Abstract
The frustrated magnet αRuCl_{3} constitutes a fascinating quantum material platform that harbors the intriguing Kitaev physics. However, a consensus on its intricate spin interactions and fieldinduced quantum phases has not been reached yet. Here we exploit multiple stateoftheart manybody methods and determine the microscopic spin model that quantitatively explains major observations in αRuCl_{3}, including the zigzag order, doublepeak specific heat, magnetic anisotropy, and the characteristic Mstar dynamical spin structure, etc. According to our model simulations, the inplane field drives the system into the polarized phase at about 7 T and a thermal fractionalization occurs at finite temperature, reconciling observations in different experiments. Under outofplane fields, the zigzag order is suppressed at 35 T, above which, and below a polarization field of 100 T level, there emerges a fieldinduced quantum spin liquid. The fractional entropy and algebraic lowtemperature specific heat unveil the nature of a gapless spin liquid, which can be explored in highfield measurements on αRuCl_{3}.
Introduction
The spinorbit magnet αRuCl_{3}, with edgesharing RuCl_{6} octahedra and a nearly perfect honeycomb plane, has been widely believed to be a correlated insulator with the Kitaev interaction^{1,2,3,4,5,6}. The compound αRuCl_{3}, and the Kitaev materials in general, have recently raised great research interest in exploring the inherent Kitaev physics^{7,8,9,10,11}, which can realize nonAbelian anyon with potential applications in topological quantum computations^{12,13}. Due to additional nonKitaev interactions in the material, αRuCl_{3} exhibits a zigzag antiferromagnetic (AF) order at sufficiently low temperature (T_{c} ≃ 7 K)^{2,14,15}, which can be suppressed by an external inplane field of 78 T^{16,17,18}. Surprisingly, the thermodynamics and the unusual excitation continuum observed in the inelastic neutron scattering (INS) measurements suggest the presence of fractional excitations and the proximity of αRuCl_{3} to a quantum spin liquid (QSL) phase^{14,15,19}. Furthermore, experimental probes including the nuclear magnetic resonance (NMR)^{17,20,21}, Raman scattering^{22}, electron spin resonance (ESR)^{23}, THz spectroscopy^{24,25}, and magnetic torque^{26,27}, etc, have been employed to address the possible Kitaev physics in αRuCl_{3} from all conceivable angles. In particular, the unusual (even halfinteger quantized) thermal Hall signal was observed in a certain temperature and field window^{28,29,30,31}, suggesting the emergent Majorana fractional excitations. However, significant open questions remain to be addressed: whether the inplane field in αRuCl_{3} induces a QSL ground state that supports the spinliquid signals in experiment, and furthermore, is there a QSL phase induced by fields along other direction?
To accommodate the QSL states in quantum materials like αRuCl_{3}, realization of magnetic interactions of Kitaev type plays a central role. Therefore, the very first step toward the precise answer to above questions is to pin down an effective lowenergy spin model of αRuCl_{3}. As a matter of fact, people have proposed a number of spin models with various couplings^{19,32,33,34,35,36,37,38,39,40,41,42}, yet even the signs of the couplings are not easy to determine and currently no single model can simultaneously cover the major experimental observations^{43}, leaving a gap between theoretical understanding and experimental observations. In this work, we exploit multiple accurate manybody approaches to tackle this problem, including the exponential tensor renormalization group (XTRG)^{44,45} for thermal states, the density matrix renormalization group (DMRG) and variational Monte Carlo (VMC) for the ground state, and the exact diagonalization (ED) for the spectral properties. Through largescale calculations, we determine an effective KitaevHeisenbergGammaGamma\(^{\prime}\) (KJΓ\({{\Gamma }}^{\prime}\)) model [cf. Eq. (1) below] that can perfectly reproduce the major experimental features in the equilibrium and dynamic measurements.
