Abstract
Recent numerical studies indicate that the antiferromagnetic Kitaev honeycomb lattice model undergoes a magneticfieldinduced quantum phase transition into a new spinliquid phase. This intermediatefield phase has been previously characterized as a gapless spin liquid. By implementing a recently developed variational approach based on the exact fractionalized excitations of the zerofield model, we demonstrate that the fieldinduced spin liquid is gapped and belongs to Kitaev’s 16fold way. Specifically, the lowfield nonAbelian liquid with Chern number C = ±1 transitions into an Abelian liquid with C = ±4. The critical field and the fielddependent behaviors of key physical quantities are in good quantitative agreement with published numerical results. Furthermore, we derive an effective field theory for the fieldinduced critical point which readily explains the ostensibly gapless nature of the intermediatefield spin liquid.
Introduction
The exactly solvable Kitaev model on the honeycomb lattice^{1} has deepened our insight into quantum spin liquids and helped us in identifying strongly spinorbitcoupled 4d and 5d materials that may host these exotic quantum phases of matter^{2,3}. Indeed, recent years have seen a flurry of such “Kitaev materials” in which the microscopic spin Hamiltonian is believed to approximately realize the Kitaev honeycomb model^{4,5,6,7}. The most famous ones include the honeycomb iridates, Na_{2}IrO_{3}^{8,9,10,11,12,13}, αLi_{2}IrO_{3}^{14,15}, and H_{3}LiIr_{2}O_{6}^{16}, as well as the honeycomb halide αRuCl_{3}^{17,18,19,20,21,22,23,24,25}.
While most of these materials are magnetically ordered at the lowest temperatures, the zigzag magnetic order in αRuCl_{3} can be suppressed with an inplane magnetic field^{26,27,28,29,30,31,32,33,34,35}. Also, there are some experimental indications for an intermediatefield spinliquid phase between the lowfield magnetically ordered phase and the highfield spinpolarized phase. Most importantly, a recent experimental work^{36} reported a halfintegerquantized thermal Hall conductivity in the intermediatefield regime just beyond the transition out of zigzag order. Though the exact nature of this regime is still an open question, the ongoing experimental efforts reveal the importance of precisely characterizing fieldinduced spinliquid phases.
Motivated in large part by the intriguing experimental observations, the behavior of the Kitaev model in a magnetic field has been extensively studied^{37} by various approaches, including exact diagonalization^{38,39,40,41,42}, densitymatrix renormalization group (DMRG)^{40,41,42,43}, infinite DMRG (iDMRG)^{44}, tensornetwork methods^{45}, continuoustime quantum Monte Carlo techniques^{46}, and slaveparticle meanfield theories^{47}. These approaches all give consistent results. While the ferromagnetic Kitaev model has a single transition into a polarized phase, the antiferromagnetic Kitaev model includes a new intermediatefield spin liquid between the lowfield nonAbelian spin liquid^{1} and the highfield polarized phase.
In this work, we implement a novel variational approach^{48} to investigate the groundstate phase diagram of the antiferromagnetic Kitaev model in a magnetic field parallel to the [111] direction. This approach is based on the exact fractionalized Majoranafermion (“spinon”) and gaugeflux (“vison”) excitations of the pure Kitaev model at zero field^{1}. It accounts for two effects of the magnetic field: the renormalization of the Majorana dispersion through a hybridization with pairs of fluxes (see Fig. 1a) and the finite dispersion acquired by the flux pairs themselves (see Fig. 1b). Remarkably, we find a continuous quantum phase transition, induced by a softening of a hybridized excitation, at a critical field h_{c} ≃ 0.50, which is very close to the critical field h_{c} ≃ 0.44 reported by a recent iDMRG study^{44}. The critical point signals the transition of the nonAbelian spin liquid^{1} with Chern number C = ± 1 into an Abelian spin liquid with C = ± 4. The predicted field dependence of the flux expectation value and the second derivative of the groundstate energy is also in good quantitative agreement with the iDMRG results. Moreover, the effective field theory of the quantum critical point, as derived from the microscopic Hamiltonian, predicts a lowenergy ring of gapped excitations in momentum space, which is difficult to be distinguished from a gapless Fermi surface in finite systems. We conjecture that this is the main reason why previous works^{38,40,41,42,43} characterized the phase at h ≳ h_{c} as a gapless spin liquid.
