Abstract
Quantum spin liquids (QSLs) have been at the forefront of correlated electron research ever since their proposal in 1973, and the realization that they belong to the broader class of intrinsic topological orders. According to received wisdom, QSLs can arise in frustrated magnets with low spin S, where strong quantum fluctuations act to destabilize conventional, magnetically ordered states. Here, we present a Z_{2} QSL ground state that appears already in the semiclassical, largeS limit. This state has both topological and symmetryrelated groundstate degeneracy, and two types of gaps, a “magnetic flux” gap that scales linearly with S and an “electric charge” gap that drops exponentially in S. The magnet is the spinS version of the spin1/2 Kitaev honeycomb model, which has been the subject of intense studies in correlated electron systems with strong spin–orbit coupling, and in optical lattice realizations with ultracold atoms.
Introduction
Quantum spin liquids (QSLs) describe systems that evade magnetic longrange order down to zero temperature, and manifest a number of remarkable phenomena, such as topological groundstate degeneracies, emergent gauge fields, and fractional excitations with nontrivial statistics^{1,2,3,4,5,6,7,8,9,10,11}. The rich phenomenology of QSLs derives from an intrinsic tendency to form massive quantum superpositions of local, productlike wavefunctions. Notable examples are the resonating valence bond (RVB) state^{1, 5, 12,13,14}, the gapped QSL of the Toric code^{6}, and the gapless QSL phase of the spin1/2 Kitaev honeycomb model^{7}. Typically, such massive superpositions arise in frustrated magnets with low spin S, which ideally have an infinite number of competing states and a strong tunneling between them^{15}.
Here, we show that the spinS version of the celebrated Kitaev honeycomb model^{7} is a topological Z_{2} QSL already in the semiclassical limit. Specifically, the leading semiclassical fluctuations give rise to an effective lowenergy description in terms of a pseudospin1/2 Toric code^{6}. The “magnetic flux” term of the Toric code arises from the zeropoint energy of spin waves above the classical ground states, while the “electric charge” term stems from the tunneling between different classical states. The ensuing Z_{2} QSL lives on top of a honeycomb superlattice of “frozen” spin dimers^{16}, which take only two possible configurations, instead of (2S + 1)^{2}. These two states are the pseudospin1/2 degrees of freedom of the Toric code. The frozen dimer pattern breaks translational symmetry, so the QSL possesses an extra degeneracy associated to symmetry breaking, besides the topological one. We also show that the largeS description breaks down around S ~ 3/2. For lower S, tunneling processes that shift the dimer positions become quickly relevant and compete with the “freezing” energy scale δE_{f}. Including these processes lead to a picture of “decorated quantum dimers”, where both the dimer positions and the orientations of the two spins in each dimer are allowed to resonate. The resulting picture for S = 1 in terms of another type of spin liquid will be discussed.
Results
Model and classical ground states
The spinS Kitaev model on the honeycomb lattice is described by the Hamiltonian
Here, \(\left\langle {ij} \right\rangle\) denotes a pair of nearestneighbor (NN) spins S_{ i } and S_{ j }. There are three types of NN bonds, depending on their orientation, which are labeled by “x”, “y” or “z” (Fig. 1a). These bonds are directed along \(\frac{{\mathbf{y  z}}}{{\sqrt 2 }}\), \(\frac{{\mathbf{z  x}}}{{\sqrt 2 }}\), and \(\frac{{\mathbf{x  y}}}{{\sqrt 2 }}\), respectively, where x, y, and z are the usual unit vectors in the Cartesian frame. The constant K denotes the Kitaev interaction. Note that there is a four sublattice duality transformation^{17} that maps the positive K to the negative K model, but we shall discuss the general case here for completeness. We shall also define κ = −sgn(K).
The classical ground states of this model were first analyzed by BSS^{16}. There, the authors identified an infinite number of socalled “Cartesian” states, which map to dimer coverings of the honeycomb lattice, modulo a factor of two for the orientation of the two spins per dimer. They further showed that the Cartesian states are connected to each other by continuous valleys of other ground states, leading to a huge groundstate degeneracy. Soon after, Chandra et al.^{18} showed that the manifold actually consists of infinitely more ground states and possesses an emergent gauge structure that leads to power law correlations.
The crucial aspect of the present study is the use of a convenient parametrization of the classical groundstate manifold, which reveals the topological terms arising from quantum fluctuations in an explicit way. This parametrization is shown in Fig. 1a. We denote the two sublattices of the honeycomb by A and B. Next, we parametrize each spin as S_{ i } = (a_{ i }, b_{ i }, c_{ i }) or κ(a_{ i }, b_{ i }, c_{ i }) for i ∈ A or B, respectively, and \(a_i^2 + b_i^2 + c_i^2 = S^2\). Then, for every pair of NN sites, S_{ i } and S_{ j }, we can minimize their mutual interaction by requiring that a_{ i } = a_{ j } or b_{ i } = b_{ j } or c_{ i } = c_{ j }, if the two sites share, respectively, an “x” or “y“ or “z” type of bond. To see if the ensuing states are ground states, we check that they saturate the lower bound of the energy per site, \(E_{{\mathrm{min}}}{\mathrm{/}}N =  \left K \rightS^2{\mathrm{/}}2\)^{16}. Indeed, the energy from the three bonds emanating from any site i add up to \( \left K \right\left( {a_i^2 + b_i^2 + c_i^2} \right) =  \left K \rightS^2\). And since each bond is shared by two sites, these configurations saturate the lower bound and are therefore ground states. The Cartesian states of BSS arise by keeping only one component of (a_{ i }, b_{ i }, c_{ i }) finite, and equal to η_{ i }S, where η_{ i } = ±1. Modulo these Isinglike variables, the Cartesian states map to dimer coverings of the lattice [Fig. 1b]. There are (1.381)^{N/2} coverings^{19,20,21}, and (1.662)^{N} Cartesian states in total^{16}.
The semiclassical analysis leading to the Toric code proceeds in three steps. The first is to show that fluctuations select the Cartesian over the nonCartesian states, which identifies the positions and spin orientations of the dimers as the relevant degrees of freedom. In the second step, which was carried out by BSS^{16}, fluctuations freeze the positions of the dimers to a given pattern, leaving their spin orientation as the only relevant degrees of freedom below the associated freezing energy scale δE_{f}. These degrees of freedom can be described by pseudospin1/2 variables residing on the bonds of a honeycomb superlattice. This parametrization reveals that the order by disorder effect has a topological structure that was not noticed previously, and which is robust to all orders in 1/S expansion. The third step is to go beyond the order by disorder effect and include quantum mechanical tunneling between different pseudospin configurations. This step is essential for restoring all local Z_{2} gauge symmetries of the original model, and for the formation of the quantum spin liquid.
