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title: Background - The Kitaev Honeycomb Model
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<title>Background - The Kitaev Honeycomb Model</title>
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<ul>
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<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
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<ul>
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<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
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<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
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<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
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<li><a href="#exact-solvability" id="toc-exact-solvability">Exact Solvability</a></li>
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<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
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<li><a href="#mapping-to-a-majorana-hamiltonian" id="toc-mapping-to-a-majorana-hamiltonian">Mapping to a Majorana Hamiltonian</a>
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<ul>
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<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single spin</a></li>
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<li><a href="#partitioning-the-hilbert-space-into-bond-sectors" id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning the Hilbert Space into Bond sectors</a></li>
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<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace" id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping back from Bond Sectors to the Physical Subspace</a></li>
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<li><a href="#open-boundary-conditions" id="toc-open-boundary-conditions">Open boundary conditions</a></li>
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</ul></li>
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<li><a href="#emergent-gauge-fields" id="toc-emergent-gauge-fields">Emergent gauge fields</a>
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<ul>
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<li><a href="#ground-state-degeneracy" id="toc-ground-state-degeneracy">Ground State Degeneracy</a></li>
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</ul></li>
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<li><a href="#bg-the-ground-state" id="toc-bg-the-ground-state">The Ground State</a></li>
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<li><a href="#phases-of-the-kitaev-model" id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<!-- Table of Contents -->
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<!-- <nav id="TOC" role="doc-toc">
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<ul>
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<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
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<ul>
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<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
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<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
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<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
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<li><a href="#exact-solvability" id="toc-exact-solvability">Exact Solvability</a></li>
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<li><a href="#glossary" id="toc-glossary">Glossary</a></li>
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<li><a href="#mapping-to-a-majorana-hamiltonian" id="toc-mapping-to-a-majorana-hamiltonian">Mapping to a Majorana Hamiltonian</a>
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<ul>
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<li><a href="#for-a-single-spin" id="toc-for-a-single-spin">For a single spin</a></li>
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<li><a href="#partitioning-the-hilbert-space-into-bond-sectors" id="toc-partitioning-the-hilbert-space-into-bond-sectors">Partitioning the Hilbert Space into Bond sectors</a></li>
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<li><a href="#mapping-back-from-bond-sectors-to-the-physical-subspace" id="toc-mapping-back-from-bond-sectors-to-the-physical-subspace">Mapping back from Bond Sectors to the Physical Subspace</a></li>
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<li><a href="#open-boundary-conditions" id="toc-open-boundary-conditions">Open boundary conditions</a></li>
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</ul></li>
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<li><a href="#emergent-gauge-fields" id="toc-emergent-gauge-fields">Emergent gauge fields</a>
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<ul>
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<li><a href="#ground-state-degeneracy" id="toc-ground-state-degeneracy">Ground State Degeneracy</a></li>
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</ul></li>
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<li><a href="#bg-the-ground-state" id="toc-bg-the-ground-state">The Ground State</a></li>
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<li><a href="#phases-of-the-kitaev-model" id="toc-phases-of-the-kitaev-model">Phases of the Kitaev Model</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<div id="page-header">
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<p>2 Background</p>
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<hr />
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</div>
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<section id="bg-hkm-model" class="level1">
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<h1>The Kitaev Honeycomb Model</h1>
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<figure>