Specifically, in our KJΓ\({{\Gamma }}^{\prime}\) model the Kitaev interaction K is much greater than other nonKitaev terms and found to play the predominant role in the intermediate temperature regime, showing that αRuCl_{3} is indeed in close proximity to a QSL. As the compound, our model also possesses a lowT zigzag order, which is melted at about 7 K. At intermediate energy scale, a characteristic M star in the dynamical spin structure is unambiguously reproduced. Moreover, we find that inplane magnetic field suppresses the zigzag order at around 7 T, and drives the system into a trivial polarized phase. Nevertheless, even above the partially polarized states, our finitetemperature calculations suggest that αRuCl_{3} could have a fractional liquid regime with exotic Kitaev paramagnetism, reconciling previous experimental debates. We put forward proposals to explore the fractional liquid in αRuCl_{3} via thermodynamic and spinpolarized INS measurements. Remarkably, when the magnetic field is applied perpendicular to the honeycomb plane, we disclose a QSL phase driven by high fields, which sheds new light on the search of QSL in Kitaev materials. Furthermore, we propose experimental probes through magnetization and calorimetry^{46} measurements under 100T class pulsed magnetic fields^{47,48}.
Results
Effective spin model and quantum manybody methods
We study the KJΓ\({{\Gamma }}^{\prime}\) honeycomb model with the interactions constrained within the nearestneighbor sites, i.e.,
where \({{\bf{S}}}_{i}=\{{S}_{i}^{x},{S}_{i}^{y},{S}_{i}^{z}\}\) are the pseudo spin1/2 operators at site i, and 〈i, j〉_{γ} denotes the nearestneighbor pair on the γ bond, with {α, β, γ} being {x, y, z} under a cyclic permutation. K is the Kitaev coupling, Γ and \({{\Gamma }}^{\prime}\) the offdiagonal couplings, and J is the Heisenberg term. The symmetry of the model, besides the lattice translation symmetries, is described by the finite magnetic point group \({D}_{{\rm{3d}}}\times {Z}_{2}^{T}\), where \({Z}_{2}^{T}=\{E,T\}\) is the timereversal symmetry group and each element in D_{3d} stands for a combination of lattice rotation and spin rotation due to the spinorbit coupling. The symmetry group restricts the physical properties of the system. For instance, the Landé g tensor and the magnetic susceptibility tensor, should be uniaxial.
We recall that the \({{\Gamma }}^{\prime}\) term is important for stabilizing the zigzag magnetic order at low temperature in the extended ferromagnetic (FM) Kitaev model with K < 0^{36,49,50}. While the zigzag order can also be induced by the thirdneighbor Heisenberg coupling J_{3}^{32,33,51}, we constrain ourselves within a minimal KJΓ\({{\Gamma }}^{\prime}\) model in the present study and leave the discussion on the J_{3} coupling in the Supplementary Note 1. In the simulations of αRuCl_{3} under magnetic fields, we mainly consider the inplane field along the \([11\bar{2}]\) direction, \({H}_{[11\bar{2}]}\parallel {\bf{a}}\), and the outofplane field along the [111] direction, H_{[111]}∥c^{*}, with the corresponding Landé factors \({g}_{{\rm{ab}}}(={g}_{[11\bar{2}]})\) and \({g}_{{\rm{{c}}^{* }}}(={g}_{[111]})\), respectively. The index [l, m, n] represents the field direction in the spin space depicted in Fig. 1b. Therefore, the Zeeman coupling between field H_{[l, m, n]} to local moments can be written as \({H}_{{\rm{Zeeman}}}={g}_{[l,m,n]}{\mu }_{{\rm{B}}}{\mu }_{0}{H}_{[l,m,n]}{S}_{i}^{[l,m,n]}\), where S^{[l, m, n]} ≡ S ⋅ d_{l,m,n} with S = (S_{x}, S_{y}, S_{z}) and \({{\bf{d}}}_{l,m,n}={(l,m,n)}^{T}/\sqrt{{l}^{2}+{m}^{2}+{n}^{2}}\). The site index i = 1, ⋯ , N, with N ≡ W × L × 2 the total site number.