Selection of Cartesian states
The first crucial ingredient of the effective description in terms of dimers is to show that fluctuations select the Cartesian over the nonCartesian states. BSS made this hypothesis based on an analogy to a related 1D problem. Here, we prove it by real space perturbation theory (RSPT)^{22,23,24,25}. We introduce local frames (u_{ i }, v_{ i }, w_{ i }), with w_{ i } along the classical directions, and write \({\bf{S}}_i = S_i^w{\bf{w}}_i + S_i^u{\bf{u}}_i + S_i^v{\bf{v}}_i\). Then, we split \({\cal H}\) into a diagonal part \({\cal H}_0 = h\mathop {\sum}\nolimits_i \left( {S  S_i^w} \right)\), describing fluctuations in the local field h = KS, and a perturbation \({\cal V} = {\cal H}  {\cal H}_0\), which couples fluctuations on different sites. The essential physics is captured by the leading, shortwavelength corrections from secondorder RSPT. The three types of bonds, say (1–10), (1–6), and (1–2) of Fig. 1, give \(\delta E_{1,10} = \xi \left( {1  \tilde a_1^2} \right)^2\), \(\delta E_{1,6} = \xi \left( {1  \tilde b_1^2} \right)^2\), \(\delta E_{1,2} = \xi \left( {1  \tilde c_1^2} \right)^2\), where \(\xi =  \left K \rightS{\mathrm{/}}8\) and \(\left( {\tilde a_i,\tilde b_i,\tilde c_i} \right) = \left( {a_i,b_i,c_i} \right){\mathrm{/}}S\). Using the spin length constraints and disregarding overall constants, gives the anisotropy term
similar to the one found in refs. ^{26, 27}. This anisotropy selects the Cartesian states, confirming the hypothesis of BSS^{16}.
Dimer freezing and ηvariables
Next, we discuss the lifting of the degeneracy within the manifold of Cartesian states, starting with the corrections from spin waves. As shown by BSS, the linear spinwave Hamiltonian splits into noninteracting modes propagating along loops without dimers, and the minimum zeropoint energy arises by maximizing the number of the shortest such “empty” loops, like the shaded hexagon of Fig. 1b. This gives the “star” or “columnar” dimer pattern of Fig. 2a, which is known from the context of the quantum dimer model and the frustrated Heisenberg model on the honeycomb lattice^{28,29,30}. In this pattern, the only dynamical degrees of freedom remaining are the Isinglike variables η = ±1, which specify the direction of the two spins shared by each given dimer.
The physics of the dimer freezing is actually more involved from what is predicted from the linear spinwave theory, but let us postpone this discussion for later and focus on the spin states associated to the “star” pattern. There are three ways to place this pattern in the lattice and each dimer has two configurations, so at first sight, the number of selected spin states is 3 × 2^{N/2}. BSS showed, however, that the minimum zeropoint energy is associated with spinwave modes that have antiperiodic boundary conditions (ABC) around the empty hexagons, which reduces the number of states to 3 × 2^{N/3}.
However, this is not the full story yet. It turns out that the boundary condition on the spinwave modes actually endows the selected manifold with a topological magnetic flux term (and, in particular, the above number of states has to be multiplied 2^{2g−1}, where g is the genus of the system). To see this, we repeat the spinwave analysis using our ηparametrization. We begin by rewriting \({\cal H}\) in the local frame. Let us take the empty hexagon h_{ α } of Fig. 2a and choose u_{ i } and v_{ i } in the following way (and similarly for every other empty hexagon):
where x, y, and z are the Cartesian unit vectors, and the product of the six ηvariables on empty hexagons,
is the magnetic flux that plays a central role in the following. With the above choice of the local frames, the couplings between empty hexagons map to terms of the type \(\kappa S_i^wS_j^w\). For example, \(S_1^xS_{10}^x \mapsto \kappa S_1^wS_{10}^w\). On the other hand, the intrahexagon terms map as follows
Thus, in the rotated frame, the only dependence of the Hamiltonian on η’s is via the fluxes \(\left\{ {B_{h_\alpha }} \right\}\) on the empty hexagons {h_{ α }}. And, since the choice of the local frame does not alter the physics, it follows that classical states that belong to the “star” pattern of Fig. 2a and have the same \(\left\{ {B_{h_\alpha }} \right\}\) share the same semiclassical spinwave spectrum, at all orders in 1/S. The same is true for the renormalization of the groundstate energy and therefore the order bydisorder effect.
Let us show the latter explicitly and we shall return to the spinwave modes further below. We introduce the usual Holstein–Primakoff bosons c_{ i } via the transformation^{31}, \(S_i^\dagger = S_i^u + iS_i^v = \left( {2S  c_i^\dagger c_i} \right)^{1/2}c_i\), \(S_i^w = S  n_i\), \(n_i = c_i^\dagger c_i\). The coupling between neighboring empty hexagons, like \(S_1^xS_{10}^x\), reduces then to
The linear spinwave theory amounts to disregarding the term κn_{1}n_{10} from the righthand side of this equation. Empty hexagons then decouple, leading to a quadratic, sixsite boson problem, with two sublattices and periodic (PBC) or antiperiodic (ABC) boundary conditions, for \(\kappa B_{h_\alpha } =  1\) or 1, respectively. So, the BSS result that ABC give the lowest zeropoint energy amounts to imposing \(\kappa B_{h_\alpha } = 1\) for all empty hexagons h_{ α }. More explicitly, by combining the zeropoint energies, δE_{PBC} and δE_{ABC} for PBC and ABC, respectively, we get, for a given h_{ α },
where \(c = \frac{{\delta E_{{\mathrm{PBC}}} + \delta E_{{\mathrm{ABC}}}}}{2}\) and \(J_{\mathrm{m}} = \frac{{\delta E_{{\mathrm{PBC}}}  \delta E_{{\mathrm{ABC}}}}}{2}\). The linear spinwave theory of BSS^{16} gives \(\delta E_{{\mathrm{PBC}}} = 2\left K \rightS\) and \(\delta E_{{\mathrm{ABC}}} = \sqrt 3 \left K \rightS\), and so \(J_{\mathrm{m}} = \frac{{2  \sqrt 3 }}{2}KS\). However, as shown in Fig. 3 and emphasized below, the linear theory overestimates \(\left {J_{\mathrm{m}}} \right\) strongly due to the presence of a large percentage (four out of six) of “spurious” zero modes.
Equation (7) confirms that the groundstate energy depends explicitly on the fluxes \(\left\{ {B_{h_\alpha }} \right\}\), and that states with the same set of fluxes have the same zeropoint energy. Importantly, the quartic terms of the type n_{1}n_{10} of Eq. (6) will give rise to interactions between the fluxes, of the form \(J_{\alpha \alpha {\prime}}B_{h_\alpha }B_{h_{\alpha {\prime}}}\) plus higherorder terms, where the couplings J_{αα′} depend on the positions of the corresponding empty hexagons h_{ α }, h_{α′}. As we demonstrate further below, these interactions are at least one order of magnitude weaker than the leading term ∝J_{m}, and in addition do not alter any of the crucial ingredients leading to a topological QSL state.