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<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig:intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom u_{jk} = \pm 1 that lives on each bond, the bond variables. These are antisymmetric, u_{jk} = -u_{kj}, so we represent them graphically with arrows on each bond that point in the direction that u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas b_i^x,\;b_i^y,\;b_i^z,\;c_i. The x, y and z Majoranas then pair along the bonds forming conserved \mathbb{Z}_2 bond operators u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle. The remaining c_i operators form an effective quadratic Hamiltonian H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j." />
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<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. <strong>(b)</strong>. After transforming to the Majorana representation we get an emergent gauge degree of freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> that lives on each bond, the bond variables. These are antisymmetric, <span class="math inline">\(u_{jk} = -u_{kj}\)</span>, so we represent them graphically with arrows on each bond that point in the direction that <span class="math inline">\(u_{jk} = +1\)</span> <strong>(c)</strong>. The Majorana transformation can be visualised as breaking each spin into four Majoranas <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span>. The x, y and z Majoranas then pair along the bonds forming conserved <span class="math inline">\(\mathbb{Z}_2\)</span> bond operators <span class="math inline">\(u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle\)</span>. The remaining <span class="math inline">\(c_i\)</span> operators form an effective quadratic Hamiltonian <span class="math inline">\(H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span>.</figcaption>
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</figure>
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<section id="the-spin-hamiltonian" class="level2">
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<h2>The Spin Hamiltonian</h2>
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<p>This section introduces the seminal Kitaev honeycomb (KH) model. The KH model is an exactly solvable microscopic model of interacting spin<span class="math inline">\(-1/2\)</span> spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label <span class="math inline">\(\alpha \in \{ x, y, z\}\)</span> and that bond couple two spins along the <span class="math inline">\(\alpha\)</span> axis. See fig. <a href="#fig:intro_figure_by_hand">1</a> for a diagram.</p>
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<p>This gives us the Hamiltonian <span id="eq:bg-kh-model"><span class="math display">\[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + \Gamma \mathrm{three spin term}, \qquad{(1)}\]</span></span> where <span class="math inline">\(\sigma^\alpha_j\)</span> is a Pauli matrix acting on site <span class="math inline">\(j\)</span> and <span class="math inline">\(\langle j,k\rangle_\alpha\)</span> is a pair of nearest-neighbour indices connected by an <span class="math inline">\(\alpha\)</span>-bond with exchange coupling <span class="math inline">\(J^\alpha\)</span> <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
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<p>The Kitaev Honeycomb model <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span> can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds <span class="citation" data-cites="Jackeli2009 HerrmannsAnRev2018 Winter2017 TrebstPhysRep2022 Takagi2019"> [<a href="#ref-Jackeli2009" role="doc-biblioref">2</a>–<a href="#ref-Takagi2019" role="doc-biblioref">6</a>]</span>. The model is highly frustrated: energetically each spin would like to align along a different direction with each of its three neighbours. This cannot be achieved even classically. This frustration leads the the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier <span class="citation" data-cites="andersonResonatingValenceBonds1973"> [<a href="#ref-andersonResonatingValenceBonds1973" role="doc-biblioref">7</a>]</span>, the KH model was one of the first concrete models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. At finite temperature the model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">8</a>]</span>. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes tied to an emergent gauge field and the remaining fermions are governed by a free fermion hamiltonian.</p>
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<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="../2_Background/2.2_HKM_Model.html#emergent-gauge-fields">emergent gauge fields</a> and the projector. We will then discuss the <a href="../2_Background/2.2_HKM_Model.html#bg-the-ground-state">ground state</a> found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.</p>
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<p>The <a href="../2_Background/2.3_Anyons.html#anyonic-statistics">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics.</p>
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</section>
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<section id="the-spin-model" class="level2">
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<h2>The Spin Model</h2>
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<p>Eq. <a href="#eq:bg-kh-model">1</a> shows</p>
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</section>
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<section id="the-majorana-model" class="level2">
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<h2>The Majorana Model</h2>
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<p>The Kitaev Honeycomb model <span class="math display">\[H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\]</span> is remarkable because it combines three key properties.</p>
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<p>First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as <span class="math inline">\(\alpha\mathrm{-RuCl}_3\)</span>, are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small corrections <span class="citation" data-cites="banerjeeProximateKitaevQuantum2016 TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>,<a href="#ref-banerjeeProximateKitaevQuantum2016" role="doc-biblioref">9</a>]</span>.</p>
