100.0%"
|
||
alt="Figure 3: (a) The density of one of the topologically protected edge states in the B phase. (Below) the log density plotted along the black path showing that the state is exponentially localised. (a)/(b) The density of states of the corresponding lattice in (a) periodic boundary conditions, (b) open boundary conditions. The colour of the bars shows the mean log IPR for each energy window. Cutting the boundary fills the gap with localised states." />
|
||
<figcaption aria-hidden="true"><span>Figure 3:</span> (a) The density of
|
||
one of the topologically protected edge states in the B phase. (Below)
|
||
the log density plotted along the black path showing that the state is
|
||
exponentially localised. (a)/(b) The density of states of the
|
||
corresponding lattice in (a) periodic boundary conditions, (b) open
|
||
boundary conditions. The colour of the bars shows the mean log IPR for
|
||
each energy window. Cutting the boundary fills the gap with localised
|
||
states.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<h2 id="anderson-transition-to-a-thermal-metal">Anderson Transition to a
|
||
Thermal Metal</h2>
|
||
<p>Previous work on the honeycomb model at finite temperature has shown
|
||
that the B phase undergoes a thermal transition from a quantum spin
|
||
liquid phase a to a <strong>thermal metal</strong> phase<span
|
||
class="citation" data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">16</a></sup></span>.</p>
|
||
<p>This happens because at finite temperature, thermal fluctuations lead
|
||
to spontaneous vortex-pair formation. As discussed previously these
|
||
fluxes are dressed by Majorana bounds states and the composite object is
|
||
an Ising-type non-Abelian anyon<span class="citation"
|
||
data-cites="Beenakker2013"><sup><a href="#ref-Beenakker2013"
|
||
role="doc-biblioref">17</a></sup></span>. The interactions between these
|
||
anyons are oscillatory similar to the RKKY exchange and decay
|
||
exponentially with separation<span class="citation"
|
||
data-cites="Laumann2012 Lahtinen_2011 lahtinenTopologicalLiquidNucleation2012"><sup><a
|
||
href="#ref-Laumann2012" role="doc-biblioref">18</a>–<a
|
||
href="#ref-lahtinenTopologicalLiquidNucleation2012"
|
||
role="doc-biblioref">20</a></sup></span>. At sufficient density, the
|
||
anyons hybridise 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">18</a></sup></span>. At close range the oscillatory
|
||
behaviour of the interactions can be modelled by a random sign which
|
||
forms the basis for a random matrix theory description of the thermal
|
||
metal state.</p>
|
||
<p>The amorphous chiral spin liquid undergoes the same form of Anderson
|
||
transition to a thermal metal state. Markov Chain Monte Carlo would be
|
||
necessary to simulate this in full detail<span class="citation"
|
||
data-cites="selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">16</a></sup></span> but in order to avoid that
|
||
complexity in the current work we instead opted to use vortex density
|
||
<span class="math inline">\(\rho\)</span> as a proxy for
|
||
temperature.</p>
|
||
<p>We simply give each plaquette probability <span
|
||
class="math inline">\(\rho\)</span> of being a vortex, possibly with one
|
||
additional adjustment to preserve overall vortex parity. 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>) while
|
||
at intermediate temperatures there may be vortex-vortex correlations
|
||
that are not captured by positioning vortices using uncorrelated random
|
||
variables.</p>
|
||
<p>First we performed a finite size scaling to that the presence of a
|
||
gap in the CSL ground state and absence of a gap in the thermal phase
|
||
are both robust as we go to larger systems, see fig. <a
|
||
href="#fig:fermion_gap_vs_L">4</a>.</p>
|
||
<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>Next we evaluated the fermionic density of states (DOS), Inverse
|
||
Participation Ratio (IPR) and IPR scaling exponent <span
|
||
class="math inline">\(\tau\)</span> as functions of the vortex density
|
||
<span class="math inline">\(\rho\)</span>, see fig. <a
|
||
href="#fig:DOS_vs_rho">5</a>. This leads to a nice picture of what
|
||
happens as we raise the temperature of the system away from the gapped,
|
||
insulating CSL phase. At small <span
|
||
class="math inline">\(\rho\)</span>, states begin to populate the gap
|
||
but they have <span class="math inline">\(\tau\approx0\)</span>,
|
||
indicating that they are localised states pinned to the vortices, 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 the thermal metal phase.</p>
