2 Background
The Kitaev Honeycomb Model
The Spin Hamiltonian
This section introduces the seminal Kitaev honeycomb (KH) model. The KH model is an exactly solvable microscopic model of interacting spin\(-1/2\) spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label \(\alpha \in \{ x, y, z\}\) and that bond couple two spins along the \(\alpha\) axis. See fig. 1 for a diagram.
This gives us the Hamiltonian \[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha} + \Gamma \mathrm{three spin term}, \qquad{(1)}\] where \(\sigma^\alpha_j\) is a Pauli matrix acting on site \(j\) and \(\langle j,k\rangle_\alpha\) is a pair of nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\) [1].
The Kitaev Honeycomb model [1] can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds [2–6]. 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 [7], 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 [8]. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field and the remaining fermions are governed by a free fermion hamiltonian.
This section will go over the standard model in detail, first discussing the spin model, then detailing the transformation to a Majorana hamiltonian that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the emergent gauge fields and the projector. We 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 phase diagram.
The next section will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics.
The Spin Model
Eq. 1 shows
The Majorana Model
The Kitaev Honeycomb model \[H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha},\] is remarkable because it combines three key properties.
First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as \(\alpha\mathrm{-RuCl}_3\), 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 [5,9].
expand later: Why do we need spin orbit coupling and what will the corrections be?
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 [10].
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 [1]. 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.
“dynamical two spin correlation functions are identically zero beyond nearest neighbor separation in the Kitaev Model” [11]
- strong spin orbit coupling yields spatial anisotropic spin exchange leading to compass models [12]
- spin model of the Kitaev model is one
- has extensively many conserved fluxes
intro - 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)
- RuCl_3 is the classic QSL candidate material
- really follows the Kitaev-Heisenberg model
- experimental probes include inelastic neutron scattering, Raman scattering
Exact Solvability
For notational brevity, it is useful to introduce the spin bond operators \(K_{ij} = \sigma_j^{\alpha}\sigma_k^{\alpha}\) where \(\alpha\) is a function of \(i,j\) that picks the correct bond type.
This Kitaev model has a set of conserved quantities that, in the spin language, take the form of Wilson loop operators \(W_p\) 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 \(\phi\) to refer to a Wilson loop operator that does not enclose any other sites, such as a single hexagon in a honeycomb lattice.
\[W_p = \prod_{\mathrm{i,j}\; \in\; p} K_{ij}\]
In closed loops, each site appears twice in the product with two of the three bond types. Applying \(\sigma^\alpha \sigma^\beta = \epsilon^{\alpha \beta \gamma} \sigma^\gamma, \alpha \neq \beta\) then gives us a product containing a single Pauli matrix associated with each site in the loop with the type of the outward pointing bond. Hence the \(W_p\) associated with hexagons or shapes with an even number of sides all square to 1 and, hence, have eigenvalues \(\pm 1\). As the honeycomb lattice is bipartite, there are no closed loops that contain an odd number of edges. On other lattices [13], plaquettes with an odd number of sides (odd plaquettes) have eigenvalues \(\pm i\).
Remarkably, all of the spin bond operators \(K_{ij}\) commute with all the Wilson loop operators \(W_p\). \[[W_p, K_{ij}] = 0\] We can prove this by considering three cases: 1. neither \(i\) nor \(j\) is part of the loop 2. one of \(i\) or \(j\) are part of the loop 3. both are part of the loop
The first case is trivial. The other two require some algebra, outlined in fig. ¿fig:visual_kitaev_2?.
Glossary
Lattice: The underlying graph on which the models are defined. Composed of sites (vertices), bonds (edges) and plaquettes (faces).
The model : Used when I refer to properties of the the Kitaev model that do not depend on the particular lattice.
The Honeycomb model : The Kitaev Model defined on the honeycomb lattice.
The Amorphous model : The Kitaev Model defined on the amorphous lattices described here.
The Spin Hamiltonian
- Spin Bond Operators: \(\hat{k}_{ij} = \sigma_i^\alpha \sigma_j^\alpha\)
- Loop Operators: \(\hat{W_p} = \prod_{<i,j>} k_{ij}\)
- Plaquette Operators: Loops that enclose a single plaquette.