Model
We consider the antiferromagnetic Kitaev model^{1} in an external magnetic field along the [111] direction,
where h is the magnetic field (in units of the Kitaev energy) and \({\hat{{{{{{\bf{r}}}}}}}}_{\alpha }\) is the nearestneighbor vector from an A site to a B site along an α bond (see Fig. 1). For the exactly solvable Kitaev model in the h = 0 limit, the lowenergy spectrum comprises gapless matter fermions (i.e., spinons) with a single Dirac cone and gapped dispersionless \({{\mathbb{Z}}}_{2}\) gauge fluxes. These elementary excitations are described in terms of four Majorana fermions \({c}_{{{{{{\bf{r}}}}}}}\) and \({b}_{{{{{{\bf{r}}}}}}}^{\alpha }\) with α = x, y, z at each site r, where \({c}_{{{{{{\bf{r}}}}}}}\) are the matter fermions, and \({b}_{{{{{{\bf{r}}}}}}}^{\alpha }\) are bond fermions associated with the \({{\mathbb{Z}}}_{2}\) gauge field \({u}_{{{{{{\bf{r}}}}}},{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha }\equiv i{b}_{{{{{{\bf{r}}}}}}}^{\alpha }{b}_{{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha }=\pm 1\). The gauge fields are conserved bond variables that commute with each other; their product around any plaquette p (see Fig. 1a) is gauge invariant and expressible in terms of the physical spins:
Thus, W_{p} = ±1 can be identified as static \({{\mathbb{Z}}}_{2}\) gauge fluxes. In each flux sector, {W_{p} = ±1}, represented with an appropriate gaugefield configuration, \(\{{u}_{{{{{{\bf{r}}}}}},{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha }=\pm 1\}\), the zerofield model then reduces to a quadratic matterfermion problem.
While the model in Eq. (1) is not exactly solvable for a finite field, we can derive a lowenergy effective model by projecting \({{{{{\mathcal{H}}}}}}\) into the lowenergy sector of the pure Kitaev model (corresponding to h = 0) generated by single matterfermion and/or fluxpair excitations^{48}. We focus on flux pairs because, unlike single fluxes, they are coherent fermionic quasiparticles^{48} and can readily hybridize with matter fermions (see Fig. 1a). The fermionic fluxpair excitations can be represented with dressed bondfermion operators \({({\tilde{\chi }}_{{{{{{\bf{r}}}}}}\in A}^{\alpha })}^{{{\dagger}} }=\frac{1}{2}({\tilde{b}}_{{{{{{\bf{r}}}}}}}^{\alpha }i{\tilde{b}}_{{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha })\) that have the same projective symmetries as the bare bondfermion operators \({({\chi }_{{{{{{\bf{r}}}}}}\in A}^{\alpha })}^{{{\dagger}} }=\frac{1}{2}({b}_{{{{{{\bf{r}}}}}}}^{\alpha }i{b}_{{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha })\). The operator \({({\tilde{\chi }}_{{{{{{\bf{r}}}}}}\in A}^{\alpha })}^{{{\dagger}} }\) turns the ground state of the pure Kitaev model into an excited state with a singleflux pair on the α bond connected to the site r ∈ A by not only creating a bond fermion but also distorting the matterfermion state: \({({\tilde{\chi }}_{{{{{{\bf{r}}}}}}\in A}^{\alpha })}^{{{\dagger}} }\left\omega \right\rangle \otimes \left0\right\rangle =\ \left{\phi }_{{{{{{\bf{r}}}}}}}^{\alpha }\right\rangle \otimes \left{\chi }_{{{{{{\bf{r}}}}}}}^{\alpha }\right\rangle\), where \(\left\omega \right\rangle\) and \(\left{\phi }_{{{{{{\bf{r}}}}}}}^{\alpha }\right\rangle\) are the matterfermion vacua of the gaugefield configurations \(\left0\right\rangle\) and \(\left{\chi }_{{{{{{\bf{r}}}}}}}^{\alpha }\right\rangle\) that correspond to the fluxfree sector and the singlefluxpair sector, respectively. (Mathematically, \(\left{\chi }_{{{{{{\bf{r}}}}}}}^{\alpha }\right\rangle ={({\chi }_{{{{{{\bf{r}}}}}}}^{\alpha })}^{{{\dagger}} }\left0\right\rangle\), while \(\left0\right\rangle\) is the barebondfermion vacuum with \({u}_{{{{{{\bf{r}}}}}},{{{{{\bf{r}}}}}}+{\hat{{{{{{\bf{r}}}}}}}}_{\alpha }}^{\alpha }=1\) for all bonds.) If we project the pure Kitaev model [i.e., the first term of Eq. (1)] to its lowenergy sector containing at most one matterfermion or fluxpair excitation, the resulting lowenergy Hamiltonian reads
where the first term is the quadratic matterfermion problem within the fluxfree sector^{1}, while the second term accounts for the finite energy (Δ_{χ} ≃ 0.26) of a flux pair. The Zeeman term [i.e., the second term of Eq. (1)] can then either hybridize a flux pair with a matter fermion (see Fig. 1a) or hop a flux pair to a neighboring bond (see Fig. 1b). By summing \({\tilde{{{{{{\mathcal{H}}}}}}}}_{h = 0}\) and the most general symmetryallowed Hamiltonians describing these two processes, the effective lowenergy Hamiltonian for the full model in Eq. (1) becomes
where R is a lattice vector, ϵ_{αβ} = ∑_{γ}ϵ_{αβγ} is an antisymmetric symbol based on the LeviCivita symbol ϵ_{αβγ}, while p_{R,α} and q are dimensionless parameters to be determined. Notice that some p_{R,α} are identical due to the threefold rotation symmetry acting simultaneously in real space and spin space.
Since the effective Hamiltonian \(\tilde{{{{{{\mathcal{H}}}}}}}\) is quadratic, it can be straightforwardly diagonalized in momentum space:
where \({\lambda }_{{{{\bf{k}}}}}={\sum }_{\alpha }{{{{\rm{e}}}}}^{{i{{{{{\bf{k}}}}}}\cdot {\hat{{{{{{\bf{r}}}}}}}_{\alpha} }}}\) and P_{k,α} = ∑_{R}p_{R,α} e^{ik⋅R}, while
are momentumspace matter and bond fermions in terms of the sublattice index ν = A, B and the system size N. By considering the matrix elements of the Zeeman term ∝ h in Eq. (1) within the lowenergy sector of the pure Kitaev model^{48}, we relate the dimensionless parameters in Eq. (5) to matterfermion matrix elements of this exactly solvable model (Note: see the Supplementary Information for more details on the dimensionless parameters of the effective Hamiltonian, the expectation value of the flux operator, the coefficients of the effective field theory, and the nonanalytic behavior of the groundstate energy):
where r = 0 is an A site, while \({\psi }_{{{{{{\bf{k}}}}}}}=({C}_{{{{{{\bf{k}}}}}},A}+i{{{{\rm{e}}}}}^{i{\varphi }_{{{{{{\bf{k}}}}}}}}{C}_{{{{{{\bf{k}}}}}},B})/\sqrt{2}\) in terms of \({{{{\rm{e}}}}}^{i{\varphi }_{{{{{{\bf{k}}}}}}}}={\lambda }_{{{{{{\bf{k}}}}}}}/ {\lambda }_{{{{{{\bf{k}}}}}}}\) are the matter fermions diagonalizing the fluxfree sector of the pure Kitaev model. For a finite honeycomb lattice with N = 121 × 121 unit cells, we numerically find q ≃ 0.0494 and P_{0,α} ≃ 0.722.