In the simulations, various quantum manybody calculation methods have been employed (see Methods). The thermodynamic properties under zero and finite magnetic fields are computed by XTRG on finitesize systems (see, e.g, YC4 systems shown in Fig. 1a). The model parameters are pinpointed by fitting the XTRG results to the thermodynamic measurements, and then confirmed by the groundstate magnetization calculations by DMRG with the same geometry and VMC on an 8 × 8 × 2 torus. Moreover, the ED calculations of the dynamical properties are performed on a 24site torus, which are in remarkable agreement to experiments and further strengthen the validity and accuracy of our spin model. Therefore, by combining these cuttingedge manybody approaches, we explain the experimental observations from the determined effective spin Hamiltonian, and explore the fieldinduced QSL in αRuCl_{3} under magnetic fields.
Model parameters
As shown in Fig. 2ab, through simulating the experimental measurements, including the magnetic specific heat and both in and outofplane susceptibility data^{3,14,15,52,53,54,55}, we accurately determine the parameters in the Hamiltonian Eq. (1), which read K = − 25 meV, Γ = 0.3∣K∣, \({{\Gamma }}^{\prime} =0.02\, K\), and J = − 0.1∣K∣. The in and outofplane Landé factors are found to be g_{ab} = 2.5 and \({g}_{{\rm{{c}}^{* }}}=2.3\), respectively. We find that both the magnetic specific heat C_{m} and the two susceptibilities (inplane χ_{ab} and outofplane \({\chi }_{{\rm{{c}}^{* }}}\)) are quite sensitive to the Γ term, and the inclusion of \({{\Gamma }}^{\prime}\)(J) term can significantly change the lowT C_{m}(χ_{ab}) data. Based on these observations, we accurately pinpoint the various couplings. The details of parameter determination, with comparisons to the previously proposed candidate models can be found in Supplementary Notes 1, 2. To check the robustness and uniqueness of the parameter fittings, we have also performed an automatic Hamiltonian searching^{56} with the Bayesian optimization combined largescale thermodynamics solver XTRG, and find that the above effective parameter set indeed locates within the optimal regime of the optimization (Supplementary Note 1). In addition, the validity of our αRuCl_{3} model is firmly supported by directly comparing the model calculations to the measured magnetization curves in Fig. 2c and INS measurements in Fig. 2d–f.
In our KJΓ\({{\Gamma }}^{\prime}\) model of αRuCl_{3}, we see a dominating FM Kitaev interaction and a subleading positive Γ term (Γ > 0), which fulfill the interaction signs proposed from recent experiments^{6,34,39} and agree with some ab initio studies^{9,33,36,37,43,57}. The strong Kitaev interaction seems to play a predominant role at intermediate temperature, which leads to the fractional liquid regime and therefore naturally explains the observed proximate spin liquid behaviors^{14,15,19}.
Magnetic specific heat and twotemperature scales
We now show our simulations of the KJΓ\({{\Gamma }}^{\prime}\) model and compare the results to the thermodynamic measurements. In Fig. 2a, the XTRG results accurately capture the prominent doublepeak feature of the magnetic specific heat C_{m}, i.e., a round highT peak at T_{h} ≃ 100 K and a lowT one at T_{l} ≃ 7 K. As T_{h} ≃ 100 K is a relatively hightemperature scale where the phonon background needs to be carefully deal with^{52}, and there exists quantitative difference among the various C_{m} measurements in the highT regime^{3,15,52}. Nevertheless, the highT scale T_{h} itself is relatively stable, and in Fig. 2a our XTRG result indeed exhibits a highT peak centered at around 100 K, in good agreement with various experiments. Note that the hightemperature crossover at T_{h} corresponds to the establishment of shortrange spin correlations, which can be ascribed to the emergence of itinerant Majorana fermions^{15,58} in the fractional liquid picture that we will discuss. Such a crossover can also be observed in the susceptibilities, which deviate the highT CurieWeiss law and exhibit an intermediateT CurieWeiss scaling below T_{h}^{59}, as shown in Fig. 2b for χ_{ab} (the same for \({\chi }_{{\rm{{c}}^{* }}}\)).