With this in mind, we are now ready to identify the first crucial ingredient of the Toric code description announced above. The η_{ i } variables live on the midpoints of the bonds of a honeycomb superlattice (Fig. 2a) and we can promote these variables to Pauli matrices \({\boldsymbol{\eta }}_i^z\). Then the dominant, nontrivial correction to the zeropoint energy—the second term of the righthand side of Eq. (7)—is the magnetic flux term of the Toric code^{6} on this superlattice.
Quantum mechanical tunneling
The second ingredient of the Toric code, the electric charge term, stems from processes that flip the three η’s around a vertex of the superlattice. Let us take, e.g., the spin coherent state of the h_{ β } hexagon of Fig. 2a,
The leading processes that transform this state to its timereversed state \(\left {\bar h_\beta } \right\rangle\), with η_{1}, η_{2}, and η_{8} flipped, appear in (6S)th order of RSPT, with \({\cal V} = K\left( {S_7^xS_8^x + S_9^yS_{10}^y + S_1^zS_2^z} \right)\). The corresponding offdiagonal matrix element J_{e} of the resulting effective Hamiltonian \({\cal H}_{{\mathrm{eff}}}\) depends, unlike J_{m}, on the choice of the local axes (u_{ i }, v_{ i }). Here, we fix J_{e} to be a real number by choosing the local axes such that \({\cal V} \mapsto K\left( {S_7^uS_8^u + S_9^uS_{10}^u + S_1^uS_2^u} \right)\). Following, e.g., the steps of the Supplementary Material of ref. ^{26}, we get
In the language of the η operators, this matrix element is represented by \(J_{\mathrm{e}}{\boldsymbol{\eta }}_1^x{\boldsymbol{\eta }}_2^x{\boldsymbol{\eta }}_8^x \equiv J_{\mathrm{e}}{\bf{A}}_v\), which involve the three η’s around the vertex v that sits at the center of h_{ β } (Fig. 2a).
Similarly to the higherorder potential terms mentioned above, there also exist higherorder tunneling terms, such as A_{ v }A_{v′}, whose amplitudes are much weaker than J_{e}.
Effective Hamiltonian of ηvariables
Collecting the potential energy (disregarding c) and the tunneling terms above gives the effective Hamiltonian for the ηvariables:
Here v and p label, respectively, the vertices and the plaquettes of the honeycomb superlattice, while the respective indices v_{1}–v_{3} and p_{1}–p_{6} are shown in Fig. 2b. In terms of the original lattice, the index v labels the nonempty hexagons of type h_{ β }, while p labels the empty hexagons of type h_{ α }. The omitted terms are the much weaker, higherorder terms, B_{ p }B_{p′}, A_{ v }A_{v′}, etc, discussed above.
The Toric code model corresponds to the two leading terms of Eq. (10). However, the remarkable properties of the Toric code are also present in the full model of Eq. (10). These properties stem from the relations \({\bf{A}}_v^2 = {\bf{B}}_p^2 = 1\) and the fact that \(\left\{ {{\bf{A}}_v,{\bf{B}}_p,{\cal H}_{{\mathrm{eff}}}} \right\}\) is a set of mutually commuting operators^{6}. This model is a Z_{2} lattice gauge theory^{32, 33}, with the local gauge transformations generated by A_{ v }. In the following, we discuss the most important of these properties^{6, 8, 10}.
Topological sectors
On a torus, \(\mathop {\prod}\nolimits_v {\kern 1pt} {\bf{A}}_v = \mathop {\prod}\nolimits_p {\kern 1pt} {\bf{B}}_p = 1\) and so there are N_{ v } = 2^{N/3−1} and N_{ p } = 2^{N/6−1} independent choices of A_{ v } and B_{ p }, respectively, leading to 2^{N/2−2} states. So the quantum numbers {A_{ v }, B_{ p }} do not exhaust all 2^{N/2} states of η’s. The missing quantum numbers are provided by the nonlocal operators \({\bf{X}}_1 = \mathop {\prod}\nolimits_{{\mathrm{C}}_{{\mathrm{X1}}}} {\kern 1pt} {\boldsymbol{\eta }}^x\) and \({\bf{X}}_2 = \mathop {\prod}\nolimits_{{\mathrm{C}}_{{\mathrm{X2}}}} {\kern 1pt} {\boldsymbol{\eta }}^x\), defined on the noncontractible loops C_{X1} and C_{X2} of Fig. 2b. These operators commute with A_{ v } and B_{ p }, and with each other, and in addition \({\bf{X}}_1^2 = {\bf{X}}_2^2 = 1\). The quantum numbers {A_{ v }, B_{ p }, X_{1}, X_{2}} then exhaust the Hilbert space of η’s.
Ground states for K < 0
Without loss of generality, we will consider the K < 0 case, where both J_{m} and J_{e} are negative. Let us first ignore J_{e}. As demonstrated below, the higherorder couplings between the fluxes are much weaker than \(\left {J_{\mathrm{m}}} \right\), so the groundstate flux sector is the one with B_{ p } = 1 for all p. This sector contains 2^{N/3−1} states, which are degenerate in the absence of J_{e}, even when we include interactions between the fluxes. The tunneling term will lift the degeneracy and will select the states with A_{ v } = 1 for all v. On a torus, there are four such states, which correspond to the choices of the winding numbers X_{1} and X_{2}. One of them is
where \({\cal N}\) is a normalization factor, and \(\left {{\mathrm{FM}}_x} \right\rangle = \left { \to \cdots \to } \right\rangle\) is the fully polarized state along x, which has A_{ v } = 1, ∀v. Expanding the product over (1 + B_{ p }) shows that this state is the equalamplitude superposition of all possible loops of overturned spins (spins pointing along −x, which correspond to electric flux lines) on top of the FM background, see Fig. 4 and refs. ^{7, 8}. The remaining three ground states of the Toric code, \(\left {X_1,X_2} \right\rangle = \left {  1,1} \right\rangle\), \(\left {1,  1} \right\rangle\) and \(\left {  1,  1} \right\rangle\), arise by replacing the reference state \(\left {{\mathrm{FM}}_x} \right\rangle\) in Eq. (11) with \({\bf{Z}}_2\left {{\mathrm{FM}}_x} \right\rangle\), \({\bf{Z}}_1\left {{\mathrm{FM}}_x} \right\rangle\), and \({\bf{Z}}_1{\bf{Z}}_2\left {{\mathrm{FM}}_x} \right\rangle\), respectively, where \({\bf{Z}}_1 = \mathop {\prod}\nolimits_{{\mathrm{C}}_{{\mathrm{Z1}}}} {\kern 1pt} {\boldsymbol{\eta }}^z\) and \({\bf{Z}}_2 = \mathop {\prod}\nolimits_{{\mathrm{C}}_{{\mathrm{Z2}}}} {\boldsymbol{\eta }}^z\), defined along C_{Z1} and C_{Z2} of Fig. 2b. These operators flip X_{2} and X_{1}, respectively, because of the anticommutation relations {Z_{1}, X_{2}} = 0 and {Z_{2}, X_{1}} = 0.