|
||
<div id="fig:DOS_vs_rho" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/results/DOS_vs_rho/DOS_vs_rho.svg"
|
||
style="width:100.0%"
|
||
alt="Figure 5: (Top) Density of states and (Bottom) scaling exponent \tau of the amorphous Kitaev model as a vortex density \rho is increased. The scaling exponent \tau is the exponent with which the inverse participation ratio scales with system size. It gives a measure of the degree of localisation of the states in each (E/J, \rho) bin. At zero \rho we have the gapped ground state. At small \rho, states begin to populate the gap. These states have \tau\approx0, indicating that they are localised states pinned to fluxes, and the system remains insulating. As \rho increases further, the in-gap states merge with the bulk band and become extensive, fully closing the gap, and the system transitions to a thermal metal phase." />
|
||
<figcaption aria-hidden="true"><span>Figure 5:</span> (Top) Density of
|
||
states and (Bottom) scaling exponent <span
|
||
class="math inline">\(\tau\)</span> of the amorphous Kitaev model as a
|
||
vortex density <span class="math inline">\(\rho\)</span> is increased.
|
||
The scaling exponent <span class="math inline">\(\tau\)</span> is the
|
||
exponent with which the inverse participation ratio scales with system
|
||
size. It gives a measure of the degree of localisation of the states in
|
||
each <span class="math inline">\((E/J, \rho)\)</span> bin. At zero <span
|
||
class="math inline">\(\rho\)</span> we have the gapped ground state. At
|
||
small <span class="math inline">\(\rho\)</span>, states begin to
|
||
populate the gap. These states have <span
|
||
class="math inline">\(\tau\approx0\)</span>, indicating that they are
|
||
localised states pinned to fluxes, and the system remains insulating. As
|
||
<span class="math inline">\(\rho\)</span> increases further, the in-gap
|
||
states merge with the bulk band and become extensive, fully closing the
|
||
gap, and the system transitions to a thermal metal phase.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<p>The thermal metal phase has a signature logarithmic divergence at
|
||
zero energy and oscillations in the DOS. These signatures can be shown
|
||
to occur by a recursive argument that involves mapping the original
|
||
model onto a Majorana model with interactions that take random signs
|
||
which can itself be mapped onto a coarser lattice with lower energy
|
||
excitations and so on. This can be repeating indefinitely, showing the
|
||
model must have excitations at arbitrarily low energies in the
|
||
thermodynamic limit<span class="citation"
|
||
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
|
||
role="doc-biblioref">21</a></sup></span>.</p>
|
||
<p>These signatures for our model and for the honeycomb model are shown
|
||
in fig. <a href="#fig:DOS_oscillations">6</a>. They do not occur in the
|
||
honeycomb model unless the chiral symmetry is broken by a magnetic
|
||
field.</p>
|
||
<div id="fig:DOS_oscillations" class="fignos">
|
||
<figure>
|
||
<img
|
||
src="/assets/thesis/figure_code/amk_chapter/results/DOS_oscillations/DOS_oscillations.svg"
|
||
style="width:100.0%"
|
||
alt="Figure 6: Density of states at high temperature showing the logarithmic divergence at zero energy and oscillations characteristic of the thermal metal state16,21. (a) shows the honeycomb lattice model in the B phase with magnetic field, while (b) shows that our model transitions to a thermal metal phase without an external magnetic field but rather due to the spontaneous chiral symmetry breaking. In both plots the density of vortices is \rho = 0.5 corresponding to the T = \infty limit." />
|
||
<figcaption aria-hidden="true"><span>Figure 6:</span> Density of states
|
||
at high temperature showing the logarithmic divergence at zero energy
|
||
and oscillations characteristic of the thermal metal state<span
|
||
class="citation"
|
||
data-cites="bocquet_disordered_2000 selfThermallyInducedMetallic2019"><sup><a
|
||
href="#ref-selfThermallyInducedMetallic2019"
|
||
role="doc-biblioref">16</a>,<a href="#ref-bocquet_disordered_2000"
|
||
role="doc-biblioref">21</a></sup></span>. (a) shows the honeycomb
|
||
lattice model in the B phase with magnetic field, while (b) shows that
|
||
our model transitions to a thermal metal phase without an external
|
||
magnetic field but rather due to the spontaneous chiral symmetry
|
||
breaking. In both plots the density of vortices is <span
|
||
class="math inline">\(\rho = 0.5\)</span> corresponding to the <span
