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title: Background - The Kitaev Honeycomb Model
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<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
<ul>
<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
<li><a href="#the-fermion-problem" id="toc-the-fermion-problem">The Fermion Problem</a></li>
<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
<li><a href="#sec:anyons" id="toc-sec:anyons">Anyons, Topology and the Chern number</a></li>
<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
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<li><a href="#bg-hkm-model" id="toc-bg-hkm-model">The Kitaev Honeycomb Model</a>
<ul>
<li><a href="#the-spin-hamiltonian" id="toc-the-spin-hamiltonian">The Spin Hamiltonian</a></li>
<li><a href="#the-spin-model" id="toc-the-spin-model">The Spin Model</a></li>
<li><a href="#the-majorana-model" id="toc-the-majorana-model">The Majorana Model</a></li>
<li><a href="#the-fermion-problem" id="toc-the-fermion-problem">The Fermion Problem</a></li>
<li><a href="#an-emergent-gauge-field" id="toc-an-emergent-gauge-field">An Emergent Gauge Field</a></li>
<li><a href="#sec:anyons" id="toc-sec:anyons">Anyons, Topology and the Chern number</a></li>
<li><a href="#ground-state-phases" id="toc-ground-state-phases">Ground State Phases</a></li>
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<p>2 Background</p>
<hr />
</div>
<section id="bg-hkm-model" class="level1">
<h1>The Kitaev Honeycomb Model</h1>
<figure>
<img src="/assets/thesis/amk_chapter/intro/honeycomb_zoom/intro_figure_by_hand.svg" id="fig-intro_figure_by_hand" data-short-caption="The Kitaev Honeycomb Model" style="width:100.0%" alt="Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom u_{jk} = \pm 1 that lives on each bond, the bond variables. These are antisymmetric, u_{jk} = -u_{kj}, so we represent them graphically with arrows on each bond that point in the direction that u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas b_i^x,\;b_i^y,\;b_i^z,\;c_i. The x, y and z Majoranas then pair along the bonds forming conserved \mathbb{Z}_2 bond operators u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle. The remaining c_i operators form an effective quadratic Hamiltonian H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j." />
<figcaption aria-hidden="true">Figure 1: <strong>(a)</strong> The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. <strong>(b)</strong>. After transforming to the Majorana representation we get an emergent gauge degree of freedom <span class="math inline">\(u_{jk} = \pm 1\)</span> that lives on each bond, the bond variables. These are antisymmetric, <span class="math inline">\(u_{jk} = -u_{kj}\)</span>, so we represent them graphically with arrows on each bond that point in the direction that <span class="math inline">\(u_{jk} = +1\)</span> <strong>(c)</strong>. The Majorana transformation can be visualised as breaking each spin into four Majoranas <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span>. The x, y and z Majoranas then pair along the bonds forming conserved <span class="math inline">\(\mathbb{Z}_2\)</span> bond operators <span class="math inline">\(u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle\)</span>. The remaining <span class="math inline">\(c_i\)</span> operators form an effective quadratic Hamiltonian <span class="math inline">\(H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span>.</figcaption>
</figure>
<section id="the-spin-hamiltonian" class="level2">
<h2>The Spin Hamiltonian</h2>
<p>The Kitaev Honeycomb (KH) model is an exactly solvable 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 couples 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 of the setup.</p>
<p>This gives us the Hamiltonian <span id="eq:bg-kh-model"><span class="math display">\[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, \qquad{(1)}\]</span></span> where <span class="math inline">\(\sigma^\alpha_j\)</span> is the <span class="math inline">\(\alpha\)</span> component of 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>. Kitaev introduced this model in his seminal 2006 paper <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>.</p>
<p>The KH 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: each spin would like to align along a different direction with each of its three neighbours but this cannot be achieved even classically <span class="citation" data-cites="chandraClassicalHeisenbergSpins2010 selaOrderbydisorderSpinorbitalLiquids2014"> [<a href="#ref-chandraClassicalHeisenbergSpins2010" role="doc-biblioref">7</a>,<a href="#ref-selaOrderbydisorderSpinorbitalLiquids2014" role="doc-biblioref">8</a>]</span>. This frustration leads the model to have a Quantum Sping 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">9</a>]</span>, the KH model was the first exactly solvable 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. 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>. At finite temperature the KH model undergoes a phase transition to a thermal metal state <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</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 with the remaining fermions are governed by a free fermion hamiltonian.</p>
<p>This section will go over the standard model in detail, first discussing <a href="../2_Background/2.2_HKM_Model.html#the-spin-model">the spin model</a>, then detailing the transformation to a <a href="../2_Background/2.2_HKM_Model.html#the-majorana-model">Majorana hamiltonian</a> that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the <a href="../2_Background/2.2_HKM_Model.html#an-emergent-gauge-field">emergent gauge fields</a> and the projector. The <a href="../2_Background/2.2_HKM_Model.html#sec:anyons">next section</a> will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics. Finally will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the <a href="../2_Background/2.2_HKM_Model.html#ground-state-phases">phase diagram</a>.</p>
</section>
<section id="the-spin-model" class="level2">