Note that the groundstate sector of the original Kitaev spin model is 12fold and not 4fold degenerate, because there are three ways to place the dimer pattern of Fig. 2a into the lattice and each sector has its own Toric code description.
Excitations of \({\cal H}_{{\mathrm{eff}}}\) for K < 0 and deconfinement
The elementary excitations are pairs of static charges (vertices with A_{ v } = −1), or pairs of static fluxes (plaquettes with B_{ p } = −1). To an excellent approximation, their energy is given by \({\mathrm{\Delta }}_e \simeq 4\left {J_{\mathrm{e}}} \right\) and \({\mathrm{\Delta }}_m \simeq 4\left {J_{\mathrm{m}}} \right\), respectively. In particular, Δ_{ m } scales roughly linearly with S (see Fig. 3), whereas Δ_{ e } is exponentially small in S, as follows from Eq. (9), and practically vanishes for S ≥ 1 and realistic values of K.
Higherorder corrections to these expressions stem from the interactions between the respective pair of fluxes, which in particular depend on the distance between them. Figure 5 shows numerical data for the binding energy of two magnetic fluxes as a function of the distance d and the spin quantum number S. The data are extracted from an iterative exact diagonalization treatment of the meanfield decoupled Hamiltonian \({\cal H}_{{\mathrm{MF}}}\) described in the “Methods” section, taking a honeycomb lattice with 180 sites (i.e., 30 empty hexagon plaquettes) on a torus. The data show consistently that: first, the energy is lowest when the two fluxes reside next to each other; second, the energy levels off very quickly with the distance d, which demonstrates that the fluxes are deconfined; and third, the binding energy scale is at least one order of magnitude weaker than the leading J_{m} term (see Fig. 3), as announced already above.
Origin of gauge structure and BSS fluxes
The local Z_{2} gauge symmetry of Eq. (10) is not an emergent property, but descends from the Z_{2} gauge structure of the original spinS model, discovered by BSS^{16}. This structure stems from the presence of local conserved operators defined on the hexagons of the original lattice, which are called BSS fluxes in the following. For the h_{ β } hexagon of Fig. 2a, the BSS flux operator reads:
Now, the BSS fluxes on nonempty hexagons have the same effect as the A_{ v } operators, e.g., \({\bf{W}}_{{\mathrm{BSS}}}\left( {h_\beta } \right)\left {h_\beta } \right\rangle \to \left {\bar h_\beta } \right\rangle\), modulo a numerical prefactor, see Eq. (13) below. So the local gauge symmetry of Eq. (10) indeed descends from that of the full model.
Let us now examine the groundstate BSS flux pattern. Unlike the original classical states associated with the “star” pattern, where only the empty hexagons have welldefined W_{BSS}^{16}, the QSL ground states of Eq. (10) have welldefined W_{BSS} on all hexagons. Indeed, using the same choice of local axes as the ones used above for the tunneling we find:
Now, the resonating QSL state \(\left {1,1} \right\rangle\) of Eq. (11) satisfies \({\mathrm{sgn}}\left( {J_{\mathrm{e}}} \right)A_v\left {1,1} \right\rangle =  \left {1,1} \right\rangle\), and therefore contains the combination \(\frac{1}{{\sqrt 2 }}\left( {\left {h_\beta } \right\rangle  {\mathrm{sgn}}\left( {J_{\mathrm{e}}} \right)\left {\bar h_\beta } \right\rangle } \right)\). So, the groundstate expectation value W_{BSS}(h_{ β }) of the operator W_{BSS}(h_{ β }) is equal to
For halfinteger S, in particular, W_{BSS}(h_{ β }) = −1, irrespective of κ. For the empty hexagons, such as h_{ α }, a welldefined flux is already fixed by the zeropoint energy, as shown by BSS^{16}. Specifically, W_{BSS}(h_{ α }) = (−1)^{λS}, where λ = κ(η_{1} + η_{3} + η_{5}) + η_{2} + η_{4} + η_{6}, which is even. So, for integer S, W_{BSS}(h_{ α }) = 1, while for halfinteger S, \({W}_{{\mathrm{BSS}}}\left( {h_\alpha } \right) =  \kappa B_{h_\alpha } =  1\), because of the ABC condition on spin waves.
The BSS fluxes are in fact well defined in all eigenstates of Eq. (10), not just in the ground states. An excited state with an electric charge sitting on h_{ β } has W_{BSS}(h_{ β }) = (−κ)^{2S+1}, opposite to the one in the ground state. On the other hand, an excited state with a magnetic charge on h_{ α } has W_{BSS}(h_{ α }) = 1 for both integer and halfinteger S. These results also mean that magnetic fluxes are related to BSS fluxes on empty hexagons for all S, and electric charges are related to BSS fluxes on nonempty hexagons for halfinteger S.
More generally, the fact that the BSS fluxes are well defined on all hexagons is consistent with Elitzur’s theorem^{34,35,36} that local gauge symmetries cannot be broken spontaneously. Following the works in refs. ^{16, 37}, this also necessitates that static and dynamic twospin correlation functions are identically zero beyond NN separation, consistent with the Toric code description.
Spinwave modes
In the frozen dimer pattern of Fig. 2a, the local Hilbert space for each spinS dimer has dimension (2S + 1)^{2}, and Eq. (10) describes the dynamics inside the subspace of \(\left {m_1,m_2} \right\rangle = \left {S,\kappa S} \right\rangle\) and \(\left {  S,  \kappa S} \right\rangle\), where the projections m_{1} and m_{2} are defined along the local quantization axes. To this Hamiltonian (10), we should also add the terms that describe the coherent spinwave bosonic modes
describing the elementary, singleparticle excursions outside this 2 × 2 manifold, with Δm = ±1. Note that the important constants arising from the spinwave theory have been assigned to J_{m} already, and that the b_{ i } bosons are the eigenmodes of the spinwave Hamiltonian, either at the quadratic or the selfconsistent quartic order (see Supplementary Note 1). Also, as mentioned above, the spinwave frequencies ω_{ i } depend on the set {B_{ p }} only, and are therefore the same for all states with the same {B_{ p }} but different {A_{ v }}. This entails a huge, \(2^{\frac{N}{3}  1}\)fold degeneracy (in the torus geometry) in the spinwave branches, for each given set of {B_{ p }}. We emphasize that the magnons discussed here do not describe the elementary excitations above some magnetically ordered state. Instead, they describe coherent excitations that are present in the spectrum independently of the elementary flux and charge excitations.