Results
We study the lowenergy effective model in Eq. (4) as a function of the magnetic field h. At zero field, the spectrum coincides with that of the pure Kitaev model and contains one dispersive matterfermion band as well as the three flat bondfermion bands (see Fig. 2a). For a small field, h ≪ Δ_{χ}, the hybridization between these four bands gives rise to a finite energy gap, Δ_{K}(h) ∝ h^{3}, at the K point of the Brillouin zone (BZ). The slow field dependence of Δ_{K}(h), which is expected from a perturbative argument by Kitaev^{1}, explains why the global minimum of the band structure remains at the K point up to a large field, h_{0} ≃ 0.46. As shown in Fig. 3a, the global minimum switches from the K point to the Γ point at h = h_{0}, and the corresponding gap, Δ_{Γ}(h), closes at a slightly larger field, h_{c} ≃ 0.50 (see Fig. 2b). Since the little group of the Γ point includes the threefold rotation C_{3}, the fermion eigenmodes at the Γ point can be classified according to their C_{3} eigenvalues. The natural bondfermion modes, corresponding to C_{3} eigenvalues 1 and e^{∓2πi/3}, respectively, are then
Since the matterfermion mode ψ_{0} is invariant under C_{3}, it can only hybridize with the bondfermion mode \({\tilde{X}}_{{{{{{\boldsymbol{0}}}}}}}^{0}\). At the critical field, \({h}_{c}=3\sqrt{{{{\Delta }}}_{\chi }/2}\ {({\sum }_{\alpha }{P}_{{{{{{\boldsymbol{0}}}}}},\alpha })}^{−1}\simeq 0.50\), one of the resulting hybridized eigenmodes is gapless. In contrast, there is a higher critical field, \(h_c^{\prime} ={{{\Delta }}}_{\chi }/(2\sqrt{3}q)\simeq 1.52\) (not shown in Fig. 3), at which the pure bondfermion eigenmode \({\tilde{X}}_{{{{{{\boldsymbol{0}}}}}}}^{+}\) has vanishing energy. We note that a complete diagonalization over the full BZ reveals yet another critical point at h_{c}″ ≃ 1.0 due to the softening of a hybridized mode at the M point. We emphasize, however, that the effective model is no longer expected to be valid when h is significantly larger than h_{c}.
Figure 3a shows the overall energy gap as a function of the magnetic field h. As expected, the gap is proportional to h^{3} at the smallest fields, h ≪ Δ_{χ}. Just below h_{c}, the global minimum of the excitation spectrum switches from the K point to the Γ point, and the gap vanishes at h_{c} ≃ 0.50^{38,40,44}. Importantly, the zeroenergy mode at h = h_{c} has dominant bondfermion character with a large bondfermion weight 6/(6 + Δ_{χ}) ≃ 0.96 (see also Fig. 2b), which is consistent with the numerical closing of the vison gap in the specific heat^{38}. In contrast, the gap reopens for h ≳ h_{c}, which appears to be in contradiction with the same numerical results and the corresponding conjecture of a gapless U(1) spin liquid at intermediate fields. However, our analytic approach can also explain the numerical similarity between the gapped spin liquid at h ≳ h_{c} and a gapless spin liquid with a circular spinon Fermi surface. Indeed, as we explain below, the phase transition at h = h_{c} gives rise to a lowenergy ring at h ≳ h_{c} (see the inset of Fig. 3a) which expands from the Γ point and corresponds to a small energy gap \(\propto {(h{h}_{c})}^{3/2}\). This lowenergy ring naturally explains the large lowenergy density of states found by exact diagonalization^{38,40}. The emergence of the lowenergy ring and the nature of the h ≳ h_{c} phase are explained in the next section, where we derive an effective field theory to describe the continuous topological phase transition at h = h_{c}.
Figures 3b and c plot the second derivative of the groundstate energy, \({E}_{G}^{^{\prime\prime} }={{{{\rm{d}}}}}^{2}{E}_{G}/{{{\rm{d}}}}{h}^{2}\), and the expectation value of the \({{\mathbb{Z}}}_{2}\) gauge flux, 〈W_{p}〉, against the magnetic field. As we explain below, the discontinuity of \({E}_{G}^{^{\prime\prime} }\) at h = h_{c} is a generic property of the corresponding phase transition. This discontinuity leads to a peak in \({E}_{G}^{^{\prime\prime} }\) at h = h_{c}, which is qualitatively and quantitatively consistent with the iDMRG results^{44}. We note that our result for 〈W_{p}〉 (See the “Note” above earlier) (see Fig. 3c) is also consistent with iDMRG.
We argue that our effective model in Eq. (4) remains valid up to a field h ≳ h_{c} just beyond the first phase transition. Indeed, the fractionalized excitations of the pure Kitaev model remain well defined throughout the lowfield phase at h < h_{c}; however, after the first phase transition induced by their softening, these original excitations are superseded by the emergent excitations of the higherfield phase. Therefore, we focus on the first phase transition at h = h_{c} throughout the rest of this work.