<h2>The Spin Model</h2>
<figure>
<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="Figure 2: A visual introduction to the Kitaev honeycomb model." />
<figcaption aria-hidden="true">Figure 2: A visual introduction to the Kitaev honeycomb model.</figcaption>
</figure>
<p>As discussed in the introduction, spin hamiltonians like that of the KH model arise in electronic systems as the result the balance of multiple effects <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. For instance, in certain transition metal systems with <span class="math inline">\(d^5\)</span> valence electrons, crystal field and spin-orbit couplings conspire to shift and split the <span class="math inline">\(d\)</span> orbitals into moments with spin <span class="math inline">\(j = 1/2\)</span> and <span class="math inline">\(j = 3/2\)</span>. Of these, the bandwidth <span class="math inline">\(t\)</span> of the <span class="math inline">\(j= 1/2\)</span> band is small, meaning that even relatively meagre electron correlations (such those induced by the <span class="math inline">\(U\)</span> term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a <span class="math inline">\(j = 1/2\)</span> Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions <span class="citation" data-cites="jackeliMottInsulatorsStrong2009 khaliullinOrbitalOrderFluctuations2005"> [<a href="#ref-jackeliMottInsulatorsStrong2009" role="doc-biblioref">12</a>,<a href="#ref-khaliullinOrbitalOrderFluctuations2005" role="doc-biblioref">13</a>]</span>. In the KH model the bond directionality refers to the fact that the coupling axis <span class="math inline">\(\alpha\)</span> in terms like <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> is strongly bond dependent.</p>
<p>In the spin hamiltonian eq. <a href="#eq:bg-kh-model">1</a> we can already tease out a set of conserved fluxes that will be key to the model’s solution. These fluxes are the expectations of Wilson loop operators</p>
<p><span class="math display">\[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]</span></p>
<p>the products of bonds winding around a closed path <span class="math inline">\(p\)</span> on the lattice. These operators commute with the Hamiltonian and so have no time dynamics. The winding direction does not matter so long as it is fixed. By convention we will always use clockwise. Each closed path on the lattice is associated with a flux. The number of conserved quantities grows linearly with system size and is thus extensive, this is a common property for exactly solvable systems and can be compared to the heavy electrons present in the Falicov-Kimball model. The square of two loop operators is one so any contractible loop can be expressed as a product of loops around plaquettes of the lattice, as in fig. <a href="#fig:stokes_theorem">3</a>. For the honeycomb lattice the plaquettes are the hexagons. The expectations of <span class="math inline">\(\hat{W}_p\)</span> through each plaquette, the fluxes, are therefore enough to describe the whole flux sector. We will focus on these fluxes, denoting them by <span class="math inline">\(\phi_i\)</span>. Once we have made the mapping to the Majorana Hamiltonian I will explain how these fluxes can be connected to an emergent <span class="math inline">\(B\)</span> field which makes their interpretation as fluxes clear.</p>
<figure>
<img src="/assets/thesis/amk_chapter/stokes_theorem/stokes_theorem.svg" id="fig-stokes_theorem" data-short-caption="We can construct arbitrary loops from plaquette fluxes." style="width:71.0%" alt="Figure 3: In the Kitaev honeycomb model, Wilson loop operators \hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha} can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \phi_i. This relationship between the u_{ij} around a region and fluxes with one is evocative of Stokes’ theorem from classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later." />
<figcaption aria-hidden="true">Figure 3: In the Kitaev honeycomb model, Wilson loop operators <span class="math inline">\(\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha}\)</span> can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette <span class="math inline">\(\phi_i\)</span>. This relationship between the <span class="math inline">\(u_{ij}\)</span> around a region and fluxes with one is evocative of Stokes’ theorem from classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later.</figcaption>
</figure>
<p>It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term <span class="math inline">\(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\)</span>, this is referred to as the Heisenberg-Kitaev model <span class="citation" data-cites="Chaloupka2010"> [<a href="#ref-Chaloupka2010" role="doc-biblioref">14</a>]</span>. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span> for a full discussion of Kitaev Materials.</p>
<p>As with the Falicov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. So again we will work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as <em>vortices</em>. In order to fully solve the model however, we must first move to a Majorana picture.</p>
</section>
<section id="the-majorana-model" class="level2">
<h2>The Majorana Model</h2>
<p>Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of <span class="math inline">\(N\)</span> fermionic creation <span class="math inline">\(f_i^\dagger\)</span> and anhilation <span class="math inline">\(f_i\)</span> operators we can construct <span class="math inline">\(2N\)</span> Majorana operators <span class="math inline">\(c_m\)</span>. We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Majorana operators square to one but otherwise have standard fermionic anti-commutation relations.</p>