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<p><strong>expand later: Why do we need spin orbit coupling and what will the corrections be?</strong></p>
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<p>Second, its ground state is the canonical example of the long sought after quantum spin liquid state. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations <span class="citation" data-cites="freedmanTopologicalQuantumComputation2003"> [<a href="#ref-freedmanTopologicalQuantumComputation2003" role="doc-biblioref">10</a>]</span>.</p>
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<p>Third, and perhaps most importantly, this model is a rare many body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many body ground states in terms of single particle states <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the Kitaev Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
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<p>“dynamical two spin correlation functions are identically zero beyond nearest neighbor separation in the Kitaev Model” <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">11</a>]</span></p>
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<ul>
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<li>strong spin orbit coupling yields spatial anisotropic spin exchange leading to compass models <span class="citation" data-cites="kugelJahnTellerEffectMagnetism1982"> [<a href="#ref-kugelJahnTellerEffectMagnetism1982" role="doc-biblioref">12</a>]</span></li>
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<li>spin model of the Kitaev model is one</li>
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<li>has extensively many conserved fluxes</li>
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<li></li>
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</ul>
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<p><strong>intro</strong> - strong spin orbit coupling leads to anisotropic spin exchange (as opposed to isotropic exchange like the Heisenberg model) - geometrical frustration leads to QSL ground state with long range entanglement (not simple paramagnet)</p>
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<ul>
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<li>RuCl_3 is the classic QSL candidate material</li>
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<li>really follows the Kitaev-Heisenberg model</li>
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<li>experimental probes include inelastic neutron scattering, Raman scattering</li>
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</ul>
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</section>
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<section id="exact-solvability" class="level2">
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<h2>Exact Solvability</h2>
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<p>For notational brevity, it is useful to introduce the spin bond operators <span class="math inline">\(K_{ij} = \sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> where <span class="math inline">\(\alpha\)</span> is a function of <span class="math inline">\(i,j\)</span> that picks the correct bond type.</p>
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<p>This Kitaev model has a set of conserved quantities that, in the spin language, take the form of Wilson loop operators <span class="math inline">\(W_p\)</span> winding around a closed path on the lattice. The direction does not matter, but we will keep to clockwise here. We will use the term plaquette and the symbol <span class="math inline">\(\phi\)</span> to refer to a Wilson loop operator that does not enclose any other sites, such as a single hexagon in a honeycomb lattice.</p>
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<p><span class="math display">\[W_p = \prod_{\mathrm{i,j}\; \in\; p} K_{ij}\]</span></p>
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<p>In closed loops, each site appears twice in the product with two of the three bond types. Applying <span class="math inline">\(\sigma^\alpha \sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq \beta\)</span> then gives us a product containing a single Pauli matrix associated with each site in the loop with the type of the <em>outward</em> pointing bond. Hence the <span class="math inline">\(W_p\)</span> associated with hexagons or shapes with an even number of sides all square to 1 and, hence, have eigenvalues <span class="math inline">\(\pm 1\)</span>. As the honeycomb lattice is bipartite, there are no closed loops that contain an odd number of edges. On other lattices <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">13</a>]</span>, plaquettes with an odd number of sides (odd plaquettes) have eigenvalues <span class="math inline">\(\pm i\)</span>.</p>
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<figure>
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<img src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg" id="fig:regular_plaquettes" data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%" alt="Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path." />
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<figcaption aria-hidden="true">Figure 2: The eigenvalues of a loop or plaquette operators depend on the number of bonds in its enclosing path.</figcaption>
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</figure>
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<p>Remarkably, all of the spin bond operators <span class="math inline">\(K_{ij}\)</span> commute with all the Wilson loop operators <span class="math inline">\(W_p\)</span>. <span class="math display">\[[W_p, K_{ij}] = 0\]</span> We can prove this by considering three cases: 1. neither <span class="math inline">\(i\)</span> nor <span class="math inline">\(j\)</span> is part of the loop 2. one of <span class="math inline">\(i\)</span> or <span class="math inline">\(j\)</span> are part of the loop 3. both are part of the loop</p>