At the temperature T_{l} ≃ 7 K, the experimental C_{m} curves of αRuCl_{3} exhibit a very sharp peak, corresponding to the establishment of a zigzag magnetic order^{2,3,14,15,52}. Such a lowT scale can be accurately reproduced by our model calculations, as shown in Fig. 2a. As our calculations are performed on the cylinders of a finite width, the height of the T_{l} peak is less prominent than experiments, as the transition in the compound αRuCl_{3} may be enhanced by the interlayer couplings. Importantly, the location of T_{l} fits excellently to the experimental results. Below T_{l} our model indeed shows significantly enhanced zigzag spin correlation, which is evidenced by the lowenergy dynamical spin structure in Fig. 2f and the lowT static structure in the inset of Fig. 3a.
Anisotropic susceptibility and magnetization curves
It has been noticed from early experimental studies of αRuCl_{3} that there exists a very strong magnetic anisotropy in the compound^{2,3,6,14,53,54,55}, which was firstly ascribed to anisotropic Landé g factor^{3,55}, and recently to the existence of the offdiagonal Γ interaction^{,6,53,60}. We compute the magnetic susceptibilities along two prominent field directions, i.e., \({H}_{[11\bar{2}]}\) and H_{[111]}, and compare them to experiments in Fig. 2b^{14,53,54}. The discussions on different inplane and tilted fields are left in the Supplementary Note 3.
In Fig. 2b, we show that both the in and outofplane magnetic susceptibilities χ_{ab} and \({\chi }_{{\rm{{c}}^{* }}}\) can be well fitted using our KJΓ\({{\Gamma }}^{\prime}\) model, with dominant Kitaev K, considerable offdiagonal Γ, as well as similar inplane (g_{ab}) and outofplane (\({g}_{{\rm{{c}}^{* }}}\)) Landé factors. Therefore, our manybody simulation results indicate that the anisotropic susceptibilities mainly originate from the offdiagonal Γ coupling (cf. Supplementary Fig. 2), in consistent with the resonant elastic Xray scattering^{6} and susceptibility measurements^{53}. Moreover, with the parameter set of K, Γ, \({{\Gamma }}^{\prime}\), J, g_{ab}, and \({g}_{{\rm{{c}}^{* }}}\) determined from our thermodynamics simulations, we compute the magnetization curves \(M({H}_{[l,m,n]})=1/N\mathop{\sum }\nolimits_{i = 1}^{N}{g}_{[l,m,n]}{\mu }_{{\rm{B}}}\langle {S}_{i}^{[l,m,n]}\rangle\) along the \([11\bar{2}]\) and [111] directions using DMRG, as shown in Fig. 2c. The two simulated curves, showing clear magnetic anisotropy, are in quantitative agreement with the experimental measurements at very low temperature^{3,55}.
Dynamical spin structure and the M star
The INS measurements on αRuCl_{3} revealed iconic dynamical structure features at low and intermediate energies^{14,15}. With the determined αRuCl_{3} model, we compute the dynamical spin structure factors using ED, and compare the results to experiments. First, we show in Fig. 2d the constant kcut at the k = Γ and M points (as indicated in Fig. 2e), where a quantitative agreement between theory and experiment can be observed. In particular, the positions of the intensity peak ω_{Γ} = 2.69 ± 0.11 meV and ω_{M} = 2.2 ± 0.2 meV from the INS measurements^{14}, are accurately reproduced with our determined model. For the Γpoint intensity, the doublepeak structure, which was observed in experimental measurements^{18}, can also be well captured.
We then integrate the INS intensity \({\mathcal{I}}({\bf{k}},\omega )\) over the low and intermediateenergy regime with the atomic form factor taken into account, and check their kdependence in Fig. 2ef. In experiment, a structure factor with bright Γ and M points was observed at low energy, and, on the other hand, a renowned sixpointed star shape (dubbed M star^{43}) was reported at intermediate energies^{14,15}. In Fig. 2ef, these two characteristic dynamical spin structures are reproduced, in exactly the same energy interval as experiments. Specifically, the zigzag order at low temperature is reflected in the bright M points in the Brillouin zone (BZ) when integrated over [2, 3] meV, and the Γ point in the BZ is also turned on. As the energy interval increases to [4.5, 7.5] meV, the M star emerges as the zigzag correlation is weakened while the continuous dispersion near the Γ point remains prominent. The round Γ peak, which also appears in the pure Kitaev model, is consistent with the strong Kitaev term in our αRuCl_{3} model.