<p><span class="math inline">\(N\)</span> spins can be mapped to <span class="math inline">\(N\)</span> fermions with the well known Jordan-Wigner transformation and indeed this approach can be used to solve the Kitaev model <span class="citation" data-cites="chenExactResultsKitaev2008"> [<a href="#ref-chenExactResultsKitaev2008" role="doc-biblioref">15</a>]</span>. Here I will introduce the method Kitaev used in the original paper as this forms the basis for the results that will be presented in this thesis. Rather than mapping to <span class="math inline">\(N\)</span> fermions, Kitaev maps to <span class="math inline">\(4N\)</span> Majoranas, effectively <span class="math inline">\(2N\)</span> fermions. In contrast to the Jordan-Wigner approach which makes fermions out of strings of spin operators in order to correctly produce fermionic commutation relations, the Kitaev transformation maps each spin locally to four Majoranas. The downside is that this enlarges the Hilbert space from <span class="math inline">\(2^N\)</span> to <span class="math inline">\(4^N\)</span>. We will have to employ a projector <span class="math inline">\(\hat{P}\)</span> to come back down to the physical Hilbert space later. As everything is local, I will drop the site indices <span class="math inline">\(ijk\)</span> in expressions that refer to only a single site.</p>
<p>The mapping is defined in terms of four Majoranas per site <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span> such that</p>
<p><span id="eq:bg-kh-mapping"><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]</span></span></p>
<p>The tildes on the spin operators <span class="math inline">\(\tilde{\sigma_i^\alpha}\)</span> emphasis that they live in this new extended Hilbert space and are only equivalent to the original spin operators after applying a projector <span class="math inline">\(\hat{P}\)</span>. The form of the projection operator can be understood in a few ways. From a group-theoretic perspective, before projection, the operators <span class="math inline">\(\{\tilde{\sigma}^x, \tilde{\sigma}^y, \tilde{\sigma}^z\}\)</span> form a representation of the gamma group <span class="math inline">\(G_{3,0}\)</span>. The gamma groups <span class="math inline">\(G_{p,q}\)</span> have <span class="math inline">\(p\)</span> generators that square to the identity and <span class="math inline">\(q\)</span> that square (roughly) to <span class="math inline">\(-1\)</span>. The generators otherwise obey standard anticommutation relations. The well known gamma matrices <span class="math inline">\(\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}\)</span> represent <span class="math inline">\(G_{1,3}\)</span> the quaternions <span class="math inline">\(G_{0,3}\)</span> and the Pauli matrices <span class="math inline">\(G_{3,0}\)</span>.</p>
<p>The Pauli matrices, however, have the additional property that the <em>chiral element</em> <span class="math inline">\(\sigma^x \sigma^y \sigma^z = \pmi\)</span> is not fully determined by the group properties of <span class="math inline">\(G_{3,0}\)</span> but it is equal to <span class="math inline">\(i\)</span> in the Pauli matrices. Therefore to fully reproduce the algebra of the Pauli matrices we must project into the subspace where <span class="math inline">\(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = +i\)</span>. The chiral element of the gamma matrices for instance <span class="math inline">\(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\)</span> is of central importance in quantum field theory. See <span class="citation" data-cites="petitjeanChiralityDiracSpinors2020"> [<a href="#ref-petitjeanChiralityDiracSpinors2020" role="doc-biblioref">16</a>]</span> for more discussion of this group theoretic view.</p>
<p>So the projector must project onto the subspace where <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde \sigma^z = i\)</span>. If we work this through we find that in general <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde\sigma^z = iD\)</span> where <span class="math inline">\(D = b^x b^y b^z c\)</span> must be the identity for every site. In other words, we can only work with <em>physical states</em> <span class="math inline">\(|\phi\rangle\)</span> that satisfy <span class="math inline">\(D_i|\phi\rangle = |\phi\rangle\)</span> for all sites <span class="math inline">\(i\)</span>. From this we construct an on-site projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> and the overall projector is simply <span class="math inline">\(P = \prod_i P_i\)</span>.</p>
<p>Another way to see what this is doing physically is to explicitly construct the two intermediate fermionic operators <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> that give rise to these four Majoranas. Denoting a fermion state by <span class="math inline">\(|n_f, n_g\rangle\)</span> the Hilbert space is the set <span class="math inline">\(\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\)</span>. We can map these to Majoranas with, for example, this definition</p>
<p><span class="math display">\[\begin{aligned}
b^x = (f + f^\dagger),\;\;&amp; b^y = -i(f - f^\dagger),\\
b^z = (g + g^\dagger),\;\;&amp; c = -i(g - g^\dagger),
\end{aligned}\]</span></p>
<p>Working through the algebra we see that the operator <span class="math inline">\(D = b^x b^y b^z c\)</span> is equal to the fermion parity <span class="math inline">\(D = -(2n_f - 1)(2n_g - 1) = \pm1\)</span> where <span class="math inline">\(n_f,\; n_g\)</span> are the number operators. So setting <span class="math inline">\(D = 1\)</span> everywhere is equivalent to restricting to the <span class="math inline">\(\{|01\rangle,|10\rangle\}\)</span> though we could equally well have used the other one.</p>
<p>Expanding the product <span class="math inline">\(\prod_i P_i\)</span> out, we find that the projector corresponds to a symmetrisation over <span class="math inline">\(\{u_{ij}\}\)</span> states within a flux sector and and overall fermion parity <span class="math inline">\(\prod_i D_i\)</span>, see <span class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a href="#ref-pedrocchiPhysicalSolutionsKitaev2011" role="doc-biblioref">17</a>]</span> or <a href="../6_Appendices/A.5_The_Projector.html#app-the-projector">appendix A.5</a> for the full derivation. The significance of this is that an arbitrary many-body state can be made to have non-zero overlap with the physical subspace via the addition or removal of just a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results</p>