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<p>The first case is trivial. The other two require some algebra, outlined in fig. <strong>¿fig:visual_kitaev_2?</strong>.</p>
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</section>
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<section id="glossary" class="level2">
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<h2>Glossary</h2>
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<ul>
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<li><p>Lattice: The underlying graph on which the models are defined. Composed of sites (vertices), bonds (edges) and plaquettes (faces).</p></li>
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<li><p>The model : Used when I refer to properties of the the Kitaev model that do not depend on the particular lattice.</p></li>
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<li><p>The Honeycomb model : The Kitaev Model defined on the honeycomb lattice.</p></li>
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<li><p>The Amorphous model : The Kitaev Model defined on the amorphous lattices described here.</p></li>
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</ul>
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<p><strong>The Spin Hamiltonian</strong></p>
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<ul>
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<li>Spin Bond Operators: <span class="math inline">\(\hat{k}_{ij} = \sigma_i^\alpha \sigma_j^\alpha\)</span></li>
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<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} k_{ij}\)</span></li>
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<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
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</ul>
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<p><strong>The Majorana Model</strong></p>
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<ul>
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<li>Majorana Operators on site <span class="math inline">\(i\)</span>: <span class="math inline">\(\hat{b}^x_i, \hat{b}^y_i, \hat{b}^z_i, \hat{c}_i\)</span></li>
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<li>Majorana Bond Operators: <span class="math inline">\(\hat{u}_{ij} = i b_i^\alpha b_j^\alpha\)</span></li>
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<li>Loop Operators: <span class="math inline">\(\hat{W_p} = \prod_{<i,j>} u_{ij}\)</span></li>
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<li>Plaquette Operators: Loops that enclose a single plaquette.</li>
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<li>Gauge Operators: <span class="math inline">\(D_i = \hat{b}^x_i \hat{b}^y_i \hat{b}^z_i \hat{c}_i\)</span></li>
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<li>The Extended Hilbert space: The larger Hilbert space spanned by the Majorana operators.</li>
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<li>The physical subspace: The subspace of the extended Hilbert space that we identify with the Hilbert space of the original spin model.</li>
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<li>The Projector <span class="math inline">\(\hat{P}\)</span>: The projector onto the physical subspace.</li>
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</ul>
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<p><strong>Flux Sectors</strong></p>
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<ul>
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<li><p>Odd/Even Plaquettes: Plaquettes with an odd/even number of sides.</p></li>
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<li><p>Fluxes <span class="math inline">\(\phi_i\)</span>: The expectation values of the plaquette operators <span class="math inline">\(\pm 1\)</span> for even and <span class="math inline">\(\pm i\)</span> for odd plaquettes.</p></li>
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<li><p>Flux Sector: A subspace of Hilbert space in which the fluxes take particular values.</p></li>
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<li><p>Ground state flux sector: The Flux Sector containing the lowest energy many body state.</p></li>
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<li><p>Vortices: Flux excitations away from the ground state flux sector.</p></li>
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<li><p>Dual Loops: A set of <span class="math inline">\(u_{jk}\)</span> that correspond to loops on the dual lattice.</p></li>
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<li><p>non-contractible loops or dual loops: The two loops topologically distinct loops on the torus that cannot be smoothly deformed to a point.</p></li>
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<li><p>Topological Fluxes <span class="math inline">\(\Phi_{x}, \Phi_{y}\)</span>: The two fluxes associated with the two non-contractible loops.</p></li>
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<li><p>Topological Transport Operators: <span class="math inline">\(\mathcal{T}_{x}, \mathcal{T}_{y}\)</span>: The two vortex-pair operations associated with the non-contractible <em>dual</em> loops.</p></li>
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</ul>
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<p><strong>Phases</strong></p>
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<ul>
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<li>The A phase: The three anisotropic regions of the phase diagram <span class="math inline">\(A_x, A_y, A_z\)</span> where <span class="math inline">\(A_\alpha\)</span> means <span class="math inline">\(J_\alpha >> J_\beta, J_\gamma\)</span>.</li>
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<li>The B phase: The roughly isotropic region of the phase diagram.</li>
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</ul>
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||