|
||
class="math inline">\(T = \infty\)</span> limit.</figcaption>
|
||
</figure>
|
||
</div>
|
||
<h1 id="conclusion">Conclusion</h1>
|
||
<p>In this chapter we have looked at an extension of the Kitaev
|
||
honeycomb model to amorphous lattices with coordination number three. We
|
||
discussed a method to construct arbitrary trivalent lattices using
|
||
Voronoi partitions, how to embed them onto the torus and how to
|
||
edge-colour them using a SAT solver.</p>
|
||
<p>We provided extensive numerical evidence that the ground state flux
|
||
sector of the model is given by a simply function of the number of sides
|
||
of each plaquette backed up by an analysis of the energetic finite size
|
||
effects.</p>
|
||
<p>We found two quantum spin liquid phases that can be distinguished
|
||
using a real-space generalisation of the Chern number. We showed that
|
||
via finite size scaling that these phases are robustly gapped. 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.</p>
|
||
<p>Finally we showed 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="limits-of-the-ground-state-conjecture">Limits of the ground
|
||
state conjecture</h2>
|
||
<p>We found a small number of lattices for which 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 amplified 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 A phase where the couplings are anisotropic. We
|
||
showed that the colouring does not matter in the B 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 toric
|
||
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>
|
||
<h1 id="outlook">Outlook</h1>
|
||
<p>This exactly solvable chiral QSL provides a first example of a
|
||
topological quantum many-body phase in amorphous magnets, which raises a
|
||
number of questions for future research.</p>
|
||
<h2 id="experimental-realisations-and-signatures">Experimental
|
||
Realisations and Signatures</h2>
|
||
<p>The obvious question is whether amorphous Kitaev materials could be
|
||
physically realised.</p>
|
||
<p>Most crystals can as exists in a metastable amorphous state if they
|
||
are cooled rapidly, freezing them into a disordered configuration<span
|
||
class="citation"
|
||
data-cites="Weaire1976 Petrakovski1981 Kaneyoshi2018"><sup><a
|
||
href="#ref-Weaire1976" role="doc-biblioref">22</a>–<a
|
||
href="#ref-Kaneyoshi2018" role="doc-biblioref">24</a></sup></span>.
|
||
Indeed quenching has been used by humans to control the hardness of
|
||
steel or iron for thousands of years. It would therefore be interesting
|
||
to study amorphous version of candidate Kitaev materials<span
|
||
class="citation" data-cites="trebstKitaevMaterials2022"><sup><a
|
||
href="#ref-trebstKitaevMaterials2022"
|
||
role="doc-biblioref">25</a></sup></span> such as <span
|
||
class="math inline">\(\alpha-\textrm{RuCl}_3\)</span> to see whether
|
||
they maintain even approximate fixed coordination number locally as is
|
||
the case with amorphous Silicon and Germanium<span class="citation"
|
||
data-cites="Weaire1971 betteridge1973possible"><sup><a
|
||
href="#ref-Weaire1971" role="doc-biblioref">26</a>,<a
|
||
href="#ref-betteridge1973possible"
|
||
role="doc-biblioref">27</a></sup></span>.</p>
|
||
<p>Looking instead at more engineered realisation, metal organic
|
||
frameworks have been shown to be capable of forming amorphous
|
||
lattices <span class="citation"
|
||
data-cites="bennett2014amorphous"><sup><a
|
||
href="#ref-bennett2014amorphous"
|
||
role="doc-biblioref">28</a></sup></span> and there are recent proposals
|
||
for realizing strong Kitaev interactions <span class="citation"
|
||
data-cites="yamadaDesigningKitaevSpin2017"><sup><a
|
||
href="#ref-yamadaDesigningKitaevSpin2017"
|
||
role="doc-biblioref">29</a></sup></span> as well as reports of QSL
|
||
behavior <span class="citation"
|
||
data-cites="misumiQuantumSpinLiquid2020"><sup><a
|
||
href="#ref-misumiQuantumSpinLiquid2020"
|
||
role="doc-biblioref">30</a></sup></span>.</p>
|
||
<h2 id="generalisations">Generalisations</h2>
|
||
<p>The model presented here could be generalized in several ways.</p>
|
||
<p>First, it would be interesting to study the stability of the chiral
|
||
amorphous Kitaev QSL with respect to perturbations <span
|
||
class="citation"
|
||
data-cites="Rau2014 Chaloupka2010 Chaloupka2013 Chaloupka2015 Winter2016"><sup><a
|
||
href="#ref-Rau2014" role="doc-biblioref">31</a>–<a
|
||
href="#ref-Winter2016" role="doc-biblioref">35</a></sup></span>.</p>
|
||
<p>Second, one could investigate whether a QSL phase may exist for for
|
||