<p>We can now rewrite the spin hamiltonian in Majorana form with the caveat that they are only strictly equivalent after projection. The Ising interactions <span class="math inline">\(\sigma_j^{\alpha}\sigma_k^{\alpha}\)</span> decouple into the form <span class="math inline">\(-i (i b^\alpha_i b^\alpha_j) c_i c_j\)</span>. We factor out the <em>bond operators</em> <span class="math inline">\(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\)</span> which are Hermitian and, remarkably, commute with the Hamiltonian and each other.</p>
<p><span class="math display">\[\begin{aligned}
\tilde{H} &amp;= - \sum_{\langle i,j\rangle_\alpha} J^{\alpha}\tilde{\sigma}_i^{\alpha}\tilde{\sigma}_j^{\alpha}\\
&amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j
\end{aligned}\]</span></p>
<p>The bond operators <span class="math inline">\(\hat{u}_{ij}\)</span> square to one so have eigenvalues <span class="math inline">\(\pm1\)</span>. As they’re conserved we will work in their eigenbasis and take off the hats in the Hamiltonian.</p>
<p><span id="eq:bg-kh-maj-model"><span class="math display">\[\begin{aligned}
H &amp;= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j
\end{aligned}\qquad{(3)}\]</span></span></p>
</section>
<section id="the-fermion-problem" class="level2">
<h2>The Fermion Problem</h2>
<p>We now have a quadratic Hamiltonian, eq. <a href="#eq:bg-kh-maj-model">3</a>, coupled to a classical field <span class="math inline">\(u_{ij}\)</span>. What follows is relatively standard theory for quadratic Hamiltonians <span class="citation" data-cites="BlaizotRipka1986"> [<a href="#ref-BlaizotRipka1986" role="doc-biblioref">18</a>]</span>.</p>
<p>Because of the antisymmetry <span class="math inline">\(J^{\alpha} u_{ij}\)</span>, the eigenvalues of eq. <a href="#eq:bg-kh-maj-model">3</a> come in pairs <span class="math inline">\(\pm \epsilon_m\)</span>. 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">\(b_m&#39;\)</span> have energies <span class="math inline">\(\epsilon_m\)</span> and <span class="math inline">\(-\epsilon_m\)</span>. The transformation <span class="math inline">\(Q\)</span></p>
<p><span class="math display">\[(c_1, c_2... c_{2N}) Q = (b_1, b_1&#39;, b_2, b_2&#39; ... b_{N}, b_{N}&#39;)\]</span></p>
<p>puts the Hamiltonian into normal mode form</p>
<p><span class="math display">\[H = \frac{i}{2} \sum_m \epsilon_m b_m b_m&#39;.\]</span></p>
<p>The determinant of <span class="math inline">\(Q\)</span> appears when evaluating the projector explicitly, otherwise, the <span class="math inline">\(b_m\)</span> are merely an intermediate step. From them, we form fermionic operators</p>
<p><span class="math display">\[ f_i = \tfrac{1}{2} (b_m + ib_m&#39;)\]</span></p>
<p>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">\[ H = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]</span></p>
<p>The energy of the ground state <span class="math inline">\(|n_m = 0\rangle\)</span> of the many-body system at fixed <span class="math inline">\(\{u_{ij}\}\)</span> is</p>
<p><span class="math display">\[E_{0} = -\frac{1}{2}\sum_m \epsilon_m \]</span></p>
<p>and we can construct any state from a particular choice of <span class="math inline">\(n_m = 0,1\)</span>. If we only care about the ground state energy <span class="math inline">\(E_{0}\)</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 set to recover <span class="math inline">\(\sum_m \epsilon_m\)</span> directly.</p>
</section>
<section id="an-emergent-gauge-field" class="level2">
<h2>An Emergent Gauge Field</h2>
<p>We have transformed the spin Hamiltonian into a Majorana hamiltonian <span class="math inline">\(H = i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\)</span> describing the dynamics of a classical field <span class="math inline">\(u_{ij}\)</span> and Majoranas <span class="math inline">\(c_i\)</span>. It is natural to ask how the classical field <span class="math inline">\(u_{ij}\)</span> relates to the fluxes of the original spin model. We can evaluate the fluxes <span class="math inline">\(\phi_i\)</span> in terms of the bond operators</p>
<p><span id="eq:flux-majorana"><span class="math display">\[\phi_i = \prod_{\langle j,k\rangle \in \mathcal{P}_i} i u_{jk}.\qquad{(4)}\]</span></span></p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/gauge_symmetries/gauge_symmetries.svg" id="fig-gauge_symmetries" data-short-caption="Gauge Symmetries" style="width:100.0%" alt="Figure 4: A honeycomb lattice with edges in grey, along with its dual, the triangle lattice in red. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator D_j at a vertex is to flip the value of the three u_{jk} variables (black lines) surrounding site j. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent D_j operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing D_j operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of u_{ij}s flipped on its edge. Therefore, all retain the same flux \phi_i." />
<figcaption aria-hidden="true">Figure 4: A honeycomb lattice with edges in grey, along with its dual, the triangle lattice in red. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator <span class="math inline">\(D_j\)</span> at a vertex is to flip the value of the three <span class="math inline">\(u_{jk}\)</span> variables (black lines) surrounding site <span class="math inline">\(j\)</span>. The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent <span class="math inline">\(D_j\)</span> operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing <span class="math inline">\(D_j\)</span> operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of <span class="math inline">\(u_{ij}\)</span>s flipped on its edge. Therefore, all retain the same flux <span class="math inline">\(\phi_i\)</span>.</figcaption>
</figure>