<p><img src="/assets/thesis/amk_chapter/visual_kitaev_1.svg" id="fig:visual_kitaev_1" data-short-caption="A Visual Intro to the Kitaev Model" style="width:100.0%" alt="A visual introduction to the Kitaev Model." /> <img src="/assets/thesis/amk_chapter/visual_kitaev_2.svg" id="fig:visual_kitaev_2" data-short-caption="Plaquette Operators are Conserved" style="width:100.0%" alt="Plaquette operators are conserved." /></p>
|
||
<p>Since the Hamiltonian is a linear combination of bond operators, it commutes with the plaquette operators. This is helpful because it leads to a simultaneous eigenbasis for the Hamiltonian and the plaquette operators. We can, thus, work in <em>or “on”???</em> a basis in which the eigenvalues of the plaquette operators take on a definite value and, for all intents and purposes, act like classical degrees of freedom. These are the extensively many conserved quantities that make the model tractable.</p>
|
||
<p>Plaquette operators measure flux. We will find that the ground state of the model corresponds to some particular choice of flux through each plaquette. We will refer to excitations which flip the expectation value of a plaquette operator away from the ground state as <strong>vortices</strong>.</p>
|
||
<p>Thus, fixing a configuration of the vortices partitions the many-body Hilbert space into a set of ‘vortex sectors’ labelled by that particular flux configuration <span class="math inline">\(\phi_i = \pm 1,\pm i\)</span>.</p>
|
||
</section>
|
||
<section id="mapping-to-a-majorana-hamiltonian" class="level2">
|
||
<h2>Mapping to a Majorana Hamiltonian</h2>
|
||
<section id="for-a-single-spin" class="level3">
|
||
<h3>For a single spin</h3>
|
||
<p>Let us start by considering only one site and its <span class="math inline">\(\sigma^x, \sigma^y\)</span> and <span class="math inline">\(\sigma^z\)</span> operators which live in a two dimensional Hilbert space <span class="math inline">\(\mathcal{L}\)</span>.</p>
|
||
<p>We will introduce two fermionic modes <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> that satisfy the canonical anticommutation relations along with their number operators <span class="math inline">\(n_f = f^\dagger f, n_g = g^\dagger g\)</span> and the total fermionic parity operator <span class="math inline">\(F_p = (2n_f - 1)(2n_g - 1)\)</span> which can be used to divide their Fock space up into even and odd parity subspaces. These subspaces are separated by the addition or removal of one fermion.</p>
|
||
<p>From these two fermionic modes, we can build four Majorana operators: <span class="math display">\[\begin{aligned}
|
||
b^x &= f + f^\dagger\\
|
||
b^y &= -i(f - f^\dagger)\\
|
||
b^z &= g + g^\dagger\\
|
||
c &= -i(g - g^\dagger)
|
||
\end{aligned}\]</span></p>
|
||
<p>The Majoranas obey the usual commutation relations, squaring to one and anticommuting with each other. The fermions and Majorana live in a four dimensional Fock space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>. We can therefore identify the two dimensional space <span class="math inline">\(\mathcal{M}\)</span> with one of the parity subspaces of <span class="math inline">\(\mathcal{\tilde{L}}\)</span> which will be called the <em>physical subspace</em> <span class="math inline">\(\mathcal{\tilde{L}}_p\)</span>. Kitaev defines the operator <span class="math display">\[D = b^xb^yb^zc\]</span> which can be expanded to <span class="math display">\[D = -(2n_f - 1)(2n_g - 1) = -F_p\]</span> and labels the physical subspace as the space spanned by states for which <span class="math display">\[ D|\phi\rangle = |\phi\rangle\]</span></p>
|
||
<p>We can also think of the physical subspace as whatever is left after applying the projector <span class="math display">\[P = \frac{1 - D}{2}\]</span> This formulation will be useful for taking states that span the extended space <span class="math inline">\(\mathcal{\tilde{M}}\)</span> and projecting them into the physical subspace.</p>
|
||
<p>So now, with the caveat that we are working in the physical subspace, we can define new Pauli operators:</p>
|
||
<p><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]</span></p>
|
||
<p>These extended space Pauli operators satisfy all the usual commutation relations. The only difference is that if we evaluate <span class="math inline">\(\sigma^x \sigma^y \sigma^z = i\)</span>, we instead get <span class="math display">\[ \tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]</span></p>
|
||
<p>This makes sense if we promise to confine ourselves to the physical subspace <span class="math inline">\(D = 1\)</span>.</p>
|
||
<section id="for-multiple-spins" class="level4">
|
||
<h4>For multiple spins</h4>
|
||
<p>This construction easily generalises to the case of multiple spins. We get a set of 4 Majoranas <span class="math inline">\(b^x_j,\; b^y_j,\;b^z_j,\; c_j\)</span> and a <span class="math inline">\(D_j = b^x_jb^y_jb^z_jc_j\)</span> operator for every spin. For a state to be physical, we require that <span class="math inline">\(D_j |\psi\rangle = |\psi\rangle\)</span> for all <span class="math inline">\(j\)</span>.</p>
|
||
<p>From these each Pauli operator can be constructed: <span class="math display">\[\tilde{\sigma}^\alpha_j = i b^\alpha_j c_j\]</span></p>
|
||
<p>This is where the magic happens. We can promote the spin hamiltonian from <span class="math inline">\(\mathcal{L}\)</span> into the extended space <span class="math inline">\(\mathcal{\tilde{L}}\)</span>, safe in the knowledge that nothing changes so long as we only actually work with physical states. The Hamiltonian <span class="math display">\[\begin{aligned}
|
||
\tilde{H} &= - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\tilde{\sigma}_j^{\alpha}\tilde{\sigma}_k^{\alpha}\\
|
||
&= \frac{i}{4} \sum_{\langle j,k\rangle_\alpha} 2J^{\alpha} (ib^\alpha_i b^\alpha_j) c_i c_j\\
|
||
&= \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j
|
||
\end{aligned}\]</span></p>
|
||
