Quantum spin liquid phases in Kitaev materials

Abstract

We develop a gauge-invariant renormalized mean-field theory (RMFT) to reliably find the quantum spin liquid (QSL) states and their field response for realistic Kitaev materials under strong magnetic fields and described by the generalized Kitaev J-K-Γ-${\Gamma }^{{\prime} }$ model. Remarkably, while our RMFT reproduces previous results based on using more complicated numerical methods, it also predicts several new stable QSL states. In particular, since Kitaev spin liquid (KSL) is no longer a saddle point solution, a new exotic 2-cone state distinct from the KSL is found to describe experimental observations well, and hence should be the candidate state realized in the Kitaev material, α-RuCl₃. We further explore the mechanism for the suppression of the observed thermal Hall conductivity at low temperatures within the fermionic framework, and show that the polar-angle dependence of the fermionic gap can distinguish the found 2-cone state from the KSL state in further experiments.

Introduction

The quantum spin liquid (QSL) is a state of quantum spins without spin ordering but spins are highly entangled¹. The original idea that interacting spins may form a liquid phase was first proposed by P. W. Anderson². It was later brought up by Anderson to construct the parent state that fosters superconductivity for doped cuprate superconductors³. The realization of Anderson’s idea has been pursued in many interacting spin systems. In particular, the quantum spin liquid phase has been demonstrated to be the ground state for a number of theoretical models. Notably, the Kitaev model on honeycomb lattice is an exactly solvable model with the quantum spin liquid phase being the ground state⁴.

Being an exactly solvable model, the Kitaev model provides a platform to understand gapped and gapless quantum spin-liquid (QSL) states with distinct anyon statistics. The gapless Kitaev spin-liquid (KSL) state possesses two kinds of basic excitation, cone-like gapless fermionic excitation formed by Majorana fermions and gapped excitation formed by flux excitations, termed as vison. Under weak magnetic fields, the Majorana-fermion excitations open up a gap with their bands possessing finite Chern number ν = ±1^4,5,6,7. The associated Majorana edge states give rise to half quantized thermal Hall conductivity^4,8 which is claimed to be observed in some experiments^9,10,11. The gapped visons together with Majorana bound states at their core obey non-Abelian statistics^4,6,12, thus providing a potential platform for topological quantum computation.

On experimental side, due to the unusual spin-spin interaction involved, it is difficult to realize the Kitaev model in pure spin systems. Instead, it was pointed out that super-exchange interactions of pseudospins formed by the combination of orbital and spin degree of freedom for d electrons may realize the required spin-spin interaction^13,14. However, in real systems, other exchange interactions are also present; the resulting model usually contains many other spin-spin interactions. The typical model that includes the Kitaev spin-spin interaction is the so-called the generalized Kitaev $J\,\text{-}K\text{-}\Gamma \text{-}\,{\Gamma }^{{\prime} }$ model in which K is the strength of the Kitaev spin-spin interaction, J is the isotropic super-exchange interaction, Γ and ${\Gamma }^{{\prime} }$ are interactions of off-diagonal components of pseudospins^{15,16,17,18,19,20}. Material candidates for realizing the model are mainly in iridium oxides, ruthenium and cobalt compounds^21,22,23,24.

Recently, a key feature of QSL was observed in the candidate Kitaev material α-RuCl₃. By applying magnetic fields, it is found that the antiferromagnetic phase in α-RuCl₃ evolves into a phase with half-integer quantized thermal Hall conductivity in some samples at a given temperature range^9,10,11. However, some other experiments reported non-quantized thermal Hall conductivity at low temperatures^25,26,27. It is also suggested that phonons make the major contribution to the thermal Hall conductivity²⁸. Although it is still under debate on whether this phase is a spin-liquid phase and on the bosonic or fermionic origin of the observed thermal Hall phenomenon^25,26,27,29, the field-angle-dependent specific heat measurement seems to support fermionic scenario, as the sign-changing of thermal Hall conductivity and the fitted band gap seem to agree with predictions by the KSL under weak field^30,31,32. Theoretically, these predictions are based on features of original KSL. As indicated in the above, however, except for the dominant ferromagnetic Kitaev interaction K, the effective spin model for α-RuCl₃ contains a large first-neighbor off-diagonal Γ interaction, possible small J and ${\Gamma }^{{\prime} }$ interactions, and third-neighbor Heisenberg interaction^{15,16,17,18,19,20}. It is known that the KSL is unstable with the presence of these extra interactions^{33,34,35,36,37,38}. Therefore, it calls for a theoretical explanation of the observed phenomenon.

On the theoretical side, to study fermionic spin liquid state, a widely-used method is first employing the Kitaev Majorana-fermion decomposition of spin to construct quadratic mean-field (MF) Hamiltonian, and then rewriting the Hamiltonian back to standard Schwinger-Fermion form. With the general variational parameters of MF Hamiltonian provided by projective symmetry group (PSG) analysis^39,40,41, the corresponding Gutzwiller projected variational wave-function is used to optimize the variational energy by variational Monte Carlo (VMC) method^35,42,43. Such a mathematical procedure, however, may become cumbersome once one wants to study the effect of external magnetic fields in arbitrary orientations, as many relevant terms arise due to the broken symmetries of the Hamiltonian. The VMC method is also limited by the system size and would not be able to find results in the thermodynamic limit. In addition, though the Kitaev Majorana-fermion decomposition works well in the Kitaev model, it suffers from the problem of being gauge non-invariant in the general spin interaction model. As we will show, the generalized Kitaev $J\,\text{-}K\text{-}\Gamma \text{-}\,{\Gamma }^{{\prime} }$ model would unphysically yield the same QSL solution in the mean-field level for both positive and negative parameters, (±J, $\pm K\pm \Gamma \pm {\Gamma }^{{\prime} }$) (as an example, one may consider the J → ∞ limit). Therefore, it also calls for a new theoretical approach to account for the observed phenomenon.

To overcome the above mentioned difficulty encountered in VMC, in this work, we apply the renormalized mean-field theory (RMFT) to find the quantum spin liquid states in the presence of magnetic fields. The renormalized mean-field theory was a reliable method introduced to find correlated electronic phases in the Hubbard model after the high Tc cuprate superconductors were discovered^44,45,46,47. The theory has achieved great success for systems being spin-SU(2) invariant and was originally aimed to replace the rigorous approach based on VMC method by a simplified approximation in which effects of the Gutzwiller projection are approximated by classical weights —namely, by the renormalized factors (such as g_J and g_m defined in in the following sections) for the coupling constants of various operators in the Hamiltonian. It was later generalized to include local order parameters and correlations in the renormalization factor^48,49. Here we shall further generalize RMFT by computing the renormalized factor directly and extend RMFT to systems without spin-SU(2) symmetry. For this purpose, we first employ the gauge invariant decomposition of the spin operator^50,51. Adopting this decomposition, a gauge invariant MF theory that includes quantum averages over all allowed local clusters of operators is derived. By comparing with the exact solution of the Kitaev model, we find that if the coupling constant is renormalized by a renormalized factor g_J, the MF solution can reproduce all results of the ground state that is based on the exact solution. We rigorously prove that g_J is exactly equal to 4 for the Kitaev model. The Kitaev’s exact solution provides an excellent example of the renormalized mean-field theory, in which we have generalized it by replacing the classical weighting factors with factors determined by quantum averages over local clusters of operators. When the interaction J,Γ,${\Gamma }^{{\prime} }$ or magnetic moments are present, the g_J factor needs to be replaced by a bond-dependent tensor ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$, and the magnetic moments ${m}_{i}^{\alpha }$ also acquire an approximated renormalized factor ${g}_{m,i}^{\alpha }$. The resulting RMFT reproduces several results previously obtained by other numerical methods, such as density matrix renormalization group(DMRG) and VMC study of Kitaev model under field^43,52, and in particular, reproduces the 8-,14-,20-cone quantum spin-liquid (QSL) states and the band Chern number under fields in the VMC study of K-Γ model^35,42. Therefore, our RMFT approach provides an economic way in the thermodynamic limit to investigate the MF spinon band properties and the vortices structure in $J\,\text{-}K\text{-}\Gamma \text{-}\,{\Gamma }^{{\prime} }$ model under field in arbitrary orientations. Here, the vortices structure refers to the vortices state generalized from the exact solution of Kitaev model⁴.

Note that a recent review article lists various types of ground states that were predicted by various numerical methods on the K-Γ model⁵³. We would like to point it out that even though these various advanced numerical methods, including DMRG and tensor network approaches, have proposed a number of ground states for K-Γ model, a definitive conclusion still remains elusive. Furthermore, to our knowledge, these methods have not yet been able to reproduce key experimentally observed spin-liquid features under experimental magnetic field orientation, such as chiral edge channels and the six-fold angular dependence of the fermionic gap. While some of the methods capture a spin-liquid ground state^34,53, they inherently struggle to describe such features due to their 2 × 2 spin representation and are limited to system sizes or geometries. For instance, the K-Γ spin-liquid reported in ref. ³⁴, where the Chern number under field is lacking. In addition, they do not report on other characterization of the phases, e.g., excitations, etc. In addition, to describe realistic Kitaev materials, one should adopt the generalized Kitaev J-K-Γ-${\Gamma }^{{\prime} }$ model instead of the K-Γ model compared in ref. ⁵³. Therefore, in the following, we direct our efforts toward identifying the robust spin-liquid phase within the J-K-Γ-${\Gamma }^{{\prime} }$ model under experimental magnetic field orientation, with all parameters chosen within the realistic material range. We then examine the magnetic response of these states to compare with experimental observation^{9,10,11,30,31}. More detailed descriptions of our results are shown in below.