<p>In addition, the bond operators form a highly degenerate description of the system. 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">\(H\)</span> forming a set of local symmetries. 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>. This changes the bond configuration <span class="math inline">\(\{u_{ij}\}\)</span> but leaves the flux configuration <span class="math inline">\(\{\phi_i\}\)</span> unchanged. Physically, we interpret <span class="math inline">\(u_{ij}\)</span> as a gauge field with a high degree of degeneracy and <span class="math inline">\(\{D_i\}\)</span> as the set of gauge symmetries. The Majorana bond operators <span class="math inline">\(u_{ij}\)</span> are an emergent, classical, <span class="math inline">\(\mathbb{Z}_2\)</span> gauge field! The flux configuration <span class="math inline">\(\{\phi_i\}\)</span> is what encodes physical information about the system without all the gauge degeneracy.</p>
<p>The ground state of the KH model is the flux configuration where all fluxes are one <span class="math inline">\(\{\phi_i = +1\; \forall \; i\}\)</span>. This can be proven via Lieb’s theorem <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">19</a>]</span> which gives the lowest energy magnetic flux configuration for a system of electrons hopping in a magnetic field. Kitaev remarks in his original paper that he was not initially aware of the relevance of Lieb’s 1994 result. This is not surprising because at first glance the two models seem quite different but the connection is quite instructive for understanding the KH and its generalisations.</p>
<p>Lieb discusses a model of mobile electrons</p>
<p><span class="math display">\[H = \sum_{ij} t_{ij} c^\dagger_i c_j\]</span></p>
<p>where the hopping terms <span class="math inline">\(t_{ij} = |t_{ij}|\exp(i\theta_{ij})\)</span> incorporate Aharanhov-Bohm (AB) phases <span class="citation" data-cites="aharonovSignificanceElectromagneticPotentials1959"> [<a href="#ref-aharonovSignificanceElectromagneticPotentials1959" role="doc-biblioref">20</a>]</span> <span class="math inline">\(\theta_{ij}\)</span>. The AB phases model the effect of a slowly varying magnetic field on the electrons through the integral of the magnetic vector potential <span class="math inline">\(\theta_{ij} = \int_i^j \vec{A} \cdot d\vec{l}\)</span>, a Peierls substitution <span class="citation" data-cites="peierlsZurTheorieDiamagnetismus1933"> [<a href="#ref-peierlsZurTheorieDiamagnetismus1933" role="doc-biblioref">21</a>]</span>. If we map the Majorana form of the Kitaev model to Lieb’s model we see that our <span class="math inline">\(t_{ij} = i J^\alpha u_{ij}\)</span>. The <span class="math inline">\(i u_{ij} = \pm i\)</span> correspond to AB phases <span class="math inline">\(\theta_{ij} = \pi/2\)</span> or <span class="math inline">\(3\pi/2\)</span> along each bond.</p>
<figure>
<img src="/assets/thesis/amk_chapter/intro/regular_plaquettes/regular_plaquettes.svg" id="fig-regular_plaquettes" data-short-caption="Plaquettes in the Kitaev Model" style="width:86.0%" alt="Figure 5: Lieb’s theorem gives the ground state flux configuration for even sided plaquettes in systems with at least one translationally invariant direction  [19]. These labels correspond to the Kitaev fluxes \phi_i rather than the magnetic fluxes Q_i of Lieb’s original paper (\phi_i = \exp{iQ_i}). Other work has extended Lieb’s theorem numerically to arbitrary plaquettes,  [22–25]. The additional twofold degeneracy of the \pm i, \mp i terms is a consequence of the odd sided plaquettes breaking chiral symmetry. Chiral symmetry is spontaneously broken in the ground state  [23]." />
<figcaption aria-hidden="true">Figure 5: Lieb’s theorem gives the ground state flux configuration for even sided plaquettes in systems with at least one translationally invariant direction <span class="citation" data-cites="lieb_flux_1994"> [<a href="#ref-lieb_flux_1994" role="doc-biblioref">19</a>]</span>. These labels correspond to the Kitaev fluxes <span class="math inline">\(\phi_i\)</span> rather than the magnetic fluxes <span class="math inline">\(Q_i\)</span> of Lieb’s original paper (<span class="math inline">\(\phi_i = \exp{iQ_i}\)</span>). Other work has extended Lieb’s theorem numerically to arbitrary plaquettes, <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">22</a>–<a href="#ref-Peri2020" role="doc-biblioref">25</a>]</span>. The additional twofold degeneracy of the <span class="math inline">\(\pm i, \mp i\)</span> terms is a consequence of the odd sided plaquettes breaking chiral symmetry. Chiral symmetry is spontaneously broken in the ground state <span class="citation" data-cites="Yao2009"> [<a href="#ref-Yao2009" role="doc-biblioref">23</a>]</span>.</figcaption>
</figure>
<p>Stokes’ theorem tells us that the magnetic flux <span class="math inline">\(Q_m\)</span> through a surface <span class="math inline">\(S\)</span> is related to the line integral of <span class="math inline">\(\vec{A}\)</span> along the boundary of the surface <span class="math inline">\(\partial S\)</span> which in the discrete case reduces to the sum of the AB phases along the path. We can thus rewrite eq. <a href="#eq:flux-majorana">4</a> as</p>
<p><span id="eq:flux-magnetic"><span class="math display">\[\begin{aligned}
\phi_i &amp;= \prod_{\mathcal{P}_i} i u_{jk}\\
&amp;= \prod_{\mathcal{P}_i} \exp{(i\theta_{jk}})\\
&amp;= \exp \left( i \sum_{\mathcal{P}_i} \theta_{jk} \right)\\
&amp;= \exp \left( i \oint_{\mathcal{P}_i} \vec{A} \cdot d\vec{l} \right)\\
&amp;= \exp \left( i Q_i \right)
\end{aligned}\qquad{(5)}\]</span></span></p>
<p>Thus we can interpret the fluxes <span class="math inline">\(\phi_i\)</span> as the exponential of magnetic fluxes <span class="math inline">\(Q_m\)</span> of some fictitious gauge field <span class="math inline">\(\vec{A}\)</span> and the bond operators as <span class="math inline">\(i u_{ij} = \exp i \int_i^j \vec{A} \cdot d\vec{l}\)</span>. In this analogy to classical electromagnetism, the sets <span class="math inline">\(\{u_{ij}\}\)</span> that correspond to the same <span class="math inline">\(\{\phi_i\}\)</span> are all gauge equivalent as we have already seen via other means. The fact that fluxes can be written as products of bond operators and composed is a consequence of eq. <a href="#eq:flux-magnetic">5</a>. If the lattice contains odd plaquettes, as in the Yao-Kivelson model <span class="citation" data-cites="yaoExactChiralSpin2007"> [<a href="#ref-yaoExactChiralSpin2007" role="doc-biblioref">26</a>]</span>, the complex fluxes that appear are a sign that chiral symmetry has been broken.</p>