Suppressing the zigzag order by inplane fields
In experiments, the lowT zigzag magnetic order has been observed to be suppressed by the inplane magnetic fields above 78 T^{3,17,18,55}. We hereby investigate this fieldinduced effect by computing the spin structure factors under finite fields. The Mpoint peak of the structure factor S(M) in the T\({H}_{[11\bar{2}]}\) plane characterizes the zigzag magnetic order as shown in Fig. 3a. The derivatives \(\frac{dS({\rm{M}})}{dT}{ }_{H = 0}\) and \(\frac{dS({\rm{M}})}{dH}{ }_{T = 1.9{\rm{K}}}\) are calculated in Fig. 3bc, which can only show a round peak at the transition as limited by our finitesize simulation. For H = 0, the turning temperature is at about 7 K, below which the zigzag order builds up; on the other hand, the isothermal MH curves in Fig. 3c suggest a transition point at \({h}_{[11\bar{2}]}^{c}={\mu }_{0}{H}_{[11\bar{2}]}\simeq 7\) T, beyond which the zigzag order is suppressed. Correspondingly, in Fig. 4c [and also in Fig. 4d], the lowtemperature scale T_{l} decreases as the fields increase, initially very slow for small fields and then quickly approaches zero only in the field regime near the critical point, again in very good consistency with experimental measurements^{3,17,20}.
Besides, from the contour plots of C_{m}/T and the isentropes in Fig. 4a, c, one can also recognize the critical temperature and field consistent with the above estimations. Moreover, when the field direction is tilted about 55^{∘} away from the a axis in the ac^{*} plane (i.e., H_{[110]} along the \(c^{\prime}\) axis), as shown in Fig. 4b, d, our model calculations suggest a critical field \({h}_{[110]}^{c}={\mu }_{0}{H}_{[110]}\simeq 10\) T with suppressed zigzag order, in accordance with recent NMR probe^{20}. Overall, the excellent agreements of the finitefield simulations with different experiments further confirm our KJΓ\({{\Gamma }}^{\prime}\) model as an accurate description of the Kitaev material αRuCl_{3}.
FiniteT phase diagram under inplane fields
Despite intensive experimental and theoretical studies, the phase diagram of αRuCl_{3} under inplane fields remains an interesting open question. The thermal Hall^{29}, Raman scattering^{22}, and thermal expansion^{61} measurements suggest the existence of an intermediate QSL phase between the zigzag and polarized phases. On the other hand, the magnetization^{3,55}, INS^{18}, NMR^{17,20,21}, ESR^{23}, Grüneisen parameter^{62}, and magnetic torque measurements^{27} support a singletransition scenario (leaving aside the transition between two zigzag phases due to different interlayer stackings^{63}). Nevertheless, most experiments found signatures of fractional excitations at finite temperature, although an alternative multimagnon interpretation has also been proposed^{23}.
Now with the accurate αRuCl_{3} model and multiple manybody computation approaches, we aim to determine the phase diagram and nature of the fielddriven phase(s). Our main results are summarized in Fig. 1c, e, where a single quantum phase transition (QPT) is observed as the inplane fields \({H}_{[11\bar{2}]}\) increases. Both VMC and DMRG calculations find a trivial polarized phase in the largefield side (\({\mu }_{0}{H}_{[11\bar{2}]}\,> \,{h}_{[11\bar{2}]}^{c}\)), as evidenced by the magnetization curve in Fig. 2c as well as the results in Supplementary Note 3.