other models defined on amorphous lattices. For example, in real
|
||
materials, there will generally be an additional small Heisenberg term
|
||
<span class="math display">\[H_{KH} = - \sum_{\langle
|
||
j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} +
|
||
\sigma_j\sigma_k\]</span> With a view to more realistic prospects of
|
||
observation, it would be interesting to see if the properties of the
|
||
Kitaev-Heisenberg model generalise from the honeycomb to the amorphous
|
||
case[<span class="citation" data-cites="Chaloupka2010"><sup><a
|
||
href="#ref-Chaloupka2010" role="doc-biblioref">32</a></sup></span>;<span
|
||
class="citation" data-cites="Chaloupka2015"><sup><a
|
||
href="#ref-Chaloupka2015" role="doc-biblioref">34</a></sup></span>;<span
|
||
class="citation" data-cites="Jackeli2009"><sup><a
|
||
href="#ref-Jackeli2009" role="doc-biblioref">36</a></sup></span>;<span
|
||
class="citation" data-cites="Kalmeyer1989"><sup><a
|
||
href="#ref-Kalmeyer1989" role="doc-biblioref">37</a></sup></span>;<span
|
||
class="citation"
|
||
data-cites="manousakisSpinTextonehalfHeisenberg1991"><sup><a
|
||
href="#ref-manousakisSpinTextonehalfHeisenberg1991"
|
||
role="doc-biblioref">38</a></sup></span>;].</p>
|
||
<p>Finally it might be possible to look at generalizations to
|
||
higher-spin models or those on 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">39</a>–<a
|
||
href="#ref-Wu2009" role="doc-biblioref">48</a></sup></span></p>
|
||
<p>Overall, there has been surprisingly little research on amorphous
|
||
quantum many body phases albeit material candidates aplenty. We expect
|
||
our exact chiral amorphous spin liquid to find many generalisation to
|
||
realistic amorphous quantum magnets and beyond.</p>
|
||
<div id="refs" class="references csl-bib-body" data-line-spacing="2"
|
||
role="doc-bibliography">
|
||
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|
||
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|
||
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|
||
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).</div>
|
||
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|
||
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|
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|
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|
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|
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|
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|
||
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|
||
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|
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order in coordinate space</a>. <em>Physical Review B</em>
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</div>
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<div class="csl-left-margin">11. </div><div
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class="csl-right-inline">Hastings, M. B. & Loring, T. A. <a
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href="https://doi.org/10.1063/1.3274817">Almost commuting matrices,
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localized <span>Wannier</span> functions, and the quantum
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||
<span>Hall</span> effect</a>. <em>Journal of Mathematical Physics</em>
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<strong>51</strong>, 015214 (2010).</div>
|
||
</div>
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<div id="ref-mitchellAmorphousTopologicalInsulators2018"
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class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">12. </div><div
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class="csl-right-inline">Mitchell, N. P., Nash, L. M., Hexner, D.,
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Turner, A. M. & Irvine, W. T. M. <a
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href="https://doi.org/10.1038/s41567-017-0024-5">Amorphous topological
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insulators constructed from random point sets</a>. <em>Nature Phys</em>
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<strong>14</strong>, 380–385 (2018).</div>
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||
</div>
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<div id="ref-peru_preprint" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">13. </div><div
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class="csl-right-inline">d’Ornellas, P., Barnett, R. & Lee, D. K. K.