We now examine the actual structure of the magnon spectrum. At the quadratic level, BSS have shown^{16} that the spectrum consists of six flat bands, with ω_{1–4}(k) = 0 and \(\omega _{5,6}({\bf{k}}) = \sqrt 3 \left K \rightS\), where the momentum k belongs to the magnetic Brillouin zone. However, the problem with the quadratic theory is that the modes 1–4 are not true zero modes, i.e., they will be gapped out by interactions. Such spurious zero modes are typical^{17, 38,39,40,41,42,43,44,45} artifacts of the harmonic theory and reflect the modes that connect different classical minima. As commented above, the large number of such spurious zero modes in the present model leads to unreliable estimates for the relevant energy scales of the problem. This necessitates that we push the semiclassical expansion to quartic order, and treat the problem via a standard selfconsistent decoupling scheme (see Supplementary Note 1).
A key finding of this analysis is that spin waves remain localized inside the empty hexagons even at the interacting spinwave level, because of the local conservation laws associated with the BSS fluxes. To see this, let us return to Eq. (6) and consider the interaction between different empty hexagons, n_{1}n_{10}, which is disregarded in the linear spinwave theory. The standard meanfield decoupling of this term gives
where \(\xi = p_1p_{10} + \left \delta \right^2 + \left m \right^2\), \(p_i = \left\langle {n_i} \right\rangle\), \(m = \left\langle {c_1^\dagger c_{10}} \right\rangle\), and \(\delta = \left\langle {c_1^\dagger c_{10}^\dagger } \right\rangle\). Now, as discussed above, the states around which we expand do not break the BSS operators on empty hexagons. For the hexagon h_{ α } of Fig. 2a, the BSS operator reads:
where λ has been defined above. Hence, the invariance of the Hamiltonian and the state around which we expand under W_{BSS}(h_{ α }) translates into the invariance of the parity of the number κ(η_{1}n_{1} + η_{3}n_{3} + η_{5}n_{5}) + η_{2}n_{2} + η_{4}n_{4} + η_{6}n_{6}. Since both κ and the ηvariables can only take the values +1 and −1, it follows that the parity of this number is the same as the parity of the total number of bosons, \(N_{h_\alpha } = \mathop {\sum}\nolimits_{i = 1}^6 {\kern 1pt} n_i\), inside the hexagon h_{ α }. This means that terms that change the parity of \(N_{h_\alpha }\) are not allowed in the expansion. As a result, the constants m and δ appearing in Eq. (16) vanish by symmetry, and magnons do not hop from one empty hexagon to another.
Let us now focus on the excitations above the groundstate sector for K < 0, where B_{ p } = 1 for all empty hexagons p. Given that the flux configuration is uniform, all constants \(p_i = \left\langle {n_i} \right\rangle\) are equal and the interacting spinwave problem then reduces to a selfconsistent problem on a single hexagon, and in addition the magnon frequencies do not depend on the position of that hexagon. The calculated frequencies are shown in Fig. 6a along with the corresponding results from the quadratic theory. All spurious modes are gapped out, and the spectrum organizes into three degenerate pairs due to symmetry (see Supplementary Note 1). This figure also tells us that all modes sit far above the energy scales \(\left {J_{\mathrm{m}}} \right\) and \(\left {J_{\mathrm{e}}} \right\) of Eq. (10). In addition, the spin length corrections δS of Fig. 6b show that spin waves do not reduce the spin length appreciably (at maximum it is about 15% for S = 1/2), so the η variables are robust degrees of freedom.
Physics at low S
We now turn our discussion to what can go wrong with the above semiclassical picture as we lower S. The dimer freezing in the star pattern of Fig. 2a stems from the zeropoint energy of spin waves. However, this analysis disregards the quantum tunneling between different dimer patterns. The leading process is the one around a hexagon (Fig. 7a). The states associated with different dimer patterns are not orthonormal, but we can estimate the relevant tunneling amplitude t_{ d } using the truncation method in ref. ^{46} (see “Methods”):
At large S, t_{ d } is extremely small, and the spinwave analysis of the dimer freezing has solid ground. This would in fact remain true down to S = 1, if we were to use linear spinwave theory. However, this theory overestimates strongly the freezing energy scale (like \(\left {J_{\mathrm{m}}} \right\)) due to the spurious zero modes mentioned above. As a result, t_{ d } becomes relevant below S ~ 3/2. To see this, let us take as a representative freezing energy scale, the energy difference \(\delta E_{\mathrm{f}}^{(6,\infty )}\) between the star pattern (shown again in Fig. 7b for convenience) and the “staggered” pattern of Fig. 7c, where the empty loops have infinite length. At the level of interacting spinwave theory, this energy difference is shown in Fig. 7d along with \(\left {t_d} \right\) (where we divide by N and by 6, respectively, so that we compare energies per site). The results show clearly that dimers become mobile below S ~ 3/2. (By contrast, linear spinwave theory gives \(\delta E_{\mathrm{f}}^{(6,\infty )}{\mathrm{/}}(NK) = \left( {\frac{{\sqrt 3 }}{6}  \frac{1}{\pi }} \right)S\)^{16}, which is much larger than \(\left {t_d} \right{\mathrm{/}}6\) down to S = 1.)
It follows that in order to understand the physics of the S = 3/2 and S = 1 cases, we need to return to the Cartesian basis, and allow both the position of the dimers and their spin orientation to resonate. Such a “decorated quantum dimer” description may appear quite more involved, but it may actually not be the case for the particular S = 1 case. The reason is that t_{ d }/6 is more than ten times larger than δE_{f}/N for S = 1 (see Fig. 7d) and, from the standard quantum dimer model on the honeycomb lattice^{28, 30}, we know that t_{ d } stabilizes a resonating “plaquette” dimer pattern, known also from the context of the frustrated Heisenberg model^{29, 47, 48}. Including the much smaller J_{e} term will include the resonances with the dimers of the opposite spin orientations. It would be interesting to check numerically this generalized semiclassical picture for S = 1, and moreover whether certain features of this picture carry over to the exactly solvable S = 1/2 case.
Discussion
It is shown that the lowenergy sector of the largeS Kitaev honeycomb model is described by a Toric code on a honeycomb superlattice. This should be contrasted with the effective squarelattice Toric code that arises in the spin1/2 model when one of the three types of bonds has much stronger coupling than the other two^{7}. Here, the magnetic and electric flux terms of the effective description arise respectively from the zeropoint energy of spin waves and quantum mechanical tunneling between different orientations of frozen dimers. This picture breaks down for S ≲ 3/2, where tunneling between different dimer patterns becomes relevant.