<p>In full, Lieb’s theorem states that the ground state has magnetic flux <span class="math inline">\(Q_i = \sum_{\mathcal{P}_i}\theta_{ij} = \pi \; (\mathrm{mod} \;2\pi)\)</span> for plaquettes with <span class="math inline">\(0 \; (\mathrm{mod}\;4)\)</span> sides and <span class="math inline">\(0 \; (\mathrm{mod}\;2\pi)\)</span> for plaquettes with <span class="math inline">\(2 \; (\mathrm{mod}\;4)\)</span> sides. In terms of our fluxes, this means <span class="math inline">\(\phi = -1\)</span> for squares, <span class="math inline">\(\phi = 1\)</span> for hexagons and so on.</p>
<p>While Lieb’s theorem is restricted to bipartite lattices with translational symmetry, other works have shown numerically that it tends to hold for more general lattices too <span class="citation" data-cites="eschmannThermodynamicClassificationThreedimensional2020 Yao2009 eschmann2019thermodynamics Peri2020"> [<a href="#ref-eschmannThermodynamicClassificationThreedimensional2020" role="doc-biblioref">22</a>–<a href="#ref-Peri2020" role="doc-biblioref">25</a>]</span>. From this we find that the generalisation to odd sided plaquettes is similar but with an additional chiral symmetry, so <span class="math inline">\(\phi = \pm i\)</span> for plaquettes with <span class="math inline">\(1 \; (\mathrm{mod}\;4)\)</span> sides and <span class="math inline">\(\mp i\)</span> for those with <span class="math inline">\(3 \; (\mathrm{mod}\;4)\)</span> sides. Overall we can write <span class="math inline">\(\phi = -(\pm i)^{n_{\mathrm{sides}}}\)</span>. Later I will present numerical evidence that this rule continues to hold for general amorphous lattices.</p>
<p>Understanding <span class="math inline">\(u_{ij}\)</span> as a gauge field provides another way to understand the action of the projector. The local projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> applied to a state constructs a superposition of the original state and the gauge equivalent state linked to it by flipping the three <span class="math inline">\(u_{ij}\)</span> around site <span class="math inline">\(i\)</span>. The overall projector <span class="math inline">\(P = \prod_i P_i\)</span> can thus be understood as a symmetrisation over all gauge equivalent states, removing the gauge degeneracy introduced by the mapping from spins to Majoranas.</p>
<!-- <figure>
<img src="../../figure_code/amk_chapter/intro/flood_fill/flood_fill.gif" style="max-width:700px;" title="Gauge Operators">
<figcaption>
A honeycomb lattice (in black) along with its dual (in red). (Left) The product of sets of $D_j$ operators (Bold Vertices) can be used to construct arbitrary contractible loops that flip $u_{ij}$ values. If we take the product of _every_ $D_j$ the boundary contracts to a point and disappears. This is a visual proof that $\prod_i D_i \propto \mathbb{1}$. This observation forms a key part of constructing an explicit expression for the projector, see [appendix A.5](#app-the-projector). (Right) In black and red the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the _plaquettes_ in the system is __not__ equivalent to the identity. Not that the edges that must be flipped can always be chosen from a spanning tree since loops can always be removed by a gauge transformation.
</figcaption>
</figure> -->
<figure>
<img src="/assets/thesis/amk_chapter/topological_fluxes.png" id="fig-topological_fluxes" data-short-caption="Topological Fluxes" style="width:57.0%" alt="Figure 6: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus (red) or through the filling (green). If they made doughnuts which had both had a jam filling and a hole, this analogy would be a lot easier to make  [27]." />
<figcaption aria-hidden="true">Figure 6: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus (red) or through the filling (green). If they made doughnuts which had both had a jam filling and a hole, this analogy would be a lot easier to make <span class="citation" data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis" role="doc-biblioref">27</a>]</span>.</figcaption>
</figure>
<p>A final but important point to mention is that the local fluxes <span class="math inline">\(\phi_i\)</span> are not quite all there is. We’ve seen that products of <span class="math inline">\(\phi_i\)</span> can be used to construct the flux associated with arbitrary contractible loops. On the plane contractible loops are all there is. However, on the torus we can construct two global fluxes <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> which correspond to paths tracing the major and minor axes. The four sectors spanned by the <span class="math inline">\(\pm1\)</span> values of these fluxes are gapped away from one another but only by virtual tunnelling processes so the gap decays exponentially with system size <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. Physically <span class="math inline">\(\Phi_x\)</span> and <span class="math inline">\(\Phi_y\)</span> could be thought of as measuring the flux that threads through the hole of the doughnut. In general, surfaces with genus <span class="math inline">\(g\)</span> have <span class="math inline">\(g\)</span> ‘handles’ and <span class="math inline">\(2g\)</span> of these global fluxes. At first glance it may seem this would not have much relevance to physical realisations of the Kitaev model that will likely have a planar geometry with open boundary conditions. However these fluxes are closely linked to topology and the existence of anyonic quasiparticle excitations in the model, which we will discuss next.</p>
</section>
<section id="sec:anyons" class="level2">