|
||
Quantised bulk conductivity as a local chern marker. <em>arXiv
|
||
preprint</em> (2022) doi:<a
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||
href="https://doi.org/10.48550/ARXIV.2207.01389">10.48550/ARXIV.2207.01389</a>.</div>
|
||
</div>
|
||
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">14. </div><div
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class="csl-right-inline">Kitaev, A. Y. <a
|
||
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-tolerant
|
||
quantum computation by anyons</a>. <em>Annals of Physics</em>
|
||
<strong>303</strong>, 2–30 (2003).</div>
|
||
</div>
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<div id="ref-qi_general_2006" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">15. </div><div class="csl-right-inline">Qi,
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||
X.-L., Wu, Y.-S. & Zhang, S.-C. <a
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href="https://doi.org/10.1103/PhysRevB.74.045125">General theorem
|
||
relating the bulk topological number to edge states in two-dimensional
|
||
insulators</a>. <em>Physical Review B</em> <strong>74</strong>, 045125
|
||
(2006).</div>
|
||
</div>
|
||
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">16. </div><div
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class="csl-right-inline">Self, C. N., Knolle, J., Iblisdir, S. &
|
||
Pachos, J. K. <a
|
||
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally induced
|
||
metallic phase in a gapped quantum spin liquid - a <span>Monte
|
||
Carlo</span> study of the <span>Kitaev</span> model with parity
|
||
projection</a>. <em>Phys. Rev. B</em> <strong>99</strong>, 045142
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||
(2019).</div>
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||
</div>
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<div id="ref-Beenakker2013" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">17. </div><div
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class="csl-right-inline">Beenakker, C. W. J. <a
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||
href="https://doi.org/10.1146/annurev-conmatphys-030212-184337">Search
|
||
for majorana fermions in superconductors</a>. <em>Annual Review of
|
||
Condensed Matter Physics</em> <strong>4</strong>, 113–136 (2013).</div>
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</div>
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<div id="ref-Laumann2012" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">18. </div><div
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class="csl-right-inline">Laumann, C. R., Ludwig, A. W. W., Huse, D. A.
|
||
& Trebst, S. <a
|
||
href="https://doi.org/10.1103/PhysRevB.85.161301">Disorder-induced
|
||
<span>Majorana</span> metal in interacting non-<span>Abelian</span>
|
||
anyon systems</a>. <em>Phys. Rev. B</em> <strong>85</strong>, 161301
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||
(2012).</div>
|
||
</div>
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<div class="csl-left-margin">19. </div><div
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class="csl-right-inline">Lahtinen, V. <a
|
||
href="https://doi.org/10.1088/1367-2630/13/7/075009">Interacting
|
||
non-<span>Abelian</span> anyons as <span>Majorana</span> fermions in the
|
||
honeycomb lattice model</a>. <em>New Journal of Physics</em>
|
||
<strong>13</strong>, 075009 (2011).</div>
|
||
</div>
|
||
<div id="ref-lahtinenTopologicalLiquidNucleation2012" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">20. </div><div
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class="csl-right-inline">Lahtinen, V., Ludwig, A. W. W., Pachos, J. K.
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& Trebst, S. <a
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||
href="https://doi.org/10.1103/PhysRevB.86.075115">Topological liquid
|
||
nucleation induced by vortex-vortex interactions in
|
||
<span>Kitaev</span>’s honeycomb model</a>. <em>Phys. Rev. B</em>
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||
<strong>86</strong>, 075115 (2012).</div>
|
||
</div>
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<div id="ref-bocquet_disordered_2000" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">21. </div><div
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class="csl-right-inline">Bocquet, M., Serban, D. & Zirnbauer, M. R.