The Majorana Model
- Majorana Operators on site \(i\): \(\hat{b}^x_i, \hat{b}^y_i, \hat{b}^z_i, \hat{c}_i\)
- Majorana Bond Operators: \(\hat{u}_{ij} = i b_i^\alpha b_j^\alpha\)
- Loop Operators: \(\hat{W_p} = \prod_{<i,j>} u_{ij}\)
- Plaquette Operators: Loops that enclose a single plaquette.
- Gauge Operators: \(D_i = \hat{b}^x_i \hat{b}^y_i \hat{b}^z_i \hat{c}_i\)
- The Extended Hilbert space: The larger Hilbert space spanned by the Majorana operators.
- The physical subspace: The subspace of the extended Hilbert space that we identify with the Hilbert space of the original spin model.
- The Projector \(\hat{P}\): The projector onto the physical subspace.
Flux Sectors
Odd/Even Plaquettes: Plaquettes with an odd/even number of sides.
Fluxes \(\phi_i\): The expectation values of the plaquette operators \(\pm 1\) for even and \(\pm i\) for odd plaquettes.
Flux Sector: A subspace of Hilbert space in which the fluxes take particular values.
Ground state flux sector: The Flux Sector containing the lowest energy many body state.
Vortices: Flux excitations away from the ground state flux sector.
Dual Loops: A set of \(u_{jk}\) that correspond to loops on the dual lattice.
non-contractible loops or dual loops: The two loops topologically distinct loops on the torus that cannot be smoothly deformed to a point.
Topological Fluxes \(\Phi_{x}, \Phi_{y}\): The two fluxes associated with the two non-contractible loops.
Topological Transport Operators: \(\mathcal{T}_{x}, \mathcal{T}_{y}\): The two vortex-pair operations associated with the non-contractible dual loops.
Phases
- The A phase: The three anisotropic regions of the phase diagram \(A_x, A_y, A_z\) where \(A_\alpha\) means \(J_\alpha >> J_\beta, J_\gamma\).
- The B phase: The roughly isotropic region of the phase diagram.
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 or “on”??? 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.
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 vortices.
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 \(\phi_i = \pm 1,\pm i\).
Mapping to a Majorana Hamiltonian
For a single spin
Let us start by considering only one site and its \(\sigma^x, \sigma^y\) and \(\sigma^z\) operators which live in a two dimensional Hilbert space \(\mathcal{L}\).
We will introduce two fermionic modes \(f\) and \(g\) that satisfy the canonical anticommutation relations along with their number operators \(n_f = f^\dagger f, n_g = g^\dagger g\) and the total fermionic parity operator \(F_p = (2n_f - 1)(2n_g - 1)\) 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.
From these two fermionic modes, we can build four Majorana operators: \[\begin{aligned} b^x &= f + f^\dagger\\ b^y &= -i(f - f^\dagger)\\ b^z &= g + g^\dagger\\ c &= -i(g - g^\dagger) \end{aligned}\]
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 \(\mathcal{\tilde{L}}\). We can therefore identify the two dimensional space \(\mathcal{M}\) with one of the parity subspaces of \(\mathcal{\tilde{L}}\) which will be called the physical subspace \(\mathcal{\tilde{L}}_p\). Kitaev defines the operator \[D = b^xb^yb^zc\] which can be expanded to \[D = -(2n_f - 1)(2n_g - 1) = -F_p\] and labels the physical subspace as the space spanned by states for which \[ D|\phi\rangle = |\phi\rangle\]
We can also think of the physical subspace as whatever is left after applying the projector \[P = \frac{1 - D}{2}\] This formulation will be useful for taking states that span the extended space \(\mathcal{\tilde{M}}\) and projecting them into the physical subspace.
So now, with the caveat that we are working in the physical subspace, we can define new Pauli operators:
\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^y = i b^y c\]
These extended space Pauli operators satisfy all the usual commutation relations. The only difference is that if we evaluate \(\sigma^x \sigma^y \sigma^z = i\), we instead get \[ \tilde{\sigma}^x\tilde{\sigma}^y\tilde{\sigma}^z = iD \]
This makes sense if we promise to confine ourselves to the physical subspace \(D = 1\).
For multiple spins
This construction easily generalises to the case of multiple spins. We get a set of 4 Majoranas \(b^x_j,\; b^y_j,\;b^z_j,\; c_j\) and a \(D_j = b^x_jb^y_jb^z_jc_j\) operator for every spin. For a state to be physical, we require that \(D_j |\psi\rangle = |\psi\rangle\) for all \(j\).