Remarkably, the critical field h_{c} ≃ 0.50 is only 10% higher than the corresponding iDMRG result, h_{c} ≃ 0.44^{44}. Also, the slight overestimation of h_{c} is not surprising because the inclusion of higherenergy (E ≃ 2Δ_{χ}) states with four fluxes and one matter fermion would lead to a reduction of h_{c}. Finally, at h = h_{c}, the dynamical spin structure factor from iDMRG indicates that the spin excitation gap closes at the Γ point, which is in agreement with our results. Indeed, since a spin excitation fractionalizes into a pair of fermion excitations, and the fermions at h = h_{c} are gapless at the Γ point (see Fig. 2b), a pair of gapless fermions has zero total momentum, corresponding to a vanishing spin gap at the Γ point. These similarities between the iDMRG results and those obtained from our effective Hamiltonian \(\tilde{{{{{{\mathcal{H}}}}}}}\) indicate that our variational lowenergy manifold captures the essence of the phase transition at h = h_{c} and the new spinliquid phase at h ≳ h_{c}.
Field theory of topological phase transition
In the vicinity of the critical field, h ≃ h_{c} ≃ 0.50, the lowenergy fermion eigenmodes belong to the trivial representation of C_{3}, and the longwavelength limit of \(\tilde{{{{{{\mathcal{H}}}}}}}\), corresponding to the region around the Γ point, can be written as
where τ_{x,y,z} are the Pauli matrices, and \({f}_{{{{{{\boldsymbol{k}}}}}}}={({f}_{1,{{{{{\boldsymbol{k}}}}}}},{f}_{2,{{{{{\boldsymbol{k}}}}}}})}^{{{{\rm{T}}}}}\) is a twocomponent fermionic operator corresponding to the two zeroenergy modes of \(\tilde{{{{{{\mathcal{H}}}}}}}\) at the critical field:
The coefficients \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{x,y,z}\) in Eq. (9) must be C_{3} invariant real polynomials. Up to cubic order in k, there are only four such polynomials: the trivial polynomial 1, the quadratic polynomial \({k}^{2}={k}_{x}^{2}+{k}_{y}^{2}\), and the cubic polynomials \({g}_{{{{{{\boldsymbol{k}}}}}}}^{x}={k}_{x}(3{k}_{y}^{2}{k}_{x}^{2})\) and \({g}_{{{{{{\boldsymbol{k}}}}}}}^{y}={k}_{y}(3{k}_{x}^{2}{k}_{y}^{2})\). Moreover, the particlehole symmetry of the original Hamiltonian \({{{{{\mathcal{H}}}}}}\) dictates that \({\tilde{{{{{{\mathcal{H}}}}}}}}_{{{{{{\rm{eff}}}}}}}\) must remain invariant under \({f}_{{{{{{\boldsymbol{k}}}}}}}\to {\tau }_{x}{({f}_{{{{{{\boldsymbol{k}}}}}}}^{{{\dagger}} })}^{{{{\rm{T}}}}}\), implying that the polynomials \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{\mu }\) must satisfy the following relationships:
These symmetry considerations then lead to the general forms
where c_{0}, c_{z}, and c_{ην} are, in general, functions of h. Since the phase transition at h = h_{c} is driven by a sign change in c_{0}, we assume that c_{z} and c_{ην} are constants, while we write \({c}_{0}=c_{0}^{\prime} (h{h}_{c})\) with a constant \(c_{0}^{\prime}\). Starting from Eqs. (5) and (7), and defining all lengths in units of the lattice vector (i.e., the distance between two neighboring A sites), the constants are derived to be \(c_{0}^{\prime} \simeq 1.00\), c_{z} ≃ 0.0125, c_{xx} ≃ − 0.00268, c_{yy} ≃ − 0.00088, and c_{xy} = c_{yx} = 0 (See the “Note” above earlier). Then, using Eq. (9), the fermion dispersion is given by
and becomes gapless at k = 0 for h = h_{c}. For h < h_{c}, the dispersion is dominated by the function \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{z}\) and is largely quadratic: \({\omega }_{{{{{{\boldsymbol{k}}}}}}}\simeq  c_{0}^{\prime}  ({h}_{c}h)+{c}_{z}{k}^{2}\). In contrast, for h > h_{c}, the function \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{z}\) vanishes for \( {{{{{\boldsymbol{k}}}}}} =\sqrt{ c_{0}^{\prime}  (h{h}_{c})/{c}_{z}}\). Thus, along this ring of radius ∣k∣, the energy gap is determined by the small cubic contributions from \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{x,y}\) and has a slow field dependence: \({{\Delta }}\propto {(h{h}_{c})}^{3/2}\). The net result is a ring of lowenergy fermions around the Γ point (see the inset of Fig. 3a).