Based on this new RMFT approach, we obtain several new results that were not found before. The main results from our RMFT are as follows: Under zero field and K = −1, Γ ≳ 0.25 there are several new stable QSL states: Z₂ 2-, 4-, 8-, 14-, 20-, 26-cone QSL states, SU(2) 16-, 32-cone QSL states and other high number cones QSL states. Some of these states remain robust in the parameter regime with large Γ(> 0.5) and small $| {\Gamma }^{{\prime} }|$, ∣J∣(< 0.1), which corresponds to the typical range for α-RuCl₃. Since their energies per site are very close to each other with difference ≲ 0.02∣K∣(2 ~ 4 K), we futher examine their low field band structure, Chern numbers, and angle-dependent gap under magnetic fields to compare with the experimental results. Remarkably, we find that only the Z₂ 2-cone state yields the Chern number ν = ±1 at low fields, and at the same time it gives the same azimuthal angle-dependent gap observed by the experiment under magnetic fields. Since this 2-cone state possesses negative Wilson loop value, it is distinct from the Kitaev spin-liquid state. When the applying field is in the $\hat{a}$-$\hat{b}$ plane, this 2-cone state exhibits opposite Chern numbers in comparison to KSL. By further examining the ground state degeneracies, we also find this 2-cone state is a topological QSL with long-range entanglement⁵⁴. Since the Kitaev spin liquid is not stable in the presence of large J, Γ, and ${\Gamma }^{{\prime} }$^{33,34,35,36,37,38}, it implies that what the experiments observed is the 2-cone state obtained in our RMFT. We further illustrate that this 2-cone state can be distinguished from the Kitaev spin-liquid state by looking into the angle-dependent gap in the polar angle (in contrast to the azimuthal angle currently observed in experiment). Finally, we discuss possible mechanism for the suppression of the observed thermal Hall conductivity at low temperatures within the Majorana-fermion framework and conclude.

Results

Theoretical model and Gauge invariant decomposition of spin

We start from the generalized Kitaev $J\,\text{-}K\text{-}\Gamma \text{-}\,{\Gamma }^{{\prime} }$ model on the honeycomb lattice^17,18, which under magnetic field $\vec{h}$, is given by

$$\begin{array}{rcl}H&=&\sum\limits_{\langle ij\rangle \in {\gamma }_{l}}J{\overrightarrow{S}}_{i}\cdot {\overrightarrow{S}}_{j}+K{S}_{i}^{\gamma }{S}_{j}^{\gamma }+\Gamma ({S}_{i}^{\alpha }{S}_{j}^{\beta }+{S}_{i}^{\beta }{S}_{j}^{\alpha })\\ &&+\,{\Gamma }^{{\prime} }({S}_{i}^{\gamma }{S}_{j}^{\alpha }+{S}_{i}^{\gamma }{S}_{j}^{\beta }+{S}_{i}^{\alpha }{S}_{j}^{\gamma }+{S}_{i}^{\beta }{S}_{j}^{\gamma })-\overrightarrow{h}\cdot \sum\limits_{i}{\overrightarrow{S}}_{i}.\end{array}$$

(1)

Here ${\overrightarrow{S}}_{j}$ is not a real spin operator and represents pseudo spin-1/2 operators at site j, γ_l denotes x_l, y_l or z_l bonds shown in Fig. 1 (a) with (α, β, γ) being the cyclic permutation of the indices (x, y, z), K is the strength of the Kitaev spin-spin interaction, J is the isotropic super-exchange interaction, and Γ and ${\Gamma }^{{\prime} }$ are interactions of off-diagonal components of pseudospins. The Hamiltonian can be cast in a more concise form as

$$H=\sum _{\langle ij\rangle ,\alpha {\alpha }^{{\prime} }}{J}_{ij}^{\alpha {\alpha }^{{\prime} }}{\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}-\frac{\overrightarrow{h}}{2}\cdot \sum _{i}{\overrightarrow{\sigma }}_{i},$$

(2)

where all spin-spin interactions are lumped into (including the 1/4 factor) an appropriate tensor ${J}_{ij}^{\alpha {\alpha }^{{\prime} }}$ and we have made use of the relation ${\overrightarrow{S}}_{j}=\frac{1}{2}{\overrightarrow{\sigma }}_{j}$.

Fig. 1: Honeycomb lattice and linking clusters used in RMFT.

a The schematic honeycomb lattice with labeled links, where $\hat{a}=[11\bar{2}]$, $\hat{b}=[\bar{1}10]$, $\hat{c}=[111]$ and all are unit vectors. b Linking cluster used to calculate ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$. c Linking cluster used to calculate ${g}_{m,i}^{\alpha }$.

Now we need to decompose the spin operator into Majorana fermions. In the standard second quantization of the spin operator in terms of fermionic operators, ${f}_{\alpha }^{\dagger }$ and f_α, we have $\overrightarrow{\sigma }={\sum }_{\alpha ,\beta }{f}_{\alpha }^{\dagger }{\overrightarrow{\sigma }}_{\alpha \beta }{f}_{\beta }$, where α and β are indices corresponding to either ↑ or ↓. To faithfully represent the spin, a projection to the Fock space with each site being singly occupied by fermions is implemented through gauge fields so that $\overrightarrow{\sigma }$ and H are gauge invariant⁵⁵. From ${f}_{\alpha }^{\dagger }$ and f_α, one can form four Majorana fermions as ${b}^{x}={f}_{\uparrow }^{\dagger }+{f}_{\uparrow }$, ${b}^{y}=-i{f}_{\uparrow }^{\dagger }+i{f}_{\uparrow }$, ${b}^{z}=-({f}_{\downarrow }^{\dagger }+{f}_{\downarrow })$, ${b}^{0}=i{f}_{\downarrow }^{\dagger }-i{f}_{\downarrow }$. We find that the spin decomposition of Majorana fermions at site j reads^41,51

$${\sigma }_{j}^{\alpha }=\frac{1}{2}(i{b}_{j}^{\alpha }{b}_{j}^{0}-i{b}_{j}^{\beta }{b}_{j}^{\gamma }),$$

(3)

where α, β, and γ are cyclic permutation of x, y, and z. Note that in the Kitaev Majorana-fermion decomposition of spin, the spin operator is expressed as ${\tilde{\sigma }}^{\alpha }=i{b}^{\alpha }{b}^{0}$ with the constraint b^xb^yb^zb⁰ = 1, where α = x, y, or z. For the Kitaev spin-spin interaction H_K, as only the same component of spins are involved for each bond (i, j) of the lattice, the associated Majorana bilinear product ${b}_{i}^{\alpha }{b}_{j}^{\alpha }$ commutes with H_K. This reduces the problem to diagonalization of a quadratic Hamiltonian and the problem is thus exactly solvable⁴. Clearly, in the presence of J, Γ, or ${\Gamma }^{{\prime} }$, the Kitaev decomposition no longer has the advantage and the Hamiltonian is not exactly solvable. Furthermore, since for general spin-spin interaction on bond (i, j), ${\sum }_{\alpha \beta }{J}^{\alpha \beta }{\sigma }_{i}^{\alpha }{\sigma }_{j}^{\beta }={\sum }_{\alpha \beta }{J}^{\alpha \beta }(i{b}_{i}^{\alpha }{b}_{i}^{0})(i{b}_{j}^{\alpha }{b}_{j}^{0})$, hence for − J^αβ, the minus sign can be absorbed into ${b}_{i}^{0}$ by redefining a new Majorana fermion as ${\tilde{b}}_{i}^{0}=-{b}_{i}^{0}$, while maintaining the constraint ${b}_{i}^{x}{b}_{i}^{y}{b}_{i}^{z}{b}_{i}^{0}=1$ at the mean-field level if no moment term presents. The Kitaev decomposition would unphysically yield the same solution in the mean-field level for both positive and negative parameters. In addition, since ${\tilde{\sigma }}^{\alpha }$ can be expressed as σ^α − Q^α with ${Q}_{i}^{\alpha }$ being the pseudospin operator given by ${Q}_{i}^{\alpha }=({f}_{i,\uparrow }^{\dagger },{f}_{i,\downarrow }){\tau }^{\alpha }{({f}_{i,\uparrow },{f}_{i,\downarrow }^{\dagger })}^{T}$, it is clear that ${Q}_{i}^{\alpha }$ rotates covariantly with the pseudospin SU(2) rotations⁵⁵ and hence ${\tilde{\sigma }}^{\alpha }$ is not gauge invariant.

From above, we conclude by using Eq.(3) that the Hamiltonian H is also gauge invariant. It is thus favorable to use Eq.(3) for the spin decomposition. So far, the consideration is for the exact Hamiltonian. In practice, one needs to keep the gauge invariance when approximations are taken. In this case, it should be kept in mind that the average of spin-spin interaction, ${\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}$, is gauge invariant only if all allowed Wick’s decompositions are taken

$$\begin{array}{rcl}\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle &=&\frac{1}{4}\left(-{\chi }_{ij}^{00}{\chi }_{ij}^{\alpha {\alpha }^{{\prime} }}+{\chi }_{ij}^{0{\alpha }^{{\prime} }}{\chi }_{ij}^{\alpha 0}+{\chi }_{ij}^{0{\gamma }^{{\prime} }}{\chi }_{ij}^{\alpha {\beta }^{{\prime} }}-{\chi }_{ij}^{0{\beta }^{{\prime} }}{\chi }_{ij}^{\alpha {\gamma }^{{\prime} }}\right.\\ &&\,+\left.\,{\chi }_{ij}^{\gamma 0}{\chi }_{ij}^{\beta {\alpha }^{{\prime} }}-{\chi }_{ij}^{\gamma {\alpha }^{{\prime} }}{\chi }_{ij}^{\beta 0}-{\chi }_{ij}^{\gamma {\gamma }^{{\prime} }}{\chi }_{ij}^{\beta {\beta }^{{\prime} }} +{\chi }_{ij}^{\gamma {\beta }^{{\prime} }}{\chi }_{ij}^{\beta {\gamma }^{{\prime} }}\right)+{m}_{i}^{\alpha }{m}_{j}^{{\alpha }^{{\prime} }},\end{array}$$

where ${\chi }_{ij}^{\mu {\mu }^{{\prime} }}=\langle i{b}_{i}^{\mu }{b}_{j}^{{\mu }^{{\prime} }}\rangle$, and ${m}_{i}^{\alpha }=\langle {\sigma }_{i}^{\alpha }\rangle$. Based on the gauge-invariant decomposition of spin, we shall derive the gauge-invariant renormalized mean field theory (RMFT), which is done in the section of methods.