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<a href="https://doi.org/10.1016/S0550-3213(00)00208-X">Disordered 2d
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||
quasiparticles in class <span>D</span>: <span>Dirac</span> fermions with
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||
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<strong>578</strong>, 628–680 (2000).</div>
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</div>
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<div id="ref-Weaire1976" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">22. </div><div
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||
href="https://doi.org/10.1080/00107517608210851">The structure of
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||
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||
173–191 (1976).</div>
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<div id="ref-Petrakovski1981" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">23. </div><div
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class="csl-right-inline">Petrakovskiı̆, G. A. <a
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||
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|
||
magnetic materials</a>. <em>Soviet Physics Uspekhi</em>
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<strong>24</strong>, 511–525 (1981).</div>
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||
</div>
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<div id="ref-Kaneyoshi2018" class="csl-entry" role="doc-biblioentry">
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class="csl-right-inline"><em>Amorphous magnetism</em>. (<span>CRC
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||
</div>
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||
<div id="ref-trebstKitaevMaterials2022" class="csl-entry"
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<div class="csl-left-margin">25. </div><div
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class="csl-right-inline">Trebst, S. & Hickey, C. <a
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||
href="https://doi.org/10.1016/j.physrep.2021.11.003">Kitaev
|
||
materials</a>. <em>Physics Reports</em> <strong>950</strong>, 1–37
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(2022).</div>
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||
</div>
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||
<div id="ref-Weaire1971" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">26. </div><div
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||
href="https://doi.org/10.1103/PhysRevB.4.2508">Electronic properties of
|
||
an amorphous solid. <span>I</span>. <span>A</span> simple tight-binding
|
||
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(1971).</div>
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||
</div>
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<div id="ref-betteridge1973possible" class="csl-entry"
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<div class="csl-left-margin">27. </div><div
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||
silicon and germanium. <em>Journal of Physics C: Solid State
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||
</div>
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||
<div id="ref-bennett2014amorphous" class="csl-entry"
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<div class="csl-left-margin">28. </div><div
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||
metalorganic frameworks. <em>Accounts of chemical research</em>
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||
</div>
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||
<div id="ref-yamadaDesigningKitaevSpin2017" class="csl-entry"
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role="doc-biblioentry">
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<div class="csl-left-margin">29. </div><div
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class="csl-right-inline">Yamada, M. G., Fujita, H. & Oshikawa, M. <a
|
||
href="https://doi.org/10.1103/PhysRevLett.119.057202">Designing
|
||
<span>Kitaev Spin Liquids</span> in <span>Metal-Organic
|
||
Frameworks</span></a>. <em>Phys. Rev. Lett.</em> <strong>119</strong>,
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057202 (2017).</div>
|
||
</div>
|
||
<div id="ref-misumiQuantumSpinLiquid2020" class="csl-entry"
|
||
role="doc-biblioentry">
|
||
<div class="csl-left-margin">30. </div><div
|
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class="csl-right-inline">Misumi, Y. <em>et al.</em> <a
|
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href="https://doi.org/10.1021/jacs.0c05472">Quantum <span>Spin Liquid
|
||
State</span> in a <span>Two-Dimensional Semiconductive Metal</span></a>.
|
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<em>J. Am. Chem. Soc.</em> <strong>142</strong>, 16513–16517
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||
(2020).</div>
|
||
</div>
|
||
<div id="ref-Rau2014" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">31. </div><div
|
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class="csl-right-inline">Rau, J. G., Lee, E. K.-H. & Kee, H.-Y. <a
|
||
href="https://doi.org/10.1103/PhysRevLett.112.077204">Generic spin model
|
||
for the honeycomb iridates beyond the kitaev limit</a>. <em>Phys. Rev.
|
||
Lett.</em> <strong>112</strong>, 077204 (2014).</div>
|
||
</div>
|
||
<div id="ref-Chaloupka2010" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">32. </div><div
|
||
class="csl-right-inline">Chaloupka, J., Jackeli, G. & Khaliullin, G.