The effective field theory in Eq. (9) describes a continuous topological phase transition. The phases on both sides of the transition belong to Kitaev’s 16fold way^{1} and are characterized by the fermion Chern number. The contribution from the lowenergy fermions to this Chern number is given by^{49}
where d_{k} = β_{k}/∣β_{k}∣ and \({{{{{{\boldsymbol{\beta }}}}}}}_{{{{{{\boldsymbol{k}}}}}}}=({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{x},{\beta }_{{{{{{\boldsymbol{k}}}}}}}^{y},{\beta }_{{{{{{\boldsymbol{k}}}}}}}^{z})\). Geometrically, C is simply the skyrmion number of the vector field d_{k}. Figure 4 depicts the vector field d_{k} around the Γ point on both sides of the phase transition at h = h_{c}. While the field configuration is topologically trivial for h < h_{c}, it includes six merons (three skyrmions) for h > h_{c}. The corresponding change in the Chern number, ΔC = 3, is then a generic property of the phase transition described by \({\tilde{{{{{{\mathcal{H}}}}}}}}_{{{{{{\rm{eff}}}}}}}\). To understand the emergence of the six merons around the Γ point, we first note that \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{\eta }\propto \,{{\mbox{Im}}}\,({k}_{+}^{3}{{{{\rm{e}}}}}^{i{\phi }_{\eta }})\) with k_{+} = k_{x} + ik_{y} and \({\phi }_{\eta }=\arctan ({c}_{\eta x}/{c}_{\eta y})\). Each function \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{\eta }\) (with η = x, y) possesses three nodal lines corresponding to \({k}_{y}/{k}_{x}=\tan ({\phi }_{\eta }/3+\varphi )\) with φ = 0, π/3, 2π/3. Ignoring the \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{y}\) function, the lowenergy spectrum then contains six Dirac nodes Q_{j} (with j = 1, 2, . . . , 6) at the intersections of the nodal lines of \({\beta }_{{{{{{\boldsymbol{k}}}}}}}^{x}\) and the ring of radius \( {{{{{\boldsymbol{k}}}}}} =\sqrt{ c_0^{\prime}  (h{h}_{c})/{c}_{z}}\). The vorticity of the vector field d_{k} around each Dirac node Q_{j} is (−1)^{j}. Assuming ϕ_{x} ≠ ϕ_{y} (which is true in our case), the finite value of \({\beta }_{{{{{{{\boldsymbol{Q}}}}}}}_{j}}^{y}\propto {(1)}^{j}\) generates a mass term for each Dirac node in such a way that the Dirac nodes all give identical contributions (+1/2 each or −1/2 each) to the change in the Chern number. The net change in the Chern number is then
Using the constants c_{z} and c_{ην} given above, we obtain ΔC = 3 at the critical field h = h_{c}. Since the lowfield phase at h < h_{c} is well known^{1} to have Chern number 1, we conclude that the higherfield phase at h ≳ h_{c} has Chern number 4.
We next consider the second derivative of the groundstate energy \({E}_{G}^{^{\prime\prime} }\) with respect to the magnetic field h. The universal critical behavior at h = h_{c} is determined by the lowenergy modes ∣k∣≤Λ, where the cutoff Λ can be made arbitrarily small (corresponding to an infrared singularity). While the contribution of these modes to \({E}_{G}^{^{\prime\prime} }\) is ∝ Λ^{2} for \(h\to {h}_{c}^{}\), it is an \({{{{{\mathcal{O}}}}}}(1)\) constant for \(h\to {h}_{c}^{+}\). In particular, there is a contribution from the neighborhood of the lowenergy ring at h ≳ h_{c} which is independent of the cutoff Λ. Therefore, we obtain a discontinuity in \({E}_{G}^{^{\prime\prime} }\) at the critical field (See the “Note” above earlier):
Remarkably, this discontinuity in \({E}_{G}^{^{\prime\prime} }\), as shown in Fig. 3b, is entirely determined by two coefficients of the effective field theory. From the constants \(c_{0}^{\prime}\) and c_{z} given above, it is found to be \({{\Delta }}{E}_{G}^{^{\prime\prime} }\simeq 5.5\), which is consistent with the corresponding result for a finite lattice (see Fig. 3b). The quantitative agreement between this value and the one obtained from iDMRG^{44} indicates that the effective field theory at h = h_{c} is both qualitatively correct and quantitatively accurate.