Comparison with early numerical works

Before presenting results for the generalized Kitaev model, we shall first compare solutions based on our RMFT with results based on other numerical results for the Kitaev model under field.

In Fig. 2, we compare our results with those based on methods of density matrix renormalization group (DMRG) and variational Monte Carlo (VMC). Clearly, good agreement is obtained for the magnetization curve in Fig. 2(a) and energy in Fig. 2(b) of Kitaev model under field. Furthermore, for some range of Γ ’s, Wang³⁵ found a 14 cone states, and therefore we would like to compare with their results in Fig. 2(c). In particular, our RMFT results also show the same pattern of changing for the topological Chern number under the magnetic field along $\hat{c}$ direction, though the exact boundary of change is different due to the energy comparison between different states. Detailed comparison of the band gap and Chern number is shown in Supplementary Fig. 6. Results in Fig. 2 and Supplementary Fig. 6 indicate that this gauge-invariant RMFT is a reliable method.

Fig. 2: Comparison of RMFT with other numerical methods.

a Comparison between RMFT and DMRG⁵² for calculated spin magnetization of the ferromagnetic or antiferromagnetic Kitaev model under field $\overrightarrow{h}=h\hat{c}$, where $\hat{c}$ is the unit vector along [111] direction. b Energy comparison between RMFT, KMF and VMC⁴³ of the K > 0 (antiferromagnetic) Kitaev model under field $\overrightarrow{h}=h\hat{c}$. The KMF denotes using Kitaev’s decomposition to do MF. c Comparison between RMFT and VMC³⁵ for calculated cone positions of the 14-cone QSL state of K-Γ model when K = −1 and Γ = 0.3. d Comparison between RMFT and VMC³⁵ of the Chern number evolution of 14-cone state under field $\overrightarrow{h}=h\hat{c}$ when K = −1 and Γ = 0.3.

It is worth to mention, if one uses MF based on Kitaev’s decomposition (KMF) in K-Γ model, one would not get listed stable QSL found by VMC or RMFT; only GKSL survives; while if one uses MF based on the gauge invariant decomposition but without including the cluster renormalization effect, for K < 0 Kitaev model, one would get a negative spin susceptibility, which is clearly wrong.

In Fig. 3, we compare our RMFT results with the well-established results of the J-K model⁵⁶. It is evident that RMFT yields relatively poor energy in the magnetically ordered phases. This discrepancy arises from the fact that the RMFT approach employs the representation in which spinons are deconfined in the sense that the spin operator is broken into two unbinded spinons. Physically, however, in a magnetically ordered phase or a polarized phase where large magnetic moments are present, spinons are expected to be confined so that two spinons recombine into the spin operator. As a result, fluctuations of the confinement potential (represented by the introduced Lagrange multipliers) cannot be neglected, and treating them merely at the mean-field level is insufficient. On the other hand, most advanced numerical methods are based on representations in terms of spin (which confines two spinons into the spin operator) and it would be more difficult for these methods to find spin-liquid. Note that in the ordered phase, the classical mean-field solution in which χ_ij vanishes is always a solution to our RMFT theory. However, it does not imply that at zero magnetic field, our RMFT solution coincides with the classical mean field solution. Instead, what we found is that at zero magnetic field, our RMFT solution does not coincide with the classical mean-field solution by the presence of finite χ_ij in the ordered phase. This is because in ordered phase, the presence of finite χ_ij makes the energy of our RMFT solution even lower than that of the classical mean-field solution. Hence even though the classical mean-field solutions in are solutions to our RMFT, they are only local minima and are not the true ground states.

Fig. 3: Comparison of RMFT results for the J-K model⁵⁶ at zero magnetic field, where the couplings are parameterized as $J=\cos \varphi$, $K=\sin \varphi$.

Here “classical” denote the classical mean-field solution in which all χ_ij's are set to zero, “ED” represents exact diagonalization, and “LSWT” refers to linear spin wave theory. Other numerical results presented in ref. ⁵⁶ are close to the ED data and are therefore not shown here.

In Fig. 4, we compare our RMFT results with the DMRG results of the K-Γ model under field³⁴. Noted that the DMRG results were obtained on a two-leg honeycomb strip—i.e., with only two unit cells in one direction—which may lead to the differences in the phase boundaries. An important similarity is that both their study and ours identify a spin-liquid phase in the large Gamma region: their KΓ spin-liquid and our 2-cone spin-liquid(see next section), respectively. Moreover, the Wilson loop value W of their K-Γ spin-liquid is approximately −1/3, while the value of W of our 2-cone spin-liquid is approximately −0.35 (see Table 2).

Fig. 4: Comparison of results for the K-Γ model under applied field.

a DMRG³⁴. b RMFT. Here the couplings are parameterized as K < 0, K² + Γ² = 1, and the magnetic field is applied along the $\hat{c}$ axis ($\overrightarrow{h}=h\hat{c}$). “GKSL” denotes the generalized Kitaev spin-liquid state, and “PS” denotes the polarized state.

QSL states with full symmetries

The symmetry group for the Hamiltonian in zero field and the corresponding projective symmetry group are analyzed in Secs. I and V of the Supplemental Material. Only six irreducible MF parameters are allowed if we require a spin liquid state to possess full symmetries; they are given by

$${\chi }_{{z}_{l}}^{00},{\chi }_{{z}_{l}}^{xx},{\chi }_{{z}_{l}}^{zz},{\chi }_{{z}_{l}}^{0b},{\chi }_{{z}_{l}}^{xy},{\chi }_{{z}_{l}}^{xz}$$

(4)

on z_l-link, where ${\chi }_{{z}_{l}}^{0b}\equiv {\chi }_{{z}_{l}}^{0x}=-{\chi }_{{z}_{l}}^{0y}$.

In refs. ^35,42 they claimed there are two terms (their η_3,5) that couple b⁰ to b^μ≠0. However, one can check that their sum results in a term that does not couple these two sets of operators. Hence, there is only one linear independent term that couples b⁰ to b^μ≠0, as in ours. Note that our number of irreducible parameters (six) is the same as Wang’s PRL work³⁵. Though in their later PRB work⁴² they claim there are seven irreducible parameters, by examining their PRB result, there are only six linear independent terms in it. Therefore, six is the correct number, also as in ours.

To search the stable quantum spin-liquid states in the accepted parameter region of α-RuCl₃ with K = −1, Γ > 0.3,$| J| ,| {\Gamma }^{{\prime} }| < 0.1$, we start from the KSL state with K = −1 and Γ = 0. When a small value of Γ is turned on, the KSL state becomes GKSL (generalized KSL). The GKSL then continuously evolves to one of 20-cone states (denoted as 20₁−) when Γ ≳ 0.25 (details in Sec. VII A of the Supplemental Material). Since the GKSL is not stable when Γ ≳ 0.25, we search for all stable QSL states in the vicinity of transition point in large systems (On a 144 × 144 × 2 lattice, where 144 × 144 corresponds to the momentum resolution, and the factor of 2 accounts for the A and B sublattice points). Note that there are two kinds of PSG setting in MF parameters that give rise to full symmetries in projected spin state. These are the homogeneous setting (all parameters are position independent) and π-flux setting which originated from the vortex-free and full-vortex state in the Kitaev’s exact solution.

For homogeneous setting and with Γ = 0.3, the stable QSL states are given by

$$\begin{array}{rcl}{Z}_{2}&:&2\,\text{-}\,,\,8\,\text{-}\,,\,1{4}_{1\text{-}2}\text{-}\,,\,2{0}_{1\text{-}4}\text{-}\,,\,26\,\text{-}\,{\rm{cone}},\\ SU(2)&:&16\,\text{-}\,,\,32\,\text{-}\,\,{\rm{cone}},\end{array}$$

and for π-flux setting, the stable QSL states are given by

$$\begin{array}{l}{Z}_{2}:\pi \,\text{-}{4}_{1\text{-}2}\text{-}\,,\,,\pi \,\text{-}40\text{-}\,,\,\pi \,\text{-}56\text{-}\,{\rm{cone}},\\ \pi \,\text{-}\,28-{\rm{cone}}\,{\rm{with}}\,2\,{\rm{fermi}}\,{\rm{rings}},\end{array}$$

where Z₂ and SU(2) refers to the invariant gauge group (IGG)³⁹ for the solution, 2, 8, 14. . . denote number of cones in the first Brillouin zone(FBZ), π-4, 40, . . . denote number of cones in the folded reduced Brillouin zone due to π-flux setting, and the subscript labels the different type QSL states with the same number of cones. The IGG determines weak gapless gauge fluctuations of the solution³⁹, where the calculation of IGG of each state is shown in Sec. V B of the Supplemental Material.

By increasing Γ, it is shown that 8-cone state are not stable when Γ is greater than 0.6, 20₄-cone state are not stable when Γ is greater than 0.53, 14₁, 26-cone states are not stable when Γ is greater than 0.43, and 20_1,3-cone, π-28-cone with 2 fermi ring state are not stable when Γ is greater than 0.38.

The other states survive for large Γ up to 1. We further examine the stability of 2-,14₂-,16-,20₂-,20₄-, 32-, π-4_1,2 states when small values of J and ${\Gamma }^{{\prime} }$ are turned on ($| J| ,| {\Gamma }^{{\prime} }| < 0.1$), and found that they are all still stable solutions.

In Fig. 5, we show the lowest fermionic energy spectrum of some of the stable QSL states we found in the homogeneous setting. The ${\chi }_{{z}_{l}}^{\mu \nu }$ of each QSL state is shown in Sec. VII B of the Supplemental Material.