The fundamental principle that prevents magnetic ordering in the semiclassical regime of the present model is the presence of an extensive number of local conservation laws, which were discovered in the seminal study by Baskaran, Sen and Shankar (BSS)^{16}. According to Elitzur’s theorem^{34}, these local symmetries cannot break spontaneously even at zero temperature, and the system fails to order magnetically even in the semiclassical limit. Given in addition that the conservation laws are not emergent, the gauge structure is not only present in the lowenergy sector, but also in the singleparticle, spinwave channel, as we analyzed in detail beyond the quadratic level.
The prospects for realizing S > 1/2 Kitaev magnets remain at present limited, although there are reports for nearly perfect honeycomb magnets with Co^{2+} ions, such as Na_{2}Co_{2}TeO_{6} and Na_{3}Co_{2}SbO_{6}^{49}, with peculiar spatial magnetic correlations^{50}. These systems show singleion anisotropy, but it is worth checking via ab initio methods if a strong Kitaev term is also present, as in the layered spin1/2 iridates and ruthenates^{51,52,53,54}. In parallel, there are proposals for emulating the model with trapped ions^{55}, superconducting quantum circuits^{56}, coupled cavity arrays^{57}, and ultracold atoms in optical lattices^{58,59,60,61}, which in particular offer the possibility for S > 1/2 extensions of the model^{59,60,61}.
Finally, we point out that the uniform^{13} or staggered^{62, 63} charge sectors of Eq. (10) describe another well known Z_{2} spin liquid, the RVB state of the spin1/2 Heisenberg kagome antiferromagnet^{46, 64,65,66,67,68}. This highlights the universal topological features of QSLs arising from very different settings, across both isotropic and highly anisotropic magnets.
Methods
Meanfield decoupled Hamiltonian \({\cal H}_{{\mathrm{MF}}}\)
Here, we discuss the exact diagonalization (ED) treatment that delivers the results shown in Figs. 3, 5, and 6b. Starting with the classical configurations of the star dimer pattern of Fig. 2, we introduce the six sublattice decomposition of Fig. 8, with a superlattice defined by the primitive translation vectors T_{1} and T_{2}. The sites i of the lattice can then be labeled as i = (R, ν), where R is a primitive vector of the superlattice, and ν = 1–6 is the sublattice index. In this parametrization, the positions of the empty hexagons h_{ α } are labeled by R. Each given classical state is parametrized in terms of the ηvariables, as shown in Fig. 2. With the local coordinate frames of Eq. (3), the Hamiltonian reads \({\cal H} = {\cal H}_0 + {\cal V}\), where \({\cal H}_0\) describes intrahexagon terms,
and \({\cal V}\) accounts for the interhexagon terms,
where T_{3} = T_{1} − T_{2}.
Now, the semiclassical analysis rests on the assumption that the expectation value of the operators \(S_i^w\) is finite and relatively close to the classical value S. The spinwave expansion, which has been discussed in the text and in the Supplementary Notes 1 and 2, is the standard way to proceed. Another way is to perform a meanfield decoupling of the interhexagon terms in \({\cal V}\), i.e., to replace
The resulting meanfield Hamiltonian \({\cal H}_{{\mathrm{MF}}}\) describes a collection of single hexagon Hamiltonians, each of which contains local Zeeman fields that depend on the state of the neighboring hexagons.
The ED results shown in Figs. 3, 5, and 6b are obtained using the following iterative procedure. We consider a honeycomb lattice on a torus with a certain number of empty hexagons. Then, we fix the numbers κB_{ R } depending on the flux configuration we are interested in. In the next step, we initialize the spin lengths to their classical values S, the same for all sites. Then, we go through all empty hexagons, one after the other, and diagonalize numerically the corresponding Hamiltonian. From the ground state we then calculate the expectation values of the six spins of the hexagon \(\left\langle {S_i^w} \right\rangle\), and then update the corresponding Zeeman terms for the neighboring hexagons. This iterative procedure converges very fast in energy, leading to a total energy of the system and the distribution of spin lengths. Note that for uniform flux configurations, such as the groundstate sector where κB_{ R } = 1 for all R, the expectation values \(\left\langle {S_i^w} \right\rangle\) are the same throughout the system, and we can therefore use a single hexagon only. For nonuniform flux configurations, on the other hand, the expectation values \(\left\langle {S_i^w} \right\rangle\) are nonuniform as well, and we need to do a selfconsistent calculation on a large enough lattice.
The ED results shown in Fig. 3 for the coupling J_{m} are obtained by comparing the energy of the groundstate flux sector, where κB_{ R } = 1 for all R, with the energy of the sector with κB_{ R } = −1 for all R. Strictly speaking, this method (called method 1 in Table 1) does not deliver the coupling J_{m}, because the energy of the second sector contains contributions from the interactions between fluxes. A more accurate determination of J_{m} arises by comparing the energy of the groundstate sector with the energy of the sector with a single flux, i.e., the sector with κB equal to −1 at a single empty hexagon and +1 elsewhere. The last two rows of Table 1 show the numerical results for J_{m} obtained using this method (called method 2 in Table 1) on a periodic lattice with 84 sites (spanning an array of 7 × 2 empty hexagons) and another with 180 sites (spanning an array of 15 × 2 empty hexagons). For comparison, we also provide in the second row the results shown in Fig. 3 using the first method. The differences between the three last rows are tiny, showing once again that the interactions between fluxes are much weaker than J_{m}.
The first row of Table 1 gives the corresponding results from nonlinear spinwave theory (NLSWT), which are shown in Fig. 3. These results are obtained by an iterative solution of the meanfield decoupled spinwave Hamiltonian, by comparing the energy of the groundstate flux sector, where κB_{ R } = 1 for all R, with the energy of the sector with κB_{ R } = −1 for all R.
Next, the results for the binding energy of two fluxes that are shown in Fig. 5 are obtained by the iterative ED procedure described above on a periodic lattice with 180 sites, that spans an array of 15 × 2 empty hexagons.
Finally, the ED results for the spin length correction shown in Fig. 6b are obtained by the iterative ED procedure above the groundstate flux configuration, which reduces to an iterative procedure on a single hexagon as noted above.
Comparison between ED and NLSWT
As we discussed in the main text, magnons do not hop from one empty hexagon to another even at the interacting spinwave level, because of the local BSS conservation laws. As a result, the constants m and δ in Eq. (16) vanish by symmetry and the same is true for all similar terms arising in the lattice. Now, Eq. (16) becomes
Written back in terms of spin operators, we recognize that this is the precisely the meanfield decoupling of \(S_i^wS_j^w\) that leads to the Hamiltonian \({\cal H}_{{\mathrm{MF}}}\) discussed above. Essentially then, the NLSWT is a truncation of the Hamiltonian \({\cal H}_{{\mathrm{MF}}}\) using the bosonic representation of spins up to a given order in 1/S. So the iterative ED results are more accurate than the ones from the iterative NLSWT. The two methods should of course agree at large enough S, which is consistent with the trend shown in Figs. 3 and 6b.