|
||
Kitaev-<span>Heisenberg</span> model on a honeycomb lattice: Possible
|
||
exotic phases in iridium oxides <span>A</span><span><span
|
||
class="math inline">\(_{2}\)</span></span><span>IrO</span><span><span
|
||
class="math inline">\(_{3}\)</span></span>. <em>Phys. Rev. Lett.</em>
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<strong>105</strong>, 027204 (2010).</div>
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</div>
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<div class="csl-left-margin">33. </div><div
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<a href="https://doi.org/10.1103/PhysRevLett.110.097204">Zigzag magnetic
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order in the iridium oxide <span>Na</span><span><span
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class="math inline">\(_{2}\)</span></span><span>IrO</span><span><span
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</div>
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<div class="csl-left-margin">34. </div><div
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class="csl-right-inline">Chaloupka, J. & Khaliullin, G. Hidden
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||
symmetries of the extended <span>Kitaev-Heisenberg</span> model:
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||
<span>Implications</span> for honeycomb lattice iridates
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||
<span>A</span><span><span
|
||
class="math inline">\(_{2}\)</span></span><span>IrO</span><span><span
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<strong>92</strong>, 024413 (2015).</div>
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</div>
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<div id="ref-Winter2016" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">35. </div><div
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class="csl-right-inline">Winter, S. M., Li, Y., Jeschke, H. O. &
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Valentí, R. <a
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href="https://doi.org/10.1103/PhysRevB.93.214431">Challenges in design
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of <span>Kitaev</span> materials: <span>Magnetic</span> interactions
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from competing energy scales</a>. <em>Phys. Rev. B</em>
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<strong>93</strong>, 214431 (2016).</div>
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</div>
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<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">36. </div><div
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href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott insulators in
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the strong spin-orbit coupling limit: From <span>Heisenberg</span> to a
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quantum compass and <span>Kitaev</span> models</a>. <em>Physical Review
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</div>
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<div class="csl-left-margin">37. </div><div
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<em>Phys. Rev. B</em> <strong>39</strong>, 11879–11899 (1989).</div>
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<div class="csl-left-margin">38. </div><div
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href="https://doi.org/10.1103/RevModPhys.63.1">The spin-\textonehalf{}
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<span>Heisenberg</span> antiferromagnet on a square lattice and its
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application to the cuprous oxides</a>. <em>Rev. Mod. Phys.</em>
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<div class="csl-left-margin">39. </div><div
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class="csl-right-inline">Baskaran, G., Sen, D. & Shankar, R. <a
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href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-<span>S
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Kitaev</span> model: <span>Classical</span> ground states, order from
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disorder, and exact correlation functions</a>. <em>Phys. Rev. B</em>
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<strong>78</strong>, 115116 (2008).</div>
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</div>
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<div id="ref-Yao2009" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">40. </div><div
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href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic spin
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liquid in an exactly solvable spin model</a>. <em>Phys. Rev. Lett.</em>
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</div>
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<div class="csl-left-margin">41. </div><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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class="math inline">\(\frac{1}{2}\)</span></span> multilayer
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(2009).</div>
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</div>
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<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">42. </div><div
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(2011).</div>
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</div>
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<div id="ref-Natori2020" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">43. </div><div
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class="csl-right-inline">Natori, W. M. H. & Knolle, J. <a
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href="https://doi.org/10.1103/PhysRevLett.125.067201">Dynamics of a
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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>
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<div id="ref-Chulliparambil2020" class="csl-entry"
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<div class="csl-left-margin">44. </div><div
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M., Janssen, L. & Tu, H.-H. <a
|
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href="https://doi.org/10.1103/PhysRevB.102.201111">Microscopic models
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<em>Phys. Rev. B</em> <strong>102</strong>, 201111 (2020).</div>
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</div>
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<div id="ref-Chulliparambil2021" class="csl-entry"
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<div class="csl-left-margin">45. </div><div
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href="https://doi.org/10.1103/PhysRevB.103.075144">Flux crystals,
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<div class="csl-left-margin">46. </div><div
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<div class="csl-left-margin">47. </div><div
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<em>Phys. Rev. B</em> <strong>104</strong>, 214422 (2021).</div>
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</div>
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<strong>79</strong>, 134427 (2009).</div>
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||
</div>
|
||
</div>
|
||
</main>
|
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