Fig. 5: Lowest fermionic excitation spectrum of different QSL states which survive for large Γ/∣K∣(up to 1).

a 2- b 14₂- c 16- d 20₂- (e)20₄- f 32-cone. Here, a is the lattice constant for the honeycomb lattice, the color scale represents the energy in unit of ∣K∣, and the region enclosed by dash line is the first Brillouin zone.

Comparison of energy obtained by RMFT and Monte-Carlo method

To verify the validity of our RMFT results, we compare our RMFT results with those obtained by using Monte-Carlo (MC) method for projection via the following procedure:

For a given RMFT solution with the RMFT energy E^MC and the corresponding RMFT Hamiltonian H^RMT in Eq. (8), we map H^RMF back to ordinary fermion basis, and obtain the corresponding ground state $\left\vert {\psi }_{0}\right\rangle$. Then we use method of Monte-Carlo to treat the projection P_G and compute ${E}^{{\rm{MC}}}=\frac{\langle {\psi }_{0}| {P}_{G}H{P}_{G}| {\psi }_{0}\rangle }{\langle {\psi }_{0}| {P}_{G}| {\psi }_{0}\rangle }$. Similarly, we can also calculate the Wilson loop value of ψ₀ by using the MC method as ${W}_{{\rm{MC}}}=\frac{\langle {\psi }_{0}| {P}_{G}\hat{W}{P}_{G}| {\psi }_{0}\rangle }{\langle {\psi }_{0}| {P}_{G}| {\psi }_{0}\rangle }$, where $\hat{W}$ is the product of six spin operators through hexagon edges⁴.

The ratio E^MC/E^RMF and W_MC of GKSL with different Γ ’s for lattice size 7 × 7 × 2 are presented in Table 1. The error is about 1% which shows that RMFT works excellently with high precision. In Table 2, we show the ratio E^MC/E^RMF and W_MC of different QSL states with homogeneous setting at Γ = 0.3. Finally, in Fig. 6(a), we show how energies of QSL states change versus Γ by plotting E^RMF relative to energy of 2-cone state versus Γ for different QSL states. The comparison of E^MC and E^RMF of different QSL states with Γ/∣K∣ = 0.3 is shown in Fig. 6 (b). It is clear that for either the RMFT energies or the MC energies, they differ only by ~0.02∣K∣. In Fig. 6 (c), the phase diagram for ${\Gamma }^{{\prime} }=-0.08$ is shown to illustrate the phase transitions between the zigzag ordered state, 2-cone spin liquid state, and the polarized state.

Table 1 Comparison of RMFT and MC results, both of which are calculated on 7 × 7 × 2 lattice

Table 2 Comparison of RMFT and MC results, both of which are calculated on 7 × 7 × 2 lattice with Γ = 0.3

Fig. 6: Energy comparison of different quantum spin liquid (QSL) states and the resulting phase diagram for fixed K = − 1.

a Energy of various QSL states relative to the 2-cone state, plotted as a function of Γ/K, calculated using RMFT. Here ΔE = E − E_2−cone. Note that π − 4₁ state at Γ = 0 corresponds to the full-vortex state in the Kitaev’s exact solution. b Energy comparison of each QSL state computed using both RMFT and Monte Carlo (MC) methods at Γ/∣K∣ = 0.3. c RMFT-calculated phase diagram for ${\Gamma }^{{\prime} }=-0.08$, with a magnetic field applied along the $\hat{a}$ axis ($\overrightarrow{h}=h\hat{a}$). Here, “zigzag” denotes the zigzag magnetically ordered state, and “PS” denotes the polarized state.

Symmetry dictated QSL states with homogeneous setting

The quantum spin liquid phases obtained may contain Dirac cones at various $\overrightarrow{k}$ points. Here we summarize the analysis of positions and shapes for Dirac cones from symmetry point of view. We first note that since the Majorana fermions mix particle and hole equally, any Hamiltonian composed by quadratic Majorana fermion terms possesses the particle-hole symmetry (PHS). Hence, energy eigenstates for particles at $-\overrightarrow{k}$ are exactly the same as the energy eigenstates for anti-particles at $\overrightarrow{k}$ with the eigen-energies satisfying ${E}_{-\overrightarrow{k}}=-{E}_{\overrightarrow{k}}$. Therefore, in the following, we shall focus on properties at $\overrightarrow{k}$ with the understanding that the same properties hold at $-\overrightarrow{k}$ as well.

In Fig. 7(a), we show the momentum ${\overrightarrow{k}}_{D}$ at which the typical found Dirac cones are located for QSL states with homogeneous setting. It is clear that these locations are either at the high symmetry points such as ${\overrightarrow{k}}_{K}$ and ${\vec{k}}_{\Gamma }$ or on high symmetry axes such as ${\vec{k}}_{M;n}^{** }$ and ${\vec{k}}_{M;n}^{**}$ with n = 1, 2, 3. Furthermore, by using the full PSG transformation, it can be proved that the QSL states characterized by Eq.(4) must have zero energy excitation at ${\overrightarrow{k}}_{K}$(see sec. VIII B of the Supplemental Material in details). In addition, we find that when ${\chi }_{{z}_{l}}^{0b}=0$, the zero energy eigenstates are formed by b⁰; while if ${\chi }_{{z}_{l}}^{0b}\,\ne\, 0$, the zero energy eigenstates are formed by the b⁰ and ${b}^{c}\equiv \overrightarrow{b}\cdot \hat{c}=({b}^{x}+{b}^{y}+{b}^{z})/\sqrt{3}$. This reflects the change of the power law of field versus gap when $\hat{h}\parallel \hat{c}$.

Fig. 7: Locations of Dirac cones in QSL states and angle dependence of gap.

a Schematic plot of typical locations of Dirac cones in momentum space for QSL states. b Exact momentum positions of the Dirac cones from 14₂-cone state under zero field, where Γ = 0.3. Note that cones at $-\overrightarrow{k}$ are omitted due to PHS. c Azimuthal angle-dependent minimum gap near ${\overrightarrow{k}}_{K}$. d Azimuthal angle-dependent minimum gap near $\vec{{{k}}^{{*}}}_{\!\!M;1,2,3}$. e Azimuthal angle-dependent minimum gap near $\vec{{{k}}^{{**}}}_{\!\!\!\!M;1,2,3}$. Here $\overrightarrow{h}=h(\cos \varphi \hat{a}+\sin \varphi \hat{b})$ and h = 0.01.

For the Dirac cones at ${\overrightarrow{k}}_{K}$ and ${\overrightarrow{k}}_{\Gamma }$, since they possess the ${C}_{3}^{\hat{c}}$ rotational symmetry with respect to c-axis and the mirror symmetry ${M}^{\hat{b}}$ with respect to the $\hat{a}$-$\hat{c}$ plane, the low energy effective Hamiltonian is given by v_F(q_aτ^a + q_bτ^b), where τ^a,b,c are the x, y, z components of Pauli-matrices formed by two low energy effective states, and $\overrightarrow{q}=\overrightarrow{k}-{\overrightarrow{k}}_{D}$. On the other hand, for the Dirac cones at $\vec{{{k}}^{{\,*}}}_{\!\!M;1}$ or $\vec{{{k}}^{{\,**}}}_{\!\!\!\!M;1}$, the ${C}_{3}^{\hat{c}}$ symmetry is broken but the ${C}_{2}^{\hat{b}}$ symmetry is preserved, hence the low energy effective Hamiltonian is given by ${v}_{F}^{a}{q}_{a}{\tau }^{a}+{v}_{F}^{b}{q}_{b}{\tau }^{b}$ with ${v}_{F}^{a}\,\ne \,{v}_{F}^{b}$. Note that the global ${C}_{3}^{\hat{c}}$ symmetry is still preserved. In this case, $\vec{{{k}}^{{\,*}}}_{\!\!M;n}$ with n = 1, 2, 3 are included as a whole so that under the ${C}_{3}^{\hat{c}}$ rotation, $\vec{{{k}}^{{\,*}}}_{\!\!M;1}$, $\vec{{{k}}^{{\,*}}}_{\!\!M;2}$, and $\vec{{{k}}^{{\,*}}}_{\!\!M;3}$ are cyclically permuted. The low energy Hamiltonians of the Dirac cones at $\vec{{{k}}^{{\,*}}}_{\!\!M;2,3}$ are thus generated by ${C}_{3}^{\hat{c}}$ rotations of the effective Hamiltonian at $\vec{{{k}}^{{\,*}}}_{\!\!M;1}$. For the Dirac cones at other low symmetry points, all q_aτ^a, q_aτ^b, q_bτ^a, and q_bτ^b terms are allowed. For more details, see sec. VIII B of the Supplemental Material.

Symmetry dictated excitation gap-field relations

In this subsection, we summarize the analysis of the shift in the location ${\overrightarrow{k}}_{D}$ of the Dirac cones and the behaviour of fermionic excitation gap under weak magnetic fields $\overrightarrow{h}$. For this purpose, we shall denote ${h}^{a}=\overrightarrow{h}\cdot \hat{a}$, ${h}^{b}=\overrightarrow{h}\cdot \hat{b}$, ${h}^{c}=\overrightarrow{h}\cdot \hat{c}$, and ${\Delta }_{M}(\overrightarrow{k})$ as the fermionic excitation gap at $\overrightarrow{k}$. For more details, see sec. VIII C of the Supplemental Material.

Dirac cones located at ${\overrightarrow{k}}_{K}$ point

For $\overrightarrow{h}\parallel \hat{c}$, the magnetic field does not cause any shift ${\overrightarrow{k}}_{D}={\overrightarrow{k}}_{K}$ and simply opens a fermionic gap ${\Delta }_{M}({\overrightarrow{k}}_{K})$. We find that if ${\chi }_{{z}_{l}}^{0b}=0$, ${\Delta }_{M}({\overrightarrow{k}}_{K})\propto {({h}^{c})}^{3}$, while if ${\chi }_{{z}_{l}}^{0b}\,\ne\, 0$, ${\Delta }_{M}({\overrightarrow{k}}_{K})\propto {h}^{c}$.