Derivation of Eq. (18)
To calculate the tunneling t_{ d } around a single hexagon, we consider the simplest 2 × 2 truncation approach described in ref. ^{46} (see also ref. ^{69}). Namely, we take a hexagon cluster and project the Hamiltonian into the 2 × 2 basis of dimer states shown in Fig. 7a:
The magnitude of the overlap Ω between the two states is
and the matrix elements of the cluster Hamiltonian are
Orthonormalizing the basis leads to the effective 2 × 2 Hamiltonian \(\left( {\begin{array}{*{20}{c}} {E_0 + v} & {t_d} \\ {t_d} & {E_0 + v} \end{array}} \right)\), where the tunneling amplitude t_{ d } and the potential energy V are given by^{46, 69}
The latter is much smaller than \(\left {t_d} \right\) and can be ignored.
Data availability
The data that support the findings of this study as well as the numerical codes are available from the corresponding author on reasonable request.
References
 1.
Anderson, P. W. Resonating valence bonds: a new kind of insulator?”. Mat. Res. Bull. 8, 153–160 (1973).
 2.
Fazekas, P. & Anderson, P. W. On the ground state properties of the anisotropic triangular antiferromagnet. Philos. Mag. 30, 423–440 (1974).
 3.
Kalmeyer, V. & Laughlin, R. B. Equivalence of the resonatingvalencebond and fractional quantum Hall states. Phys. Rev. Lett. 59, 2095–2098 (1987).
 4.
Wen, X. G., Wilczek, F. & Zee, A. Chiral spin states and superconductivity. Phys. Rev. B 39, 11413–11423 (1989).
 5.
Moessner, R., Sondhi, S. L. & Fradkin, E. Shortranged resonating valence bond physics, quantum dimer models, and Ising gauge theories. Phys. Rev. B 65, 024504 (2001).
 6.
Kitaev, A. Y. Faulttolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
 7.
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
 8.
Levin, M. A. & Wen, X.G. Stringnet condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005).
 9.
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
 10.
Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2017).
 11.
Zhou, Y., Kanoda, K. & Ng, T.K. Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).
 12.
Rokhsar, D. S. & Kivelson, S. A. Superconductivity and the quantum hardcore dimer gas. Phys. Rev. Lett. 61, 2376–2379 (1988).
 13.
Misguich, G., Serban, D. & Pasquier, V. Quantum dimer model on the kagome lattice: solvable dimerliquid and ising gauge theory. Phys. Rev. Lett. 89, 137202 (2002).
 14.
Hao, Z., Inglis, S. & Melko, R. Destroying a topological quantum bit by condensing Ising vortices. Nat. Commun. 5, 5781–5787 (2014).
 15.
Lacroix, C., Mendels, P., Mila, F. eds. Introduction to Frustrated Magnetism: Materials, Experiments, Theory (Springer Series in SolidState Sciences, Berlin, 2011).
 16.
Baskaran, G., Sen, D. & Shankar, R. SpinS Kitaev model: classical ground states, order from disorder, and exact correlation functions. Phys. Rev. B 78, 115116 (2008).
 17.
Rousochatzakis, I., Reuther, J., Thomale, R., Rachel, S. & Perkins, N. B. Phase diagram and quantum order by disorder in the Kitaev K _{ 1 } – K _{2} honeycomb magnet. Phys. Rev. X 5, 041035 (2015).
 18.
Chandra, S., Ramola, K. & Dhar, D. Classical Heisenberg spins on a hexagonal lattice with Kitaev couplings. Phys. Rev. E 82, 031113 (2010).
 19.
Wu, F. Y. Dimers on twodimensional lattices. Int. J. Mod. Phys. B 20, 5357–5371 (2006).
 20.
Baxter, R. J. Colorings of a hexagonal lattice. J. Math. Phys. 11, 784–789 (1970).
 21.
Kasteleyn, P. W. Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963).
 22.
Lindgård, P.A. Theory of adiabatic nuclear magnetic ordering in Cu. Phys. Rev. Lett. 61, 629–632 (1988).
 23.
Long, M. W. Effects that can stabilise multiple spindensity waves. J. Phys. 1, 2857–2874 (1989).
 24.
Heinilä, M. T. & Oja, A. S. Selection of the ground state in typeI fcc antiferromagnets in an external magnetic field. Phys. Rev. B 48, 7227–7237 (1993).
 25.
Chernyshev, A. L. & Zhitomirsky, M. E. Quantum selection of order in an XXZ antiferromagnet on a kagome lattice. Phys. Rev. Lett. 113, 237202 (2014).
 26.
Rousochatzakis, I. & Perkins, N. B. Classical spin liquid instability driven by offdiagonal exchange in strong spinorbit magnets. Phys. Rev. Lett. 118, 147204 (2017).
 27.
Jackeli, G. & Avella, A. Quantum order by disorder in the Kitaev model on a triangular lattice. Phys. Rev. B 92, 184416 (2015).
 28.
Moessner, R., Sondhi, S. L. & Chandra, P. Phase diagram of the hexagonal lattice quantum dimer model. Phys. Rev. B 64, 144416 (2001).
 29.
Albuquerque, A. F. et al. Phase diagram of a frustrated quantum antiferromagnet on the honeycomb lattice: magnetic order versus valencebond crystal formation. Phys. Rev. B 84, 024406 (2011).
 30.
Schlittler, T., Barthel, T., Misguich, G., Vidal, J. & Mosseri, R. Phase diagram of an extended quantum dimer model on the hexagonal lattice. Phys. Rev. Lett. 115, 217202 (2015).
 31.
Holstein, T. & Primakoff, H. Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098–1113 (1940).
 32.
Wegner, F. J. Duality in generalized ising models and phase transitions without local order parameters. J. Math. Phys. 12, 2259–2272 (1971).
 33.
Kogut, J. B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979).
 34.
Elitzur, S. Impossibility of spontaneously breaking local symmetries. Phys. Rev. D 12, 3978–3982 (1975).
 35.
Fradkin, E. Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge, 2013).
 36.
Batista, C. D. & Nussinov, Z. Generalized Elitzur’s theorem and dimensional reductions. Phys. Rev. B 72, 045137 (2005).
 37.
Baskaran, G., Mandal, S. & Shankar, R. Exact results for spin dynamics and fractionalization in the Kitaev model. Phys. Rev. Lett. 98, 247201 (2007).
 38.
Chandra, P. & Doucot, B. Possible spinliquid state at large S for the frustrated square Heisenberg lattice. Phys. Rev. B 38, 9335–9338 (1988).
 39.
Harris, A. B., Kallin, C. & Berlinsky, A. J. Possible Néel orderings of the Kagomé antiferromagnet. Phys. Rev. B 45, 2899–2919 (1992).
 40.
Chubukov, A. Order from disorder in a kagomé antiferromagnet. Phys. Rev. Lett. 69, 832–835 (1992).
 41.