From these each Pauli operator can be constructed: \[\tilde{\sigma}^\alpha_j = i b^\alpha_j c_j\]
This is where the magic happens. We can promote the spin hamiltonian from \(\mathcal{L}\) into the extended space \(\mathcal{\tilde{L}}\), safe in the knowledge that nothing changes so long as we only actually work with physical states. The Hamiltonian \[\begin{aligned} \tilde{H} &= - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\tilde{\sigma}_j^{\alpha}\tilde{\sigma}_k^{\alpha}\\ &= \frac{i}{4} \sum_{\langle j,k\rangle_\alpha} 2J^{\alpha} (ib^\alpha_i b^\alpha_j) c_i c_j\\ &= \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} \hat{u}_{ij} \hat{c}_i \hat{c}_j \end{aligned}\]
We can factor out the Majorana bond operators \(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\). Note that these bond operators are not equal to the spin bond operators \(K_{ij} = \sigma^\alpha_i \sigma^\alpha_j = - \hat{u}_{ij} c_i c_j\). 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.
Similarly to the argument with the spin bond operators \(K_{ij}\), we can quickly verify by considering three cases that the Majorana bond operators \(u_{ij}\) all commute with one another. They square to one, so have eigenvalues \(\pm 1\). They also commute with the \(c_i\) operators.
Importantly, the operators \(D_i = b^x_i b^y_i b^z_i c_i\) commute with \(K_{ij}\) and, therefore, with \(\tilde{H}\). We will show later that the action of \(D_i\) on a state is to flip the values of the three \(u_{ij}\) bonds that connect to site \(i\). Physically, this indicates that \(u_{ij}\) is a gauge field with a high degree of degeneracy.
In summary, Majorana bond operators \(u_{ij}\) are an emergent, classical, \(\mathbb{Z_2}\) gauge field!
Partitioning the Hilbert Space into Bond sectors
Similarly to the story with the plaquette operators from the spin language, we can divide the Hilbert space \(\mathcal{L}\) into sectors labelled by a set of choices \(\{\pm 1\}\) for the value of each \(u_{ij}\) operator which we denote by \(\mathcal{L}_u\). Since \(u_{ij} = -u_{ji}\), we can represent the \(u_{ij}\) graphically with an arrow that points along each bond in the direction in which \(u_{ij} = 1\).
Once confined to a particular \(\mathcal{L}_u\), we can ‘remove the hats’ from the \(\hat{u}_{ij}\). The hamiltonian becomes a quadratic, free fermion problem \[\tilde{H_u} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\] The ground state, \(|\psi_u\rangle\) can be found easily as will be explained in the next part. At this point, we may wonder whether the \(\mathcal{L}_u\) are confined entirely within the physical subspace \(\mathcal{L}_p\) and, indeed, we will see that they are not. However, it will be helpful to first develop the theory of the Majorana Hamiltonian further.
The Majorana Hamiltonian
We now have a quadratic Hamiltonian \[ \tilde{H} = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} c_i c_j\] in which most of the Majorana degrees of freedom have paired along bonds to become a classical gauge field \(u_{ij}\). What follows is relatively standard theory for quadratic Majorana Hamiltonians [14].
Because of the antisymmetry of the matrix with entries \(J^{\alpha} u_{ij}\), the eigenvalues of the Hamiltonian \(\tilde{H}_u\) come in pairs \(\pm \epsilon_m\). This redundant information is a consequence of the doubling of the Hilbert space which occurred when we transformed to the Majorana representation.
If we organise the eigenmodes of \(H\) into pairs, such that \(b_m\) and \(b_m'\) have energies \(\epsilon_m\) and \(-\epsilon_m\), we can construct the transformation \(Q\) \[(c_1, c_2... c_{2N}) Q = (b_1, b_1', b_2, b_2' ... b_{N}, b_{N}')\] and put the Hamiltonian into the form \[\tilde{H}_u = \frac{i}{2} \sum_m \epsilon_m b_m b_m'\]
The determinant of \(Q\) will be useful later when we consider the projector from \(\mathcal{\tilde{L}}\) to \(\mathcal{L}\). Otherwise, the \(b_m\) are merely an intermediate step. From them, we form fermionic operators \[ f_i = \tfrac{1}{2} (b_m + ib_m')\] with their associated number operators \(n_i = f^\dagger_i f_i\). These let us write the Hamiltonian neatly as
\[ \tilde{H}_u = \sum_m \epsilon_m (n_m - \tfrac{1}{2}).\]
The ground state \(|n_m = 0\rangle\) of the many body system at fixed \(u\) is then \[E_{u,0} = -\frac{1}{2}\sum_m \epsilon_m \] We can construct any state from a particular choice of \(n_m = 0,1\).