For $\overrightarrow{h}\parallel \hat{b}$, ${\overrightarrow{k}}_{D}$ gets shifted along $\hat{a}$ axis by the fields such that $\vec{{{k}}^{{\prime}}}_{\!\!D}\propto {\vec{k}}_{D}+{c}_{1}{({h}^{b})}^{2}\hat{a}$, where c₁ is a constant. However, the fermionic gap is not opened until two Dirac cones merge.

For $\overrightarrow{h}\parallel \hat{a}$, ${\overrightarrow{k}}_{D}$ also gets shifted along $\hat{a}$ axis such that $\vec{{{k}}^{{\prime}}}_{\!\!D}\propto {\vec{k}}_{D}-{c}_{1}{({h}^{a})}^{2}{\hat{a}}$ with c₁ being the same coefficient as in $\overrightarrow{h}\parallel \hat{b}$ case. We found that the fermionic gap follows ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})\propto {({h}^{a})}^{3}$. Note that for the applied weak fields along $\hat{a}$, in solving self-consistent solution in RMFT, we find that additional terms along c-axis such as ${g}_{m,i}^{c}$, and m^c will be generally induced.

By combining results obtained for $\overrightarrow{h}\parallel \hat{a}$ and $\overrightarrow{h}\parallel \hat{b}$ and using ${C}_{3}^{\hat{c}}$ symmetry, we find that for $\overrightarrow{h}=h(\cos \varphi \hat{a}+\sin \varphi \hat{b})$, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})\propto {h}^{3}$, and when φ = π/2, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})=0$. Thus, the azimuthal angle-dependent gap Δ_M(φ) must possess six-fold symmetry with the nodal line along $\hat{b}$.

When both h^a and h^c exist, we find that ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ are composed by ${({h}^{a})}^{3}$, ${({h}^{a})}^{2}{h}^{c}$, ${({h}^{c})}^{2}{h}^{a}$, and ${({h}^{c})}^{3}$. Note that ${({h}^{c})}^{2}{h}^{a}$ is forbidden by symmetry analysis at ${\overrightarrow{k}}_{K}$, but it can be generated by the shifted of Dirac momentum ${\vec{{{k}}^{{\prime}}}_{\!\!D}}-{\vec{k}}_{D}\propto {h}^{a}{h}^{c}$, in which $\vec{{{k}}^{{\prime}}}_{\!\!D}$ breaks the ${C}_{3}^{\hat{c}}$ symmetry.

Dirac cones located at $\vec{{{k}}^{{\,*}}}_{\!\!M;1,2,3}$ or $\vec{{{k}}^{{\,**}}}_{\!\!M;1,2,3}$

Since at $\vec{{{k}}^{{\,*}}}_{\!\!M;1}$ or $\vec{{{k}}^{{\,**}}}_{\!\!M;1}$, the ${C}_{3}^{\hat{c}}$ symmetry is broken but the ${C}_{2}^{\hat{b}}$ symmetry is preserved, there are more h term-dependence allowed for the gap ${\Delta }_{M}(\vec{{{k}}^{{\,\prime}}}_{\!\!D})$. Here we shall focus on ${\vec{k}}_{D}=\vec{{{k}}^{{\,*}}}_{\!\!M;1}$ and find:

For $\overrightarrow{h}\parallel \hat{c}$, the location of the Dirac cone is shifted along $\hat{a}$ axis with ${\vec{{{k}}^{{\prime}}}_{\!\!D}}-{\vec{k}}_{D}\propto {({h}^{c})}^{2}$ and the electronic excitation gap follows as ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})\propto {h}^{c}$.

For $\overrightarrow{h}\parallel \hat{b}$, the location of the Dirac cone is shifted along $\hat{a}$ axis with ${\vec{{{k}}^{{\prime}}}_{\!\!D}}-{\vec{k}}_{D}\propto {({h}^{b})}^{2}$. However, the fermionic gap is not opened until two Dirac cones merge.

For $\overrightarrow{h}\parallel \hat{a}$, the location of the Dirac cone is shifted along $\hat{a}$ axis with ${\vec{{{k}}^{{\prime}}}_{\!\!D}}-{\vec{k}}_{D}\propto {({h}^{a})}^{2}$. We found that the fermionic gap follows ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})\propto {h}^{a}$.

For $\overrightarrow{h}=h(\cos \varphi \hat{a}+\sin \varphi \hat{b})$, we find that ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})\propto h$ and when φ = π/2, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})=0$. Furthermore, due to the broken ${C}_{3}^{\hat{c}}$ symmetry, the azimuthal angle-dependent gap Δ_M(φ) possess two-fold reflection symmetry through $\hat{b}$ with the nodal line being along $\hat{b}$. Note that the global ${C}_{3}^{\hat{c}}$ symmetry is restored if we treat the gap at $\vec{{{k}}^{{*}}}_{\!\!M;2,3}$ as a whole. These three gaps give rise to the six-fold symmetry versus the azimuthal angle.

Dirac cones at low symmetry points

Generally, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ is proportional to h. For individual Dirac cone at low symmetry point, there is no special symmetry pattern in the corresponding azimuthal angle-dependent gap Δ_M(φ). However, if we include all Dirac cones at all ${C}_{3}^{\hat{c}}$ and M symmetry partner points of $\vec{{{k}}^{{\,\prime}}}_{\!\!D}$, the symmetry will restore.

Verification of symmetry dictated gap-field relations in RMFT

To illustrate the symmetry dictated gap-field relations in RMFT, we first choose the 14₂-cone state and examine the angle-dependent gap of each Dirac cone under weak field. As shown in Fig. 7(b–e), when all ${C}_{3}^{\hat{c}}$ and M symmetry partner points to $\vec{{{k}}^{{\prime}}}_{\!\!D}$ are included, the fermionic gaps ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ exhibit the six-fold symmetry pattern, in agreement with the analysis.

To check the gap-field relation, we consider the 8-cone state and the 2-cone state, in which ${\chi }_{{z}_{l}}^{0b}\,\ne\, 0$ for 8-cone state and ${\chi }_{{z}_{l}}^{0b}=0$ for 2-cone state. As shown in Fig. 8(a), for $\overrightarrow{h}\parallel \hat{c}$, we find that ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ from the 2-cone state is proportional to ${({h}^{c})}^{3}$, while for other cone state, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ is proportional to h^c. On the other hand, as shown in Fig. 8(b), for $\overrightarrow{h}\parallel \hat{a}$, we find that the fermionic gap for both states are proportional to ${({h}^{a})}^{3}$, for other cone-states, ${\Delta }_{M}(\vec{{{k}}^{{\prime}}}_{\!\!D})$ is proportional to h^a. These and all other RMFT results are consistent with the symmetry analysis.

Fig. 8: Minimum gap near ${\overrightarrow{k}}_{D}$ from 2-cone and 8-cone states.

a $\overrightarrow{h}\parallel \hat{c}$, b $\overrightarrow{h}\parallel \hat{a}$. Here, Γ = 0.5, h = 0.01 ~ 0.05, h₀ = 0.01, Δ_M,0 is the gap value when h = h₀.

The observed QSL state under magnetic fields in the $\hat{a}$-$\hat{b}$ plane

To explain the experimental observation on the QSL state in the candidate Kitaev material α-RuCl₃^9,10,11, we investigate the band Chern number and fermionic excitation gap for different QSL states under general magnetic fields $\overrightarrow{h}$.

We first note that, for a given magnetic field, different fermionic gaps may open at different $\overrightarrow{k}$ points, but the low-temperature behavior of the specific heat is governed by the smallest gap. For example, from symmetry analysis on a single cone, only the gap of cone at ${\overrightarrow{k}}_{K}$ is proportional to ${({h}^{a})}^{3}$, all other cones’ gaps is proportional to h^a. The experimentally observed gap, however, is well fitted by ${({h}^{a})}^{3}$³⁰. Hence it may seem that multi-cone states are excluded. However, as explained, the low temperature behaviour of specific heat is dominated by the minimum gap in FBZ, hence, if the gap near ${\overrightarrow{k}}_{K}$ is much smaller than gaps of other cones, the specific heat that is proportional to ${({h}^{a})}^{3}$ is still possible. Since in the real material, the strength of Kitaev’s interaction is estimated in 100–200 K^18,57, the half-quantized thermal Hall phase occurs around h/∣K∣ ≈ 0.05–0.1^10,58. We examine gaps of cones from multi-cone QSL states, and find that around this field range the different cones’ gap are in the same order. Hence, multi-cone states are generally excluded (except for π-4_1,2-cone state).

Secondly, we find that when the applied field is along $\hat{a}$ direction, only the 2-cone state has non-trivial topology with Chern number ν = 1. At Γ = 0.3, the ν = 1 phase of the 2-cone state is present in the range h/∣K∣ ≲ 0.1, and for h/∣K∣ > 0.1, the system becomes a polarized state. We have also checked these properties under small ∣J∣ and $| {\Gamma }^{{\prime} }|$ (<0.1), where only minor quantitative differences are observed. For the real material, the upper boundary of half-quantized thermal Hall phase is also around h/∣K∣ ~ 0.1⁵⁹. Furthermore, as shown in Fig. 9(a), the field-azimuthal-angle dependence of the fermionic gap for the 2-cone state also shows good agreement with the experimental observation³⁰. While for other QSL states, the field-azimuthal-angle dependence does not show simple patterns. Therefore we conclude that the system should be in this 2-cone state we found. We further characterize the found 2-cone state by examining its ground state degeneracy (GSD) on a torus. The GSD on a torus refers to the fact that inserting a global Z₂π-flux into either hole of a torus costs no energy in the thermodynamic limit⁵⁴. This procedure is equivalent to changing the boundary condition of the mean-field Hamiltonian from periodic to anti-periodic. Accordingly, we can label the corresponding many-body states as $\left\vert {\psi }_{\pm \pm }\right\rangle$, where the subscripts denote the boundary conditions along the x- and y-directions, respectively. After projecting these four states into the physical spin Hilbert space, the number of linearly independent states determines to the GSD on a torus. In practical computations, we first transform the Majorana fermion mean-field Hamiltonian into an ordinary fermionic basis. We then obtain the Gutzwiller-projected many-body spin states corresponding to the four boundary conditions on the 8 × 8 × 2 lattice and compute the GSD. For the 2-cone state under a weak magnetic field, the GSD on a torus is found to be 3, since $\left\vert {\psi }_{++}\right\rangle$ has odd fermionic parity and thus vanishes after Gutzwiller projection. This result indicates that the 2-cone state is a topologically ordered quantum spin liquid with long-range entanglement^35,54.