Discussion
Our simple and accurate variational approach to extended Kitaev models^{48} indicates that the antiferromagnetic (AFM) Kitaev model undergoes a continuous quantum phase transition driven by a magnetic field parallel to the [111] direction. According to this approach, the new phase, which has been reported in previous numerical works^{38,39,40,41,42,43,44}, is a gapped chiral spin liquid with a ring of lowenergy excitations. Due to its large lowenergy density of states, it is difficult for numerical simulations to distinguish this lowenergy ring from a gapless Fermi surface. In particular, while DMRG may, in principle, detect gapless modes via a finite value of the central charge^{41,42,43,44}, different studies find conflicting values^{43} or even unphysical noninteger values^{44}, thereby indicating that the currently available system sizes cannot be used to determine whether the new phase is gapped or gapless^{44}.
In contrast to the nonAbelian lowfield phase, the new phase at higher fields possesses Abelian topological order with four distinct types of anyons: 1 (vacuum), ε (fermion), as well as e and m (vortices). The two phases can then be distinguished numerically by computing the entanglement spectrum^{50} or the topological entanglement entropy for a bipartition of an infinite cylinder^{51,52,53}, readily available in iDMRG^{44}. However, due to the challenges mentioned above, such a numerical confirmation of our predictions may require the addition of irrelevant Hamiltonian terms that increase the gap in the higherfield phase without generating new phase transitions.
From an experimental perspective, it is important to note that the higherfield spin liquid is known to be stable against both Heisenberg and Gamma interactions^{38}, making it more likely to emerge in real materials. Also, in the presence of ferromagnetic Heisenberg terms, a fieldinduced transition between the higherfield spin liquid and a lowerfield zigzag order, potentially relevant for αRuCl_{3}, has been reported^{42}. According to our theory, the key experimental signature of the higherfield spin liquid is a specific quantized value of the thermal Hall conductivity, \({\kappa }_{xy}=\pi {k}_{B}^{2}T/(3\hslash )\), which is four times larger than for the lowfield nonAbelian spin liquid.
We next remark that our variational approach is still approximately valid in the presence of both a matterfermion and a fluxpair excitation and that, in the presence of nonKitaev interactions, it can also be used to describe bound states between these two types of excitations^{48}. Since such a bound state corresponds to a spin excitation, its softening leads to a divergent magnetic susceptibility for some wave vector and thus signals the onset of magnetic ordering.
We also emphasize that our approach straightforwardly generalizes to the ferromagnetic (FM) Kitaev model. In this case, the first term in Eq. (3) has a negative sign, and the fluxpairhopping parameter in Eq. (7) is found to be q ≃ 1.35, i.e., about 30 times larger than for the AFM Kitaev model. Therefore, the lowestfield phase transition is driven by a softening of a pure fluxpair mode and happens at a much smaller critical field, \(h^{\prime} ={{{\Delta }}}_{\chi }/(2\sqrt{3}q)\simeq 0.056\). The strong asymmetry between the FM and AFM Kitaev models is due to opposite (constructive and destructive) interference effects between the two processes contributing to fluxpair hopping^{48}. We note that this asymmetry is not apparent in the simplified perturbative analysis of ref. ^{1} because it neglects the energy dispersions of the intermediate states. We further remark that our results for the FM Kitaev model are also consistent with numerical studies that report a single firstorder transition into a trivial polarized phase at a critical field h_{p} ≃ 0.028^{44}. At this firstorder phase transition, corresponding to \({h}_{p}\lesssim h^{\prime}\), the fluxes suddenly proliferate and confine all fractionalized excitations.
Finally, going back to the AFM Kitaev model, it is interesting to note that a recent work^{54} has also found a fieldinduced chiral spin liquid phase with Chern number C = 4 through a completely different approach.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors thank Matthias Gohlke, Frank Pollmann, and Federico Becca for useful discussions. SS.Z. and C.D.B. are supported by funding from the Lincoln Chair of Excellence in Physics. G.B.H. was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center.
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Zhang, SS., Halász, G.B. & Batista, C.D. Theory of the Kitaev model in a [111] magnetic field. Nat Commun 13, 399 (2022). https://doi.org/10.1038/s41467022280143
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DOI: https://doi.org/10.1038/s41467022280143
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