<p>We can factor out the Majorana bond operators <span class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span>. Note that these bond operators are not equal to the spin bond operators <span class="math inline">\(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = - \hat{u}_{ij} c_i c_j\)</span>. In what follows, we will work much more frequently with the Majorana bond operators. Therefore, when we refer to bond operators without qualification, we are referring to the Majorana variety.</p>
|
||
<p>Similarly to the argument with the spin bond operators <span class="math inline">\(K_{ij}\)</span>, we can quickly verify by considering three cases that the Majorana bond operators <span class="math inline">\(u_{ij}\)</span> all commute with one another. They square to one, so have eigenvalues <span class="math inline">\(\pm 1\)</span>. They also commute with the <span class="math inline">\(c_i\)</span> operators.</p>
|
||
<p>Importantly, the operators <span class="math inline">\(D_i = b^x_i b^y_i b^z_i c_i\)</span> commute with <span class="math inline">\(K_{ij}\)</span> and, therefore, with <span class="math inline">\(\tilde{H}\)</span>. We will show later that the action of <span class="math inline">\(D_i\)</span> on a state is to flip the values of the three <span class="math inline">\(u_{ij}\)</span> bonds that connect to site <span class="math inline">\(i\)</span>. Physically, this indicates that <span class="math inline">\(u_{ij}\)</span> is a gauge field with a high degree of degeneracy.</p>
|
||
<p>In summary, Majorana bond operators <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(\mathbb{Z_2}\)</span> gauge field!</p>
|
||
</section>
|
||
</section>
|
||
<section id="partitioning-the-hilbert-space-into-bond-sectors" class="level3">
|
||
<h3>Partitioning the Hilbert Space into Bond sectors</h3>
|
||
<p>Similarly to the story with the plaquette operators from the spin language, we can divide the Hilbert space <span class="math inline">\(\mathcal{L}\)</span> into sectors labelled by a set of choices <span class="math inline">\(\{\pm 1\}\)</span> for the value of each <span class="math inline">\(u_{ij}\)</span> operator which we denote by <span class="math inline">\(\mathcal{L}_u\)</span>. Since <span class="math inline">\(u_{ij} = -u_{ji}\)</span>, we can represent the <span class="math inline">\(u_{ij}\)</span> graphically with an arrow that points along each bond in the direction in which <span class="math inline">\(u_{ij} = 1\)</span>.</p>
|
||
<p>Once confined to a particular <span class="math inline">\(\mathcal{L}_u\)</span>, we can ‘remove the hats’ from the <span class="math inline">\(\hat{u}_{ij}\)</span>. The hamiltonian becomes a quadratic, free fermion problem <span class="math display">\[\tilde{H_u} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> The ground state, <span class="math inline">\(|\psi_u\rangle\)</span> can be found easily as will be explained in the next part. At this point, we may wonder whether the <span class="math inline">\(\mathcal{L}_u\)</span> are confined entirely within the physical subspace <span class="math inline">\(\mathcal{L}_p\)</span> and, indeed, we will see that they are not. However, it will be helpful to first develop the theory of the Majorana Hamiltonian further.</p>
|
||
<p><strong>The Majorana Hamiltonian</strong></p>
|
||
<p>We now have a quadratic Hamiltonian <span class="math display">\[ \tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\]</span> in which most of the Majorana degrees of freedom have paired along bonds to become a classical gauge field <span class="math inline">\(u_{ij}\)</span>. What follows is relatively standard theory for quadratic Majorana Hamiltonians <span class="citation" data-cites="BlaizotRipka1986"> [<a href="#ref-BlaizotRipka1986" role="doc-biblioref">14</a>]</span>.</p>
|
||
<p>Because of the antisymmetry of the matrix with entries <span class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of the Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> come in pairs <span class="math inline">\(\pm \epsilon_m\)</span>. This redundant information is a consequence of the doubling of the Hilbert space which occurred when we transformed to the Majorana representation.</p>
|
||
<p>If we organise the eigenmodes of <span class="math inline">\(H\)</span> into pairs, such that <span class="math inline">\(b_m\)</span> and <span class="math inline">\(b_m'\)</span> have energies <span class="math inline">\(\epsilon_m\)</span> and <span class="math inline">\(-\epsilon_m\)</span>, we can construct the transformation <span class="math inline">\(Q\)</span> <span class="math display">\[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2, b_2' ... b_{N}, b_{N}')\]</span> and put the Hamiltonian into the form <span class="math display">\[\tilde{H}_u = \frac{i}{2} \sum_m \epsilon_m b_m b_m'\]</span></p>
|
||
<p>The determinant of <span class="math inline">\(Q\)</span> will be useful later when we consider the projector from <span class="math inline">\(\mathcal{\tilde{L}}\)</span> to <span class="math inline">\(\mathcal{L}\)</span>. Otherwise, the <span class="math inline">\(b_m\)</span> are merely an intermediate step. From them, we form fermionic operators <span class="math display">\[ f_i = \tfrac{1}{2} (b_m + ib_m')\]</span> with their associated number operators <span class="math inline">\(n_i = f^\dagger_i f_i\)</span>. These let us write the Hamiltonian neatly as</p>
|
||
<p><span class="math display">\[ \tilde{H}_u = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]</span></p>
|
||
<p>The ground state <span class="math inline">\(|n_m = 0\rangle\)</span> of the many body system at fixed <span class="math inline">\(u\)</span> is then <span class="math display">\[E_{u,0} = -\frac{1}{2}\sum_m \epsilon_m \]</span> We can construct any state from a particular choice of <span class="math inline">\(n_m = 0,1\)</span>.</p>
|
||
<p>If we only care about the value of <span class="math inline">\(E_{u,0}\)</span>, it is possible to skip forming the fermionic operators. The eigenvalues obtained directly from diagonalising <span class="math inline">\(J^{\alpha} u_{ij}\)</span> come in <span class="math inline">\(\pm \epsilon_m\)</span> pairs. We can take half the absolute value of the whole set to recover <span class="math inline">\(\sum_m \epsilon_m\)</span> easily.</p>