Fig. 9: Comparison of GKSL state with the found 2-cone state.

a Fermionic gap of 2-cone state versus orientation of magnetic fields in the $\hat{a}$-$\hat{b}$ plane. b Chern number of GKSL versus orientation of magnetic fields in the $\hat{a}$-$\hat{b}$ plane. c Chern number of 2-cone state versus orientation of magnetic fields in the $\hat{a}$-$\hat{b}$ plane. d Spectrum of GKSL at Γ = 0.16. e Spectrum of 2-cone state at Γ = 0.16.

Because only GKSL and the found 2-cone state hold Chern number ν = ±1, and both states show simple patterns of the field-azimuthal-angle dependence of the gap, we will focus on further comparisons of these two states. As shown in Fig. 9 (e), the found 2-cone state has different lowest spinon spectrum from that of the GKSL 2-cone state analytically continued from the original Kitaev solution shown in Fig. 9(d). Furthermore, as shown in Fig. 9(b) and (c), while the found 2-cone state gives the same angle-dependent energy gap observed by experiments, the Chern number pattern for six sectors of angles is opposite in sign in comparison to those of the original Kitaev spin liquid. In addition, the value of the Wilson loop for the found 2-cone state with Γ = 0.3 is −0.353 in contrast to the positive values of the Wilson loop of GKLS state (see Table 2).

Experimentally, the sign of the Chern number determines the sign of the thermal Hall conductivity κ_xy. Hence when the applied field is along $\hat{a}$, while our theory predicts a negative κ_xy for GKSL, it predicts a positive κ_xy for the found 2-cone state. This difference should serve as a check point in experiments. Nonetheless, the determination of sign for experimentally observed κ_xy is ambiguous. In particular, when the applied field is along $\hat{a}$, some experiments reported negative κ_xy¹⁰, while some experiments reported positive κ_xy^26,27. The difference may result from that either the applied field is not in the correctly aligned in the $+\hat{a}$ direction or when extracting the thermal Hall conductivity from experimental data, some additional minus sign might be included. Hence, the correct sign of the Chern number is an issue to be clarified by future experiments.

Finally, we investigate how gaps close nearby the polarized phase boundary for these two states. As shown in Fig. 10, we find that the fermionic gap for the GKSL state closes at ${\overrightarrow{k}}_{\Gamma }$, while as shown in Fig. 11, the fermionic gap for the found 2-cone state closes at ${\overrightarrow{k}}_{M}$ and ${\overrightarrow{k}}_{Y}$. At the fields beyond those plotted in Figs. 10 and 11, the states are topologically trivial, ν = 0 and with no long range entanglements, so they are just trivial states, similar to the spin-polarized state. Clearly, we find that the critical field for the GKSL state is much smaller than that for the 2-cone state, which can be attributed the fact that the GKSL state is unstable when Γ ≳ 0.25. Therefore, being different from the original Kitaev spin liquid, the 2-cone state predicted by our RMFT theory is a new state emerging for Kitaev materials in external magnetic fields for large Γ.

Fig. 10: Fermionic gap versus magnetic field $\overrightarrow{h}$ nearby ${\overrightarrow{k}}_{K}$ and ${\overrightarrow{k}}_{\Gamma }$ for GKSL state when $\overrightarrow{h}={h}^{a}\hat{a}$.

a, b Γ = 0.16 and the corresponding spinon spectrum near the critical field when ${\Delta }_{M}^{min} \sim 0$ at ${\overrightarrow{k}}_{\Gamma }$. c, d Γ = 0.2 and the corresponding spinon spectrum near the critical field. e, f Γ = 0.24 and the corresponding spinon spectrum near the critical field.

Fig. 11: Fermionic gap versus magnetic field $\overrightarrow{h}$ nearby ${\overrightarrow{k}}_{K}$ and ${\overrightarrow{k}}_{\Gamma }$ for the found 2-cone state when $\overrightarrow{h}={h}^{a}{\hat{a}}$.

a, b = 0.16 and the corresponding spinon spectrum near the critical field when ${\Delta }_{M}^{min} \sim 0$ at ${\overrightarrow{k}}_{M}$ and ${\overrightarrow{k}}_{Y}$. c, d Γ = 0.3 and the corresponding spinon spectrum near the critical field. e, f Γ = 0.5 and the corresponding spinon spectrum near the critical field.

The above analysis suggests that instead of the GKLS state being the state observed in experiments, the found 2-cone state should be the state observed by experiments. To further check and confirm this conclusion, we propose to check the field-direction-dependence of the fermionic excitation gap in polar angles. In Fig. 12(a), we compare the polar-angle θ dependence of fermionic gap for the 2-cone state and the GKLS state. Clearly, the polar-angle dependence for the 2-cone state is more symmetric, while the asymmetric feature of GKSL state is originated from the KSL state. In Fig. 12(b), we show how the polar-angle dependence for the 2-cone state change for large field and different Γs, which can serve as an important check in experiments.

Fig. 12: Comparison of field polar-angle dependence of the fermionic gap.

a the GKSL state and the 2-cone state with Γ = 0.16 and h = 0.02, where the black dot line denotes the schematic fermionic gap of the KSL state, which is proportional to ∣h^xh^yh^z∣ under the field. b 2-cone state with different Γ at h = 0.08.

Discussion

Although many experimentally observed features such as sign-changing of thermal Hall conductivity and the fitted band gap can be naturally explained in terms of quantum spin liquids formed by Majorana fermions^30,31, the key feature, the half-integer thermal Hall conductivity, was only observed in certain magnetic-field and temperature regimes. In particular, excessive thermal Hall conductivity was observed in high temperatures, while the thermal Hall conductivity is suppressed in low temperature regime. Theoretically, while excess thermal Hall conductivity could be attributed to additional channels of contribution from phonons or magnons, the suppression of the thermal Hall conductivity in low temperatures requires an explanation^{25,26,27,28,29}. The phonon-edge channel decoupling mechanism was proposed to explain the non-quantized thermal Hall conductivity at very low temperature^60,61. In this scenario, the measured thermal Hall conductivity can be either larger or smaller than the half-quantized value, depending on the thermal contact properties. However, this mechanism does not provide a quantitative description that can be directly compared with experimental data. In contrast, both experimental groups reported a suppression of the thermal Hall conductivity in the 3–10 K temperature range^25,26. While the decoupling effect might emerge below 3 K, as the measurements become relatively unreliable in this regime. In the fermion-interpretation scenario, the suppression could be due to the opening of a fermionic gap in the edge states. Here we provide an explanation to quantitatively describe the observed thermal Hall conductivity in the 3 ~ 10K temperature range. Indeed, the thermal Hall current is given by I_E = ∫_ϵ(q)≥0n(q)ϵ(q)v(q)dq/2π⁴, where $n(q)=1/(1+{e}^{\epsilon (q)/{k}_{B}T})$ is the Fermi-Dirac distribution function and $v(q)=\frac{d\epsilon (q)}{dq}$ with ϵ(q) being the energy of the quasi-particle in the edge state. If the energy dispersion of the edge state opens a gap Δ and the edge state is cutoff by the bulk state at ϵ = Λ, the finite temperature thermal Hall current is given by

$${I}_{E}=\frac{{k}_{B}^{2}}{2\pi \hslash }\mathop{\int}\nolimits_{\Delta /{k}_{B}T}^{\Lambda /{k}_{B}T}\frac{x}{1+{e}^{x}}dx.$$

(5)

After performing the integration and using ${\kappa }_{xy}^{2D}=\frac{d{I}_{E}}{dT}$, we find that the thermal Hall conductance is given by

$${\kappa }_{xy}^{2D}=T\left(\frac{\pi }{6}\frac{{k}_{B}^{2}}{\hslash }\right)\left(F\left(\frac{\Lambda }{{k}_{B}T}\right)-F\left(\frac{\Delta }{{k}_{B}T}\right)\right),$$

(6)

where $F(x)=\frac{6}{{\pi }^{2}}\left[L{i}_{2}\left.(-{e}^{-x})\right)-x\ln (1+{e}^{-x})-\frac{1}{2}\frac{{x}^{2}}{1+{e}^{x}}\right]$. As shown in Fig. 13 (a),(b) experimental ${\kappa }_{xy}^{exp}/T$ data from two different groups^26,27 both can be well fitted by using Eq.(6) plus T² contribution from phonons^{28,61,62,63,64}, even though the fitted gap is a bit large and the fitted phonon contribution could also be overestimated in current theory suggestion. We speculate therefore that, due to enhanced density of Majorana fermions and additional interactions not included in the present paper such as interlayer coupling, edge states become gapped with thermal Hall conductivity suppressed at low temperatures, while the bulk gap for thermodynamics remains the one described by the model described in the main text. Note that the fitted Λ values exhibit non-monotonic behavior versus magnetic fields, which can understood by examining how the minimum gap at k_K, k_M and k_Y points vary versus magnetic fields in Fig. 11 and Supplementary Fig. 14. Since Λ is determined by how the edge state merges into the bulk bands and are determ,ined by the bulk gap. The non-monotonic behavior originates from the non-monotonic behavior of the bulk gap determined by the minimum gap at k_K, k_M and k_Y of the found 2-cone state.

Fig. 13: Fitting the experimental data by using gapped edge state.