<h2>Anyons, Topology and the Chern number</h2>
<figure>
<img src="/assets/thesis/amk_chapter/braiding.png" id="fig-braiding" data-short-caption="Braiding in Two Dimensions" style="width:71.0%" alt="Figure 7: Worldlines of particles in 2D can become tangled or braided with one another." />
<figcaption aria-hidden="true">Figure 7: Worldlines of particles in 2D can become tangled or <em>braided</em> with one another.</figcaption>
</figure>
<p>To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of <span class="math inline">\(\pm1\)</span> if two identical particles are swapped. Only <span class="math inline">\(\pm1\)</span> are allowed since swapping twice must correspond to the identity. This argument works in 3D for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.</p>
<p>In gapped condensed matter systems, all equal time correlators decay exponentially with distance <span class="citation" data-cites="hastingsLiebSchultzMattisHigherDimensions2004"> [<a href="#ref-hastingsLiebSchultzMattisHigherDimensions2004" role="doc-biblioref">28</a>]</span>. Put another way, gapped systems support quasiparticles with a definite location in space and finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding. Recently, braiding in topological systems has attracted interest because of proposals to use ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations <span class="citation" data-cites="kitaev_fault-tolerant_2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>–<a href="#ref-hastingsDynamicallyGeneratedLogical2021" role="doc-biblioref">31</a>]</span>.</p>
<p>First we realise that in 2D, swapping identical particles twice is not topologically equivalent to the identity, see fig. <a href="#fig:braiding">7</a>. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase <span class="math inline">\(e^{i\theta}\)</span> upon exchange, hence the name <strong>any</strong>-ons. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms an Abelian group.</p>
<p>The KH model has a topologically degenerate ground state with sectors labelled by the values of the topological fluxes <span class="math inline">\((\Phi_x\)</span>, <span class="math inline">\(\Phi_y)\)</span>. Consider the operation in which a quasiparticle pair is created from the ground state, transported around one of the non-contractible loops and then annihilated together, call them <span class="math inline">\(\mathcal{T}_{x}\)</span> and <span class="math inline">\(\mathcal{T}_{y}\)</span>. These operations move the system around within the ground state manifold and they need not commute. This leads to non-Abelian anyons. As Kitaev points out, these operations are not specific to the torus: the operation <span class="math inline">\(\mathcal{T}_{x}\mathcal{T}_{y}\mathcal{T}_{x}^{-1}\mathcal{T}_{y}^{-1}\)</span> corresponds to an operation in which none of the particles crosses the torus, rather one simply winds around the other, hence these effects are relevant even for the planar case.</p>
<!-- <figure>
<img src="../../figure_code/amk_chapter/intro/types_of_dual_loops_animated/types_of_dual_loops_animated.gif" style="max-width:700px;" title="Dual Loops and Vortex Pairs">
<figcaption>
The different kinds of strings and loops that we can make by flipping bond variables or transporting vortices around. (a) Flipping a single bond $u_{ij}$ makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path (in red) on the dual lattice. (c) If we create a vortex-vortex pair, transport one of them around a loop and then annihilate them, we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus, we create a non-contractable loop of bonds $\hat{\mathcal{T}}_{x/y}$. Unlike all the other dual loops, These operators cannot be constructed from the contractable loops created by $D_j$. operators and they flip the value of the topological fluxes.
This all works the same way for the amorphous lattice but the diagram is a lot messier so I've stuck with the honeycomb here.
</figcaption>
</figure> -->
<p>In condensed matter systems, the existence of anyonic excitations automatically implies the system has topological ground state degeneracy on the torus <span class="citation" data-cites="einarssonFractionalStatisticsTorus1990"> [<a href="#ref-einarssonFractionalStatisticsTorus1990" role="doc-biblioref">32</a>]</span> and indeed anyons and topology are intimately linked <span class="citation" data-cites="oshikawaTopologicalDegeneracyNonAbelian2007 Chung_Topological_quantum2010 yaoAlgebraicSpinLiquid2009"> [<a href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007" role="doc-biblioref">33</a>–<a href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">35</a>]</span>. The Chern number <span class="math inline">\(\nu\)</span>, a concept originally used to describe complex vector bundles in algebraic topology <span class="citation" data-cites="chernCharacteristicClassesHermitian1946"> [<a href="#ref-chernCharacteristicClassesHermitian1946" role="doc-biblioref">36</a>]</span>, has found use in physics as a powerful diagnostic tool for topological systems. Kitaev showed that there is a full classification of anyonic statistics in terms of <span class="math inline">\(\nu\;(\mathrm{mod}\;16)\)</span> but relevant to us is that vortices in the KH model are Abelian when the Chern number is even and non-Abelian when the Chern number is odd. Non-Abelian statistics of the vortices in the KH model arise due to unpaired Majorana modes that are bound to them.</p>
</section>
<section id="ground-state-phases" class="level2">
<h2>Ground State Phases</h2>
<figure>
<img src="/assets/thesis/background_chapter/KH_phase_diagram.svg" id="fig-KH_phase_diagram" data-short-caption="Kitaev Honeycomb Model Phase Diagram" style="width:100.0%" alt="Figure 8: Setting the energy scale of the KH model with the constraint that J_x + J_y + J_z = 1 yields a triangular phase diagram where each of the corners represents J_\alpha = 1. For each corner \alpha the region |J_\alpha &gt; |J_\beta| + |J_\gamma| supports a gapped non-Abelian phase equivalent to that of the Toric code  [29,37]. The point around equal coupling J_x = J_y = J_z, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene." />