Despite the QSL phase is absent under inplane fields, we nevertheless find a Kitaev fractional liquid at finite temperature, whose properties are determined by the fractional excitations of the system. For the pure Kitaev model, it has been established that the itinerant Majorana fermions and Z_{2} fluxes each releases half of the entropy at two crossover temperature scales^{15,58}. Such an intriguing regime is also found robust in the extended Kitaev model with additional nonKitaev couplings^{59}. Now for the realistic αRuCl_{3} model in Eq. (1), we find again the presence of fractional liquid at intermediate T. As shown in Fig. 2b (zero field) and Fig. 3d (finite inplane fields), the intermediateT CurieWeiss susceptibility can be clearly observed, with the fitted Curie constant \(C^{\prime}\) distinct from the highT paramagnetic constant C. This indicates the emergence of a novel correlated paramagnetism—Kitaev paramagnetism—in the material αRuCl_{3}. The fractional liquid constitutes an exotic finitetemperature quantum state with disordered fluxes and itinerant Majorana fermions, driven by the strong Kitaev interaction that dominates the intermediateT regime^{59}.
In Fig. 4cd of the isentropes, we find that the Kitaev fractional liquid regime is rather broad under either inplane (\({H}_{[11\bar{2}]}\)) or tilted (H_{[110]}) fields. When the field is beyond the critical value, the fractional liquid regime gradually gets narrowed, from highT scale T_{h} down to a new lower temperature scale \({T}_{{\rm{l}}}^{\prime}\), below which the fieldinduced uniform magnetization builds up (see Supplementary Note 3). From the specific heat and isentropes in Fig. 4, we find in the polarized phase \({T}_{{\rm{l}}}^{\prime}\) increases linearly as field increases, suggesting that such a lowT scale can be ascribed to the Zeeman energy. At the intermediate temperature, the thermal entropy is around \(({\mathrm{ln}}\,2)/2\) [see Fig. 4cd], indicating that “onehalf” of the spin degree of freedom, mainly associated with the itinerant Majorana fermions, has been gradually frozen below T_{h}.
Besides, we also compute the spin structure factors \({S}^{\gamma \gamma }({\bf{k}})=1/{N}_{{\rm{Bulk}}}\ {\sum }_{i,j\in {\rm{Bulk}}}{e}^{i{\bf{k}}({{\bf{r}}}_{j}{{\bf{r}}}_{i})}\langle {S}_{i}^{\gamma }{S}_{j}^{\gamma }\rangle\) under an inplane field of \({\mu }_{0}{H}_{[11\bar{2}]}=4.2\) T in the fractional liquid regime, where N_{Bulk} is the number of bulk sites (with left and right outmost columns skipped), i, j run over the bulk sites, and γ = x, y, z. Except for the bright spots at Γ and M points, there appears stripy background in Fig. 3eg very similar to that observed in the pure Kitaev model^{59}, which reflects the extremely shortrange and bonddirectional spin correlations there. The stripe rotates as the spin component γ switches, because the γtype spin correlations \({\langle {S}_{i}^{\gamma }{S}_{j}^{\gamma }\rangle }_{\gamma }\) are nonzero only on the nearestneighbor γtype bond. As indicated in the realistic model calculations, we propose such distinct features in S^{γγ}(k) can be observed in the material αRuCl_{3} via the polarized neutron diffusive scatterings.
Signature of Majorana fermions and the Kitaev fractional liquid
It has been highly debated that whether there exists a QSL phase under intermediate inplane fields. Although more recent experiments favor the singletransition scenario^{27,62}, there is indeed signature of fractional Majorana fermions and spin liquid observed in the intermediatefield regime^{18,21,22,27,29}. Based on the model simulations, here we show that our finiteT phase diagram in Fig. 1 provides a consistent scenario that reconciles these different inplane field experiments.
For example, large^{28,64} or even halfquantized thermal Hall conductivity was observed at intermediate fields and between 4 and 6 K^{30,31,65}. However, it has also been reported that the thermal Hall conductivity vanishes rapidly when the field further varies or the temperature lowers below ~2 K^{66}. Therefore, one possible explanation, according to our model calculations, is that the ground state under inplane fields above 7 T is a trivial polarized phase [see Fig. 1c], while the large thermal Hall conductivity at intermediate fields may originate from the Majorana fermion excitations in the finiteT fractional liquid^{58}.