Here we fit the data by using the expression $({\kappa }_{xy}/T)/(\pi {k}_{B}^{2}/6\hslash)=F(\Lambda /{k}_{B}T)-F(\Delta /{k}_{B}T)+\alpha {T}^{2}$. Experimental data taken from a Czajka’s group²⁷, b Bruin’s group²⁶. To exhibit data points and the curves clearly, data points and curves are shifted by 0.1 in y-axis successively for each B field in (a), and are shifted by 0.3 for each B field in (b). The detail of fitting parameter is shown in sec.X of the Supplemental Material.

In conclusion, we develop a gauge-invariant renormalized mean-field theory that is applicable to realistic Kitaev materials and yields reliable phase diagram in thermodynamic limit. In particular, while our RMFT reproduces previous results based on numerical methods, it also predicts several new stable QSL states with energies being very close and differing by about 0.01–0.02∣K∣. Among those stable QSL states, we find that a new exotic 2-cone state can account for the major experimental observations. This 2-cone state holds negative values for the Wilson loop and in the presence of magnetic fields in the $\hat{a}$-$\hat{b}$ plane, it gives rise to the same field-angle-dependent energy gap observed by experiments. However, the Chern number for six sectors of angles that it holds has opposite sign pattern in comparison to those of the original Kitaev spin liquid. The found 2-cone state also displays non-Abelian statistics in weak magnetic fields when gap is opened and thus appears to be the state observed by experiments. We further explore more connections of our results with experiments and speculate that due to enhanced density of Majorana fermions and additional interactions not included, such as interlayer coupling, edge states become gapped so that the thermal Hall conductivity gets suppressed at low temperatures, as observed in experiments. In addition, we show that the polar-angle dependence of fermionic gap can be used to distinguish the found 2-cone state from the KSL state.

Methods

As mentioned in the introduction, the method of the renormalized mean-field theory was a reliable method introduced to find correlated electronic phases in the Hubbard model for high-Tc cuprate superconductors. The theory has achieved great success for systems being spin-SU(2) invariant. Here we shall further generalize RMFT to systems without spin-SU(2) symmetry by combining RMFT with the gauge-invariant decomposition of spin, and derive a gauge-invariant RMFT that goes beyond the usual approach based on the Gutzwiller approximation, using local classical weight factors⁴⁴. The idea is to replace classical weights by quantum averages over local clusters of operators. More precisely, the quantum average is performed in the unprojected Fock space so that the effects of the projection are included in the effective Hamiltonian in unprojected Fock space through renormalized factors. If $\left\vert {\psi }_{0}\right\rangle$ is in the class of wavefunctions in the unprojected Fock space that we look for optimizing the ground state energy, the average of the Hamiltonian $\frac{\langle {\psi }_{0}| {P}_{G}H{P}_{G}| {\psi }_{0}\rangle }{\langle {\psi }_{0}| {P}_{G}| {\psi }_{0}\rangle }$ in the projected Fock space is approximated by

$${E}_{v}^{{\rm{RMF}}}=\sum _{\langle ij\rangle ,\alpha {\alpha }^{{\prime} }}{g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}{J}_{ij}^{\alpha {\alpha }^{{\prime} }}{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}-\sum _{i\alpha }\frac{{h}^{\alpha }}{2}{g}_{m,i}^{\alpha }{m}_{i}^{\alpha }.$$

(7)

Here ${{g}^{{\prime} }}_{O}s$ are renormalized factors defined as the ratio of the average of the corresponding operator O in projected Fock space to the average in unprojected Fock space as 〈O〉/〈O〉₀, where the average in 〈O〉₀ is the average over unprojected state $\left\vert {\psi }_{0}\right\rangle$, while the average in 〈O〉 is the average over projected state ${P}_{G}\left\vert {\psi }_{0}\right\rangle$ with P_G being the Gutzwiller projector given by ${P}_{G}=\mathop{\prod }\nolimits_{j = 1}^{{N}_{s}}\frac{1+{D}_{j}}{2}$ with N_s being number of sites and ${D}_{j}={b}_{j}^{x}{b}_{j}^{y}{b}_{j}^{z}{b}_{j}^{0}=(2{n}_{j\uparrow }-1)(1-2{n}_{j\downarrow })$ ^65,66. The self-consistent RMFT Hamiltonian H^RMF is obtained by the minimization of ${E}_{v}^{{\rm{RMF}}}$⁶⁷ with constraint $\langle {Q}_{i}^{\alpha }\rangle =0$³⁹, given by

$${H}^{{\rm{RMF}}}=\sum _{ij,\mu v}\frac{\partial {E}_{v}}{\partial {\chi }_{ij}^{\mu v}}{\hat{\chi }}_{ij}^{\mu v}+\sum _{i,\alpha }\frac{\partial {E}_{v}}{\partial {m}_{i}^{\alpha }}{\sigma }_{i}^{\alpha }+\sum _{i,\alpha }{\lambda }_{i,Q}^{\alpha }{Q}_{i}^{\alpha },$$

(8)

where ${\chi }_{ij}^{\mu v}={\langle {\hat{\chi }}_{ij}^{\mu v}\rangle }_{0}$, ${\hat{\chi }}_{ij}^{\mu v}=i{b}_{i}^{\mu }{b}_{j}^{v}$, ${m}_{i}^{\alpha }={\langle {\sigma }_{i}^{\alpha }\rangle }_{0}$, and ${\lambda }_{i,Q}^{\alpha }$ are the Lagrangian multipliers that enforce ${\langle {Q}_{i}^{\alpha }\rangle }_{0}=0$. Note that taking $\left\vert {\psi }_{0}\right\rangle$ as the ground-state wavefunction of the mean-field Hamiltonian in the absence of magnetization, it generally has the property that averages of unpaired Majorana Fermion ${\langle {b}_{i}^{\alpha }\rangle }_{0}$ and pairing on the same site ${\langle {b}_{i}^{\mu }{b}_{i}^{\nu }\rangle }_{0}$ vanish.

To find the renormalized g_O factors, we note that using the identity ${D}_{j}{\sigma }_{j}^{\alpha }={\sigma }_{j}^{\alpha }{D}_{j}$, we have ${P}_{G}{\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}$ = ${\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}$, and thus ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ and ${g}_{m,i}^{\alpha }$ factors can be expressed as

$${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}=\frac{{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}\rangle }_{0}}{{\langle {P}_{G}\rangle }_{0}{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}},{g}_{m,i}^{\alpha }=\frac{{\langle {\sigma }_{i}^{\alpha }{P}_{G}\rangle }_{0}}{{\langle {P}_{G}\rangle }_{0}{m}_{i}^{\alpha }},$$

(9)

where ${m}_{i}^{\alpha }={\langle {\sigma }_{i}^{\alpha }\rangle }_{0}$. We note in passing that in general, the renormalized g_O factors are combinations of more fundamental renormalization of the field σ^α and the spin-spin interaction J.

As an illustration, we will first evaluate ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ in the Kitaev model. First, by using the Wick’s theorem, the average of products of D_j on any finite subset of the lattice, $\langle {\psi }_{0}| {\prod }_{{j}_{1},{j}_{2},...}{D}_{{j}_{1}}{D}_{{j}_{2}}...| {\psi }_{0}\rangle$, must contain ${\langle {b}_{i}^{\mu }{b}_{i}^{\nu }\rangle }_{0}$, ${\langle {b}_{i}^{\alpha }\rangle }_{0}$, ${\langle {b}_{i}^{0}{b}_{j}^{\alpha }\rangle }_{0}$, ${\langle {b}_{i}^{\alpha }{b}_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}$($\alpha \ne {\alpha }^{{\prime} }$), and thus they must vanish. Only the average 〈ψ₀∣D∣ψ₀〉 survives and is equal to $\langle {\psi }_{0}| \mathop{\prod }\nolimits_{j}^{{N}_{s}\to \infty }{D}_{j}| {\psi }_{0}\rangle =1$⁶⁶. This results from D is the total fermion parity operators, and thus commutes with Kitaev model under Kitaev’s decomposition, gives D = ±1 for eigenstates and then gives D = 1 for physical spin ground state. Hence ${\langle {P}_{G}\rangle }_{0}=(1+\langle {\psi }_{0}| D| {\psi }_{0}\rangle)/({2}^{{N}_{s}})=1/({2}^{{N}_{s}-1})$. To evaluate ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}\rangle }_{0}$, we focus on projection operators D_i and D_j and the projection operators on complementary lattice points $\bar{{D}_{i}}$ and $\bar{{D}_{j}}$ (complementary lattice points to lattice point i is the collection of whole lattice points that removes lattice point i)so that P_G can be generally rewritten as

$$\begin{array}{rcl}{2}^{{N}_{s}}{P}_{G}&=&1+{D}_{i}+{D}_{j}+{D}_{i}{D}_{j}+\overline{{D}_{i}}+\overline{{D}_{j}}+\overline{{D}_{i}{D}_{j}}+\sum\limits_{\{{i}_{1},{i}_{2},\cdots \,\}}\prod\limits_{i1,i2,...\ne i,j}{D}_{{i}_{1}}{D}_{{i}_{2}}\cdots \\ &=&(1+{D}_{i})(1+{D}_{j})(1+D)+{R}_{G},\end{array}$$

(10)

where ${R}_{G}\equiv {\sum }_{\{{i}_{1},{i}_{2},\cdots \}}{\prod }_{i1,i2,...\ne i,j}{D}_{{i}_{1}}{D}_{{i}_{2}}\cdots \,$ is the remaining products of D_j operators and the second equation follows by using $D{D}_{i}={D}_{i}D=\overline{{D}_{i}}$ and $D{D}_{i}{D}_{j}={D}_{i}{D}_{j}D=\overline{{D}_{i}{D}_{j}}$. For the gauge-invariant decomposition of spin, we have ${\sigma }_{i}^{\alpha }{D}_{i}={\sigma }_{i}^{\alpha }$. Therefore, we obtain

$${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}\rangle }_{0}=\frac{1}{{2}^{{N}_{s}}}{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}({2}^{3}+{R}_{G})\rangle }_{0}.$$