If we only care about the value of \(E_{u,0}\), it is possible to skip forming the fermionic operators. The eigenvalues obtained directly from diagonalising \(J^{\alpha} u_{ij}\) come in \(\pm \epsilon_m\) pairs. We can take half the absolute value of the whole set to recover \(\sum_m \epsilon_m\) easily.
Takeaway: the Majorana Hamiltonian is quadratic within a Bond Sector.
Mapping back from Bond Sectors to the Physical Subspace
At this point, given a particular bond configuration \(u_{ij} = \pm 1\), we can construct a quadratic Hamiltonian \(\tilde{H}_u\) in the extended space and diagonalise it to find its ground state \(|\vec{u}, \vec{n} = 0\rangle\). This is not necessarily the ground state of the system as a whole, it is just the lowest energy state within the subspace \(\mathcal{L}_u\)
However, \(|u, n_m = 0\rangle\) does not lie in the physical subspace. As an example, consider the lowest energy state associated with \(u_{ij} = +1\). This state satisfies \[u_{ij} |\vec{u}=1, \vec{n} = 0\rangle = |\vec{u}=1, \vec{n} = 0\rangle\] for all bonds \(i,j\).
If we act on it, this state with one of the gauge operators \(D_j = b_j^x b_j^y b_j^z c_j\), we see that \(D_j\) flips the value of the three bonds \(u_{ij}\) that surround site \(k\):
\[ |u'\rangle = D_j |u=1, n_m = 0\rangle\]
\[ \begin{aligned} \langle u'|u_{ij}|u'\rangle &= \langle u| b_j^x b_j^y b_j^z c_j \;ib^x_i b^x_j\; b_j^x b_j^y b_j^z c_j|u\rangle\\ &= -1 \end{aligned}\]
Since \(D_j\) commutes with the Hamiltonian in the extended space \(\tilde{H}\), the fact that \(D_j\) flips the value of bond operators indicates that there is a gauge degeneracy between the ground state of \(\tilde{H}_u\) and the set of \(\tilde{H}_{u'}\) related to it by gauge transformations \(D_j\). Thus, we can flip any three bonds around a vertex and the physics will stay the same.
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 \({D_0, D_1 ... D_n}\) which can be neatly expressed as \[|\phi_w\rangle = \prod_i \left( \frac{1 + D_i}{2}\right) |\tilde{\phi}_u\rangle\] This is convenient because the quantity \(\frac{1 + D_i}{2}\) is also the local projector onto the physical subspace. Here \(|\phi_w\rangle\) is a gauge invariant state that lives in \(\mathcal{L}_p\) which has been constructed from a set of states in different \(\mathcal{L}_u\).
This gauge degeneracy leads us to the next topic of discussion, namely how to construct a set of gauge invariant quantities out of the \(u_{ij}\), these will turn out to just be the plaquette operators.
Takeaway: The Bond Sectors overlap with the physical subspace but are not contained within it.
Open boundary conditions
Care must be taken when defining open boundary conditions. Simply removing bonds from the lattice leaves behind unpaired \(b^\alpha\) 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 \(J^{\alpha}_{ij} = 0\) for sites joined by bonds \((i,j)\) that we want to remove. This creates fermionic zero modes \(u_{ij}\) associated with these cut bonds which we set to 1 when calculating the projector.
Alternatively, since all the fermionic zero modes are degenerate anyway, an arbitrary pairing of the unpaired \(b^\alpha\) operators could be performed.
Emergent gauge fields
Ground State Degeneracy
More general arguments [15,16] imply that \(det(Q^u) \prod -i u_{ij}\) 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.
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. Whether this analysis generalises to the amorphous case is unclear.
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 both something strange happens. When we transport a vortex around both 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.
NOTE to self: This argument seems to involve adiabatic insertion of the fluxes \(\Phi_{x,y}\) as the operations that undo vortex transport around the lattice. I don’t understand why that part is necessary
The Ground State
Discuss Lieb’s theorem and generalisations for other lattices
Phases of the Kitaev Model
discuss the Abelian A phase / toric code phase / anisotropic phase
the isotropic gapless phase of the standard model
The isotropic gapped phase with the addition of a magnetic field </i,j></i,j>
Next Section: Anyonic Statistics