In the intermediateT fractional liquid regime, the Kitaev interaction is predominating and the system resembles a pure Kitaev model under external fields and at a finite temperature. This effect is particularly prominent as the field approaches the intermediate regime, i.e., near the quantum critical point, where the fractional liquid can persist to much lower temperature. Matter of fact, given a fixed low temperature, when the field is too small or too large, the system leaves the fractional liquid regime (cf. Fig. 4) and the signatures of the fractional excitation become blurred, as if there were a finitefield window of “intermediate spin liquid phase”. Such fractional liquid constitutes a Majorana metal state with a Fermi surface^{58,59}, accounting possibly for the observed quantum oscillation in longitudinal thermal transport^{66}. Besides thermal transport, the fractional liquid dominated by fractional excitations can lead to rich experimental ramifications, e.g., the emergent CurieWeiss susceptibilities in susceptibility measurements [see Fig. 3d] and the stripy spin structure background in the spinresolved neutron or resonating Xray scatterings [Fig. 3eg], which can be employed to probe the finiteT fractional liquid in the compound αRuCl_{3}.
Quantum spin liquid induced by outofplane fields
Now we apply the H_{[111]}∥c^{*} field out of the plane and investigate the fieldinduced quantum phases in αRuCl_{3}. As shown in the phase diagram in Fig. 1d, f, under the H_{[111]} fields a fieldinduced QSL phase emerges at intermediate fields between the zigzag and the polarized phases, confirmed in both the thermal and groundstate calculations. The existence of two QPTs and an intermediate phase can also be seen from the color maps of C_{m}, Z_{2} fluxes, and thermal entropies shown in Figs. 5ab and 6a.
To accurately nail down the two QPTs, we plot the c^{*} DMRG magnetization curve M(H_{[111]}) and the derivative dM/dH_{[111]} in Fig. 5c, together with the ED energy spectra in Fig. 5d, from which the groundstate phase diagram can be determined (cf. Fig. 1f). In particular, the lower transition field, \({h}_{[111]}^{c1}\simeq 35\) T, estimated from both XTRG and DMRG, is in excellent agreement with recent experiment through measuring the magnetotropic coefficient^{27}. The existence of the upper critical field \({h}_{[111]}^{c2}\) at 100 T level can also be probed with current pulsed high field techniques^{48}.
Correspondingly, we find in Fig. 5b that the Z_{2} flux \({W}_{{\rm{p}}}={2}^{6}\langle {S}_{i}^{x}{S}_{j}^{y}{S}_{k}^{z}{S}_{l}^{x}{S}_{m}^{y}{S}_{n}^{z}\rangle\) (where i, j, k, l, m, n denote the six vertices of a hexagon p) changes its sign from negative to positive at \({h}_{[111]}^{c1}\), then to virtually zero at \({h}_{[111]}^{c2}\), and finally converge to very small (positive) values in the polarized phase. These observations of flux signs in different phases are consistent with recent DMRG and tensor network studies on a KΓ\({{\Gamma }}^{\prime}\) model^{49,50}, and it is noteworthy that the flux is no longer a strictly conserved quantity as in the pure Kitaev model, the lowT expectation values of ∣W_{p}∣ thus would be very close to 1 only deep in the Kitaev spin liquid phase^{49,59,67}.
From the C_{m} color map in Fig. 5a and C_{m} curves in Fig. 6c, we find doublepeaked specific heat curves in the QSL phase, which clearly indicate the two temperature scales, e.g., T_{h} ≃ 105 K and \({T}_{{\rm{l}}}^{^{\prime\prime} }\simeq 10\) K for μ_{0}H_{[111]} = 45.5 T. They correspond to the establishment of spin correlations at T_{h} and the alignment of Z_{2} fluxes at \({T}_{{\rm{l}}}^{^{\prime\prime} }\), respectively, as shown in Fig. 5f. As a result, in Fig. 6ab the system releases \(({\mathrm{ln}}\,2)/2\) thermal entropy around T_{h}, and the rest half is released at around \({T}_{{\rm{l}}}^{^{\prime\prime} }\). The magnetic susceptibility curves in Fig. 5e fall into an intermediateT CurieWeiss behavior below T_{h} when the spin correlations are established, and deviate such emergent universal behavior when approaching \({T}_{{\rm{l}}}^{^{\prime\prime} }\) as the gauge degree of freedom (flux) gradually freezes.