(11)

Clearly, by using the Wick’s theorem to decompose the second term on the right-hand side into linked clusters that contain the factor ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}$, the coefficient to ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}$ contains averages of products of D_j on finite subset of the lattice and thus must vanish. Therefore, we obtain

$${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{P}_{G}\rangle }_{0}=\frac{1}{{2}^{{N}_{s}-3}}\left({\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}+{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{R}_{G}\rangle }_{c}\right),$$

(12)

where ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{R}_{G}\rangle }_{c}$ denotes terms with projectors D_j connected to σ_i or σ_j in the decomposition through the Wick’s theorem. For the Kitaev model, because R_G excludes $\overline{{D}_{i}}$, $\overline{{D}_{j}}$ and $\overline{{D}_{i}{D}_{j}}$, products of D_j that connect to σ_i or σ_j only occupy on finite subset of the lattice and their averages contain ${\langle {b}_{i}^{\mu }{b}_{i}^{\nu }\rangle }_{0}$ or ${\langle {b}_{i}^{\alpha }\rangle }_{0}$, and thus they must vanish. As a result, we find that ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}=4$ for the Kitaev model. The above evaluation of ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ for the Kitaev model can be generalized to evaluate ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ for the generalized Kitaev model in the presence of magnetic field. In this case, ${\langle {b}_{i}^{\mu }{b}_{i}^{\nu }\rangle }_{0}$ is nonzero due to the presence of magnetization but ${\langle {b}_{i}^{\alpha }\rangle }_{0}$ still vanishes. Furthermore, ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}{R}_{G}\rangle }_{c}$ in Eq. (12) no longer vanishes. To take it into account, we include more lattice points in P_G such that in the minimum inclusion, nearest neighboring points shown in Fig. 1 (b) and (c) are included. As a result, after rearranging P_G, Eq. (10) is rewritten as

$$\begin{array}{rcl}{2}^{{N}_{s}}{P}_{G}&=&(1+{D}_{i})(1+{D}_{j})(1+{D}_{{i}_{1}})(1+{D}_{{i}_{2}})(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})(1+D)+{R}_{{G}_{1}},\\ &=&(1+{D}_{i})(1+{D}_{j})(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})(1+{D}_{{j}_{3}})(1+D)+{R}_{{G}_{2}},\end{array}$$

(13)

where the first equation is for the evaluation of ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ and the second equation is for the evaluation of ${g}_{m,i}^{\alpha }$. Hence by using ${\sigma }_{i}^{\alpha }{D}_{i}={\sigma }_{i}^{\alpha }$, ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ and ${g}_{m,i}^{\alpha }$ are given by

$$\displaystyle\begin{array}{rcl}{g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}&=&\displaystyle\frac{4{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}(1+{D}_{{i}_{1}})(1+{D}_{{i}_{2}})(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})\rangle }_{0}}{{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}{\langle (1+{D}_{i})(1+{D}_{j})(1+{D}_{{i}_{1}})(1+{D}_{{i}_{2}})(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})\rangle }_{0}},\\ \displaystyle{g}_{m,i}^{\alpha }&=&\displaystyle\frac{2{\langle {\sigma }_{i}^{\alpha }(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})(1+{D}_{{j}_{3}})\rangle }_{0}}{{\langle {\sigma }_{i}^{\alpha }\rangle }_{0}{\langle (1+{D}_{i})(1+{D}_{{j}_{1}})(1+{D}_{{j}_{2}})(1+{D}_{{j}_{3}})\rangle }_{0}}.\end{array}$$

(14)

By using the Wick’s theorem, ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ and ${g}_{m,i}^{\alpha }$ are evaluated in details in Sec. IV E of the Supplemental Material and are given by

$${g}_{m,i}^{\alpha }=\frac{2}{{m}_{i}^{\alpha }}\times \frac{{\tilde{m}}_{i}^{\alpha }}{{\tilde{\rho }}_{i}},\,\,{g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}=\frac{4}{{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}}\times \frac{{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{{\chi }^{2}}+{\tilde{m}}_{i}^{\alpha }{\tilde{m}}_{j}^{{\alpha }^{{\prime} }}-{m}_{ij}^{dc}}{{\langle {D}_{i}{D}_{j}\rangle }_{\bar{\chi }}+{\tilde{\rho }}_{i}{\tilde{\rho }}_{j}-{\rho }_{ij}^{dc}},$$

(15)

where ${m}_{i}^{\alpha }$ is the bare moment, ${\rho }_{i}=1+{m}_{i}^{2}={\langle 1+{D}_{i}\rangle }_{0}$ is twice of the bare local spin wave-function weight,

$${\tilde{m}}_{i}^{\alpha }={m}_{i}^{\alpha }+\sum _{\langle i,{j}_{n}\rangle }\frac{{\langle {\sigma }_{i}^{\alpha }{D}_{{j}_{n}}\rangle }_{\bar{\chi }}}{{\rho }_{{j}_{n}}}\,\,\,{\rm{and}}\,\,\,{\tilde{\rho }}_{i}={\rho }_{i}+\sum _{\langle i,{j}_{n}\rangle }\frac{{\langle {D}_{i}{D}_{{j}_{n}}\rangle }_{\bar{\chi }}}{{\rho }_{{j}_{n}}}$$

(16)

are the renormalized moment ${\tilde{m}}_{i}^{\alpha }$ and the renormalized local spin wave-function weight ${\tilde{\rho }}_{i}$ by including the correlation from all adjacent sites. ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}={\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{{\chi }^{2}}+{m}_{i}^{\alpha }{m}_{j}^{{\alpha }^{{\prime} }}$, and ${\langle {\sigma }_{i}^{\alpha }{D}_{j}\rangle }_{\bar{\chi }}=2{\sum }_{{\alpha }^{{\prime} }}{m}_{j}^{{\alpha }^{{\prime} }}{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{{\chi }^{2}}$ with

$$\begin{array}{lll}{\langle {D}_{i}{D}_{j}\rangle }_{\bar{\chi }}\,=\,4\mathop{\sum}\limits_{\alpha {\alpha }^{{\prime} }}{m}_{i}^{\alpha }{m}_{j}^{{\alpha }^{{\prime} }}{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{{\chi }^{2}}+\left| \begin{array}{cccc}{\chi }_{ij}^{00}\,{\chi }_{ij}^{0x}\,{\chi }_{ij}^{0y}\,{\chi }_{ij}^{0z}\\ {\chi }_{ij}^{x0}\,{\chi }_{ij}^{xx}\,{\chi }_{ij}^{xy}\,{\chi }_{ij}^{xz}\\ {\chi }_{ij}^{y0}\,{\chi }_{ij}^{yx}\,{\chi }_{ij}^{yy}\,{\chi }_{ij}^{yz}\\ {\chi }_{ij}^{z0}\,{\chi }_{ij}^{zx}\,{\chi }_{ij}^{zy}\,{\chi }_{ij}^{zz}\end{array}\right| ,\\ \,{\rm{and}}\,{\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{{\chi }^{2}}=\displaystyle\frac{1}{4}\left(-{\chi }_{ij}^{00}{\chi }_{ij}^{\alpha {\alpha }^{{\prime} }}+{\chi }_{ij}^{0{\alpha }^{{\prime} }}{\chi }_{ij}^{\alpha 0}+{\chi }_{ij}^{0{\gamma }^{{\prime} }}{\chi }_{ij}^{\alpha {\beta }^{{\prime} }}-{\chi }_{ij}^{0{\beta }^{{\prime} }}{\chi }_{ij}^{\alpha {\gamma }^{{\prime} }}+{\chi }_{ij}^{\gamma 0}{\chi }_{ij}^{\beta {\alpha }^{{\prime} }}\right.\\\qquad\qquad\qquad\quad\left.-{\chi }_{ij}^{\gamma {\alpha }^{{\prime} }}{\chi }_{ij}^{\beta 0}-{\chi }_{ij}^{\gamma {\gamma }^{{\prime} }}{\chi }_{ij}^{\beta {\beta }^{{\prime} }}+{\chi }_{ij}^{\gamma {\beta }^{{\prime} }}{\chi }_{ij}^{\beta {\gamma }^{{\prime} }}\right).\end{array}$$

(17)

The ${m}_{ij}^{dc}$ and ${\rho }_{ij}^{dc}$ denotes the double-counting and high χ order (χ⁶, χ⁸, . . . ) term. Note that the above averages ${\langle {\sigma }_{i}^{\alpha }{\sigma }_{j}^{{\alpha }^{{\prime} }}\rangle }_{0}$, ${\langle {\sigma }_{i}^{\alpha }{D}_{j}\rangle }_{\bar{\chi }}$, and ${\langle {D}_{i}{D}_{j}\rangle }_{\bar{\chi }}$ are all gauge invariant, hence ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ and ${g}_{m,i}^{\alpha }$ are also gauge invariant. In QSL phase under zero magnetic field without spontaneous magnetic ordering, the ${g}_{J,ij}^{\alpha {\alpha }^{{\prime} }}$ factor becomes simpler and is independent of $\alpha {\alpha }^{{\prime} }$ as follows

$${g}_{J,ij}^{SL}=\frac{4}{1+{\langle {D}_{i}{D}_{j}\rangle }_{\bar{\chi }}+{\langle {D}_{i}{D}_{{j}_{1}}\rangle }_{\bar{\chi }}+{\langle {D}_{i}{D}_{{j}_{2}}\rangle }_{\bar{\chi }}+{\langle {D}_{j}{D}_{{i}_{1}}\rangle }_{\bar{\chi }}+{\langle {D}_{j}{D}_{{i}_{2}}\rangle }_{\bar{\chi }}}.$$

(18)

Clearly, it reduces back to 4 for the Kitaev model since all ${\langle {D}_{i}{D}_{j}\rangle }_{\bar{\chi }}$ equal to zero for Kitaev’s exact ground state.