<figcaption aria-hidden="true">Figure 8: Setting the energy scale of the KH model with the constraint that <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram where each of the corners represents <span class="math inline">\(J_\alpha = 1\)</span>. For each corner <span class="math inline">\(\alpha\)</span> the region <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span> supports a gapped non-Abelian phase equivalent to that of the Toric code <span class="citation" data-cites="kitaev1997quantum kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>,<a href="#ref-kitaev1997quantum" role="doc-biblioref">37</a>]</span>. The point around equal coupling <span class="math inline">\(J_x = J_y = J_z\)</span>, the B phase, is gapless. The B phase is known a Majorana metal and on the honeycomb lattice it has a Dirac cone dispersion similar to that of graphene.</figcaption>
</figure>
<p>Setting the overall energy scale with the constraint <span class="math inline">\(J_x + J_y + J_z = 1\)</span> yields a triangular phase diagram. In each of the corners one of the spin-coupling directions dominates, <span class="math inline">\(|J_\alpha &gt; |J_\beta| + |J_\gamma|\)</span>, yielding three equivalent <span class="math inline">\(A_\alpha\)</span> phases while the central triangle around <span class="math inline">\(J_x = J_y = J_z\)</span> is called the B phase. Both phases support two kinds of quasiparticles, fermions and <span class="math inline">\(\mathbb{Z}_2\)</span>-vortices. In the A phases, the vortices have bosonic statistics with respect to themselves but act like fermions with respect to the fermions, hence they are Abelian anyons, This phase has the same anyonic structure as the Toric code <span class="citation" data-cites="kitaev_fault-tolerant_2003"> [<a href="#ref-kitaev_fault-tolerant_2003" role="doc-biblioref">29</a>]</span>. The B phase can be described as a semi-metal of the Majorana fermions <span class="citation" data-cites="TrebstPhysRep2022"> [<a href="#ref-TrebstPhysRep2022" role="doc-biblioref">5</a>]</span>. Since the B phase is gapless, the quasiparticles aren’t localised and so don’t have braiding statistics.</p>
<p>An external magnetic can be used to break chiral symmetry. The lowest order term that breaks chiral symmetry but retains the solvability of the model is the three spin term <span class="math display">\[
\sum_{(i,j,k)} \sigma_i^{\alpha} \sigma_j^{\beta} \sigma_k^{\gamma}
\]</span> where the sum <span class="math inline">\((i,j,k)\)</span> runs over consecutive indices around plaquettes. The addition of this to the spin model leads to two bond terms in the corresponding Majorana model. The effect of breaking chiral symmetry is to open a gap in the B phase. The vortices of the gapped B phase are non-Abelian anyons. This phase has the same anyonic exchange statistics as <span class="math inline">\(p_x + ip_y\)</span> superconductor <span class="citation" data-cites="readPairedStatesFermions2000"> [<a href="#ref-readPairedStatesFermions2000" role="doc-biblioref">38</a>]</span>, the Moore-Read state for the <span class="math inline">\(\nu = 5/2\)</span> fractional quantum Hall state <span class="citation" data-cites="mooreNonabelionsFractionalQuantum1991"> [<a href="#ref-mooreNonabelionsFractionalQuantum1991" role="doc-biblioref">39</a>]</span> and many other systems <span class="citation" data-cites="aliceaNonAbelianStatisticsTopological2011 fuSuperconductingProximityEffect2008 lutchynMajoranaFermionsTopological2010 oregHelicalLiquidsMajorana2010 sauGenericNewPlatform2010"> [<a href="#ref-aliceaNonAbelianStatisticsTopological2011" role="doc-biblioref">40</a>–<a href="#ref-sauGenericNewPlatform2010" role="doc-biblioref">44</a>]</span>. Collectively these systems have attracted interest as possible physical realisations for braiding based quantum computers.</p>
<p>At finite temperatures, recent work has shown that the KH model undergoes a transition to a thermal metal phase. Vortex disorder causes the fermion gap to fill up and the DOS has a characteristic logarithmic divergence at zero energy which can be understood from random matrix theory <span class="citation" data-cites="selfThermallyInducedMetallic2019"> [<a href="#ref-selfThermallyInducedMetallic2019" role="doc-biblioref">11</a>]</span>.</p>
<p>To surmise, the KH model is remarkable because it combines three key properties. First, the form of the Hamiltonian can 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 KH 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">45</a>]</span>. Second, its ground state is the canonical example of the long sought after QSL state, its dynamical spin-spin correlation functions are zero beyond nearest neighbour separation <span class="citation" data-cites="baskaranExactResultsSpin2007"> [<a href="#ref-baskaranExactResultsSpin2007" role="doc-biblioref">46</a>]</span>. Its excitations are anyons, particles that can only exist in 2D that break the normal fermion/boson dichotomy.</p>
<p>Third, and perhaps most importantly, this model is a rare many-body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many-body ground states in terms of single particle states <span class="citation" data-cites="kitaevAnyonsExactlySolved2006"> [<a href="#ref-kitaevAnyonsExactlySolved2006" role="doc-biblioref">1</a>]</span>. The solubility of the KH model, like the Falicov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.</p>
<p>Next Section: <a href="../2_Background/2.4_Disorder.html">Disorder and Localisation</a></p>
</section>
</section>
<section id="bibliography" class="level1 unnumbered">
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<div class="csl-left-margin">[46] </div><div class="csl-right-inline">G. Baskaran, S. Mandal, and R. Shankar, <em><a href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys. Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
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