Khaliullin, G. Order from disorder: quantum spin gap in magnon spectra of LaTiO_{3}. Phys. Rev. B 64, 212405 (2001).
 42.
Dorier, J., Becca, F. & Mila, F. Quantum compass model on the square lattice. Phys. Rev. B 72, 024448 (2005).
 43.
Mulder, A., Ganesh, R., Capriotti, L. & Paramekanti, A. Spiral order by disorder and lattice nematic order in a frustrated Heisenberg antiferromagnet on the honeycomb lattice. Phys. Rev. B 81, 214419 (2010).
 44.
Chalker, J. T. in Introduction to Frustrated Magnetism: Materials, Experiments, Theory (eds Lacroix, C., Mendels, P. & Mila, F.), Ch. 1 (Springer Series in SolidState Sciences, Berlin, 2011).
 45.
Rousochatzakis, I., Richter, J., Zinke, R. & Tsirlin, A. A. Frustration and DzyaloshinskyMoriya anisotropy in the kagome francisites Cu_{3}Bi(SeO_{3})_{2}O_{2} X(X = Br; Cl). Phys. Rev. B 91, 024416 (2015).
 46.
Rousochatzakis, I., Wan, Y., Tchernyshyov, O. & Mila, F. Quantum dimer model for the spin1/2 kagome Z2 spin liquid. Phys. Rev. B 90, 100406(R) (2014).
 47.
Ganesh, R., van den Brink, J. & Nishimoto, S. Deconfined criticality in the frustrated Heisenberg honeycomb antiferromagnet. Phys. Rev. Lett. 110, 127203 (2013).
 48.
Zhu, Z., Huse, D. A. & White, S. R. Weak plaquette valence bond order in the S = 1/2 honeycomb J _{1}−J_{2} Heisenberg model. Phys. Rev. Lett. 110, 127205 (2013).
 49.
Viciu, L. et al. Structure and basic magnetic properties of the honeycomb lattice compounds Na_{2}Co_{2}TeO_{6} and Na_{3}Co_{2}SbO_{6}. J. Solid State Chem. 180, 1060–1067 (2007).
 50.
Lefrançois, E. et al. Magnetic properties of the honeycomb oxide Na_{2}Co_{2}TeO_{6}. Phys. Rev. B 94, 214416 (2016).
 51.
Jackeli, G. & Khaliullin, G. Mott insulators in the strong spinorbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).
 52.
Chaloupka, J., Jackeli, G. & Khaliullin, G. KitaevHeisenberg model on a honeycomb lattice: possible exotic phases in iridium oxides A_{2}IrO_{3}. Phys. Rev. Lett. 105, 027204 (2010).
 53.
WitczakKrempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated quantum phenomena in the strong spinorbit regime. Annu. Rev. Condens. Matter Phys. 5, 57–82 (2014).
 54.
S. Trebst Kitaev Materials. Preprint at https://arxiv.org/abs/1701.07056 (2017).
 55.
Schmied, R., Wesenberg, J. H. & Leibfried, D. Quantum simulation of the hexagonal Kitaev model with trapped ions. New J. Phys. 13, 115011 (2011).
 56.
You, J. Q., Shi, X.F., Hu, X. & Nori, F. Quantum emulation of a spin system with topologically protected ground states using superconducting quantum circuits. Phys. Rev. B 81, 014505 (2010).
 57.
Xiang, Z., Yu, T., Zhang, W., Hu, X. & You, J. Implementing a topological quantum model using a cavity lattice. Sci. China Phys. 55, 1549–1556 (2012).
 58.
Duan, L.M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).
 59.
Micheli, A., Brennen, G. K. & Zoller, P. A toolbox for latticespin models with polar molecules. Nat. Phys. 2, 341–347 (2006).
 60.
Gorshkov, A. V., Hazzard, K. R. A. & Rey, A. M. Kitaev honeycomb and other exotic spin models with polar molecules. Mol. Phys. 111, 1908–1916 (2013).
 61.
Manmana, S. R., Stoudenmire, E. M., Hazzard, K. R. A., Rey, A. M. & Gorshkov, A. V. Topological phases in ultracold polarmolecule quantum magnets. Phys. Rev. B 87, 081106 (2013).
 62.
Wan, Y. & Tchernyshyov, O. Phenomenological Z _{2} lattice gauge theory of the spinliquid state of the kagome Heisenberg antiferromagnet. Phys. Rev. B 87, 104408 (2013).
 63.
Hwang, K., Huh, Y. & Kim, Y. B. Z _{2} gauge theory for valence bond solids on the kagome lattice. Phys. Rev. B 92, 205131 (2015).
 64.
Sachdev, S. Kagome and triangularlattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantumdisordered ground states with unconfined bosonic spinons. Phys. Rev. B 45, 12377–12396 (1992).
 65.
Yan, S., Huse, D. A. & White, S. R. Spinliquid ground state of the S = 1/2 kagome Heisenberg antiferromagnet. Science 332, 1173–1176 (2011).
 66.
Depenbrock, S., McCulloch, I. P. & Schollwöck, U. Nature of the spinliquid ground state of the S = 1/2 Heisenberg model on the kagome lattice. Phys. Rev. Lett. 109, 067201 (2012).
 67.
HongChen, J., Zhenghan, W. & Leon, B. Identifying topological order by entanglement entropy. Nat. Phys. 8, 902–905 (2012).
 68.
Ralko, A., Mila, F. & Rousochatzakis, I. Microscopic theory of the nearestneighbor valence bond sector of the spin1/2 kagome antiferromagnet. Phys. Rev. B 97, 104401 (2018).
 69.
Schwandt, D., Mambrini, M. & Poilblanc, D. Generalized hardcore dimer model approach to lowenergy Heisenberg frustrated antiferromagnets: general properties and application to the kagome antiferromagnet. Phys. Rev. B 81, 214413 (2010).
Acknowledgements
We thank G. Baskaran, A. Ralko, A. Tsirlin, Y. Wan, and M. D. Schulz for fruitful discussions. Part of this work was done at the Perimeter Institute in Waterloo, which is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award #DESC0018056.
Author information
Affiliations
Contributions
All authors contributed to the analysis and interpretation of the results, and the preparation of the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rousochatzakis, I., Sizyuk, Y. & Perkins, N.B. Quantum spin liquid in the semiclassical regime. Nat Commun 9, 1575 (2018). https://doi.org/10.1038/s41467018039341
Received:
Accepted:
Published:
Further reading

Thermodynamic behavior of modified integerspin Kitaev models on the honeycomb lattice
Physical Review E (2021)

Flux mobility delocalization in the Kitaev spin ladder
Physical Review B (2021)

Quantumclassical crossover in the spin 12 HeisenbergKitaev kagome magnet
Physical Review Research (2020)

HeisenbergKitaev model in a magnetic field: 1/S expansion
Physical Review B (2020)

Order by singularity in Kitaev clusters
Physical Review Research (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.