|
||
<p>Takeaway: the Majorana Hamiltonian is quadratic within a Bond Sector.</p>
|
||
</section>
|
||
<section id="mapping-back-from-bond-sectors-to-the-physical-subspace" class="level3">
|
||
<h3>Mapping back from Bond Sectors to the Physical Subspace</h3>
|
||
<p>At this point, given a particular bond configuration <span class="math inline">\(u_{ij} = \pm 1\)</span>, we can construct a quadratic Hamiltonian <span class="math inline">\(\tilde{H}_u\)</span> in the extended space and diagonalise it to find its ground state <span class="math inline">\(|\vec{u}, \vec{n} = 0\rangle\)</span>. This is not necessarily the ground state of the system as a whole, it is just the lowest energy state within the subspace <span class="math inline">\(\mathcal{L}_u\)</span></p>
|
||
<p><strong>However, <span class="math inline">\(|u, n_m = 0\rangle\)</span> does not lie in the physical subspace</strong>. As an example, consider the lowest energy state associated with <span class="math inline">\(u_{ij} = +1\)</span>. This state satisfies <span class="math display">\[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle = |\vec{u}=1, \vec{n} = 0\rangle\]</span> for all bonds <span class="math inline">\(i,j\)</span>.</p>
|
||
<p>If we act on it, this state with one of the gauge operators <span class="math inline">\(D_j = b_j^x b_j^y b_j^z c_j\)</span>, we see that <span class="math inline">\(D_j\)</span> flips the value of the three bonds <span class="math inline">\(u_{ij}\)</span> that surround site <span class="math inline">\(k\)</span>:</p>
|
||
<p><span class="math display">\[ |u'\rangle = D_j |u=1, n_m = 0\rangle\]</span></p>
|
||
<p><span class="math display">\[ \begin{aligned}
|
||
\langle u'|u_{ij}|u'\rangle &= \langle u| b_j^x b_j^y b_j^z c_j \;ib^x_i b^x_j\; b_j^x b_j^y b_j^z c_j|u\rangle\\
|
||
&= -1
|
||
\end{aligned}\]</span></p>
|
||
<p>Since <span class="math inline">\(D_j\)</span> commutes with the Hamiltonian in the extended space <span class="math inline">\(\tilde{H}\)</span>, the fact that <span class="math inline">\(D_j\)</span> flips the value of bond operators indicates that there is a gauge degeneracy between the ground state of <span class="math inline">\(\tilde{H}_u\)</span> and the set of <span class="math inline">\(\tilde{H}_{u'}\)</span> related to it by gauge transformations <span class="math inline">\(D_j\)</span>. Thus, we can flip any three bonds around a vertex and the physics will stay the same.</p>
|
||
<p>We can turn this into a symmetrisation procedure by taking a superposition of every possible gauge transformation. Every possible gauge transformation is just every possible subset of <span class="math inline">\({D_0, D_1 ... D_n}\)</span> which can be neatly expressed as <span class="math display">\[|\phi_w\rangle = \prod_i \left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\]</span> This is convenient because the quantity <span class="math inline">\(\frac{1 + D_i}{2}\)</span> is also the local projector onto the physical subspace. Here <span class="math inline">\(|\phi_w\rangle\)</span> is a gauge invariant state that lives in <span class="math inline">\(\mathcal{L}_p\)</span> which has been constructed from a set of states in different <span class="math inline">\(\mathcal{L}_u\)</span>.</p>
|
||
<p>This gauge degeneracy leads us to the next topic of discussion, namely how to construct a set of gauge invariant quantities out of the <span class="math inline">\(u_{ij}\)</span>, these will turn out to just be the plaquette operators.</p>
|
||
<p>Takeaway: The Bond Sectors overlap with the physical subspace but are not contained within it.</p>
|
||
</section>
|
||
<section id="open-boundary-conditions" class="level3">
|
||
<h3>Open boundary conditions</h3>
|
||
<p>Care must be taken when defining open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired <span class="math inline">\(b^\alpha\)</span> operators that must be paired in some way to arrive at fermionic modes. To fix a pairing, we always start from a lattice defined on the torus and generate a lattice with open boundary conditions by defining the bond coupling <span class="math inline">\(J^{\alpha}_{ij} = 0\)</span> for sites joined by bonds <span class="math inline">\((i,j)\)</span> that we want to remove. This creates fermionic zero modes <span class="math inline">\(u_{ij}\)</span> associated with these cut bonds which we set to 1 when calculating the projector.</p>
|
||
<p>Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired <span class="math inline">\(b^\alpha\)</span> operators could be performed.</p>
|
||
</section>
|
||
</section>
|
||
<section id="emergent-gauge-fields" class="level2">
|
||
<h2>Emergent gauge fields</h2>
|
||
<figure>
|
||
<img src="/assets/thesis/amk_chapter/torus.jpeg" id="fig:torus" data-short-caption="Loops on the Torus" style="width:86.0%" alt="Figure 3: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically, the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labelled x and y, which cannot be smoothly deformed to a point. These two non-contractible loops can be used to construct two special pairs of operators: The two topological fluxes \Phi_x and \Phi_y that are the expectation values of u_{jk} loops around each path. There are also two operators \hat{\mathcal{T}}_x and \hat{\mathcal{T}}_y that transform one half of a vortex pair around the loop before annihilating them together again, see later." />
|
||
<figcaption aria-hidden="true">Figure 3: In periodic boundary conditions the Kitaev model is defined on the surface of a torus. Topologically, the torus is distinct from the sphere in that it has a hole that cannot be smoothly deformed away. Associated with each such hole are two non-contractible loops on the surface, here labelled <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span>, which cannot be smoothly deformed to a point. These two non-contractible loops can be used to construct two special pairs of operators: The two topological fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> that are the expectation values of <span class="math inline">\(u_{jk}\)</span> loops around each path. There are also two operators <span class="math inline">\(\hat{\mathcal{T}}_x\)</span> and <span class="math inline">\(\hat{\mathcal{T}}_y\)</span> that transform one half of a vortex pair around the loop before annihilating them together again, see later.</figcaption>
|
||
</figure>
|
||
<section id="ground-state-degeneracy" class="level3">
|
||
<h3>Ground State Degeneracy</h3>
|
||
<figure>
|
||
<img src="/assets/thesis/amk_chapter/loops_and_dual_loops/loops_and_dual_loops.svg" id="fig:loops_and_dual_loops" data-short-caption="Topological Loops and Dual Loops" style="width:100.0%" alt="Figure 4: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again." />
|
||
<figcaption aria-hidden="true">Figure 4: (Left) The two topological flux operators of the toroidal lattice. These do not correspond to any face of the lattice, but rather measure flux that threads through the major and minor axes of the torus. This shows a particular choice. Yet, any loop that crosses the boundary is gauge equivalent to one of or the sum of these two loop. (Right) The two ways to transport vortices around the diameters. These amount to creating a vortex pair, transporting one of them around the major or minor diameters of the torus and, then, annihilating them again.</figcaption>
|
||
</figure>
|
||
<p>More general arguments <span class="citation" data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a href="#ref-chungExplicitMonodromyMoore2007" role="doc-biblioref">15</a>,<a href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" role="doc-biblioref">16</a>]</span> imply that <span class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an interesting relationship to the topological fluxes. In the non-Abelian phase, we expect that it will change sign in exactly one of the four topological sectors.</p>
|
||
<p>This means that the lowest state in three of the topological sectors contain no fermions, while in one of them there must be one fermion to preserve product of fermion vortex parity. So overall the non-Abelian model has a three-fold degenerate ground state rather than the fourfold of the Abelian case (and of my intuition!). In the Abelian phase, this does not happen and we get a fourfold degenerate ground state. <strong>Whether this analysis generalises to the amorphous case is unclear.</strong></p>
|
||
<p>An alternative way to view this is to imagine we start in one state of the ground state manifold. We then attempt to construct other ground states by creating vortex pairs, transporting one vortex around one or both non-contractible loops and then annihilating them. This works for either of the two non-contractible loops but when we try to do it for <em>both</em> something strange happens. When we transport a vortex around <strong>both</strong> the major and minor axes of the torus this changes its fusion channel. Normally two vortices fuse to the vacuum but after this operation they fuse into a fermion excitation. And hence our attempt to construct that last ground state doesn’t yield a ground state at all, leaving us with just three.</p>
|
||
<p><strong>NOTE to self: This argument seems to involve adiabatic insertion of the fluxes <span class="math inline">\(\Phi_{x,y}\)</span> as the operations that undo vortex transport around the lattice. I don’t understand why that part is necessary</strong></p>
|
||
<figure>
|
||
<img src="/assets/thesis/amk_chapter/threefold_degeneracy.png" id="fig:threefold_degeneracy" data-short-caption="Ground State Degeneracy in the Abelian and Non-Abelian Phases" style="width:86.0%" alt="Figure 5: In the non-Abelian phase one of the lowest energy state in one of the topological sectors contains a fermion and hence is slightly higher in energy than the other three. This manifests as a fourfold ground state degeneracy in the Abelian phase and a threefold degeneracy in the non-Abelian phase." />
|
||
<figcaption aria-hidden="true">Figure 5: In the non-Abelian phase one of the lowest energy state in one of the topological sectors contains a fermion and hence is slightly higher in energy than the other three. This manifests as a fourfold ground state degeneracy in the Abelian phase and a threefold degeneracy in the non-Abelian phase.</figcaption>
|
||
</figure>
|
||
</section>
|
||
</section>
|
||
<section id="bg-the-ground-state" class="level2">
|
||
<h2>The Ground State</h2>
|
||
<p>Discuss Lieb’s theorem and generalisations for other lattices</p>
|
||
</section>
|
||
<section id="phases-of-the-kitaev-model" class="level2">
|
||
<h2>Phases of the Kitaev Model</h2>
|
||
<p>discuss the Abelian A phase / toric code phase / anisotropic phase</p>
|
||
<p>the isotropic gapless phase of the standard model</p>
|
||
<p>The isotropic gapped phase with the addition of a magnetic field </i,j></i,j></p>
|
||
<p>Next Section: <a href="../2_Background/2.3_Anyons.html">Anyonic Statistics</a></p>
|
||
</section>
|
||
</section>
|
||
<section id="bibliography" class="level1 unnumbered">
|
||
<h1 class="unnumbered">Bibliography</h1>
|
||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A. Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons in an Exactly Solved Model and Beyond</a></em>, Annals of Physics <strong>321</strong>, 2 (2006).</div>
|
||
</div>
|
||
<div id="ref-Jackeli2009" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">G. Jackeli and G. Khaliullin, <em><a href="https://doi.org/10.1103/PhysRevLett.102.017205">Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models</a></em>, Physical Review Letters <strong>102</strong>, 017205 (2009).</div>
|
||
</div>
|
||
<div id="ref-HerrmannsAnRev2018" class="csl-entry" role="doc-biblioentry">
|
||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">M. Hermanns, I. Kimchi, and J. Knolle, <em><a href="https://doi.org/10.1146/annurev-conmatphys-033117-053934">Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections</a></em>, Annual Review of Condensed Matter Physics <strong>9</strong>, 17 (2018).</div>
|
||
</div>
|
||
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