Furthermore, we study the properties of the QSL phase. We find very peculiar spin correlations as evidenced by the (modified) structure factor
As shown in the inset of Fig. 5b, there appears no prominent peak in \({\tilde{S}}^{_{zz}}({\bf{k}})\) at both finite and zero temperatures, for a typical field of 45.5 T in the QSL phase. As shown in Fig. 5f, we find the dominating nearestneighboring correlations at T > 0 are bonddirectional and the longerrange correlations are rather weak, same for the QSL ground state at T = 0. This is in sharp contrast to the spin correlations in the zigzag phase. More spin structure results computed with both XTRG and DMRG can be found in Supplementary Note 4. Below the temperature \({T}_{{\rm{l}}}^{^{\prime\prime} }\), we observe an algebraic specific heat behavior as shown in Fig. 6cd, which strongly suggests a gapless QSL. We remark that the gapless spin liquid in the pure Kitaev model also has bonddirectional and extremely shortrange spin correlations^{13}. The similar features of spin correlations in this QSL state may be owing to the predominant Kitaev interaction in our model.
Overall, under the intermediate fields between \({h}_{[111]}^{c1}\) and \({h}_{[111]}^{c2}\), and below the lowtemperature scale \({T}_{{\rm{l}}}^{^{\prime\prime} }\), our model calculations predict the presence of the longsought QSL phase in the compound αRuCl_{3}.
Discussion
First, we discuss the nature of the QSL driven by the outofplane fields. In Fig. 5d, the ED calculation suggests a gapless spectrum in the intermediate phase, and the DMRG simulations on long cylinders find the logarithmic correction in the entanglement entropy scaling (see Supplementary Note 4), which further supports a gapless QSL phase. On the other hand, the VMC calculations identify the intermediate phase as an Abelian chiral spin liquid, which is topologically nontrivial with a quantized Chern number ν = 2. Overall, various approaches consistently find the same scenario of two QPTs and an intermediatefield QSL phase under high magnetic fields [cf. Fig. 1d, f]. It is worth noticing that a similar scenario of a gapless QSL phase induced by outofplane fields has also been revealed in a KitaevHeisenberg model with K > 0 and J < 0^{68}, where the intermediate QSL was found to be smoothly connected to the fieldinduced spin liquid in a pure AF Kitaev model^{67,69,70}. We note the extended AF Kitaev model in Ref. ^{68} can be transformed into a KJΓ\({{\Gamma }}^{\prime}\) model with FM Kitaev term (and other nonKitaev terms still distinct from our model) through a global spin rotation^{71}. Therefore, despite the rather different spin structure factors of the fieldinduced QSL in the AF Kitaev model from ours, it is interesting to explore the possible connections between the two in the future.
Based on our αRuCl_{3} model and precise manybody calculations, we offer concrete experimental proposals for detecting the intermediate QSL phase via the magnetothermodynamic measurements under high magnetic fields. The two QPTs are within the scope of contemporary technique of pulsed high fields, and can be confirmed by measuring the magnetization curves^{47,48}. The specific heat measurements can also be employed to confirm the twotransition scenario and the highfield gapless QSL states. As field increases, the lower temperature scale T_{l} first decreases to zero as the zigzag order is suppressed, and then rises up again (i.e., \({T}_{{\rm{l}}}^{^{\prime\prime} }\)) in the QSL phase [cf. Fig. 5a]. As the specific heat exhibits a doublepeak structure in the highfield QSL regime, the thermal entropy correspondingly undergoes a twostep release in the QSL phase and exhibits a quasiplateau near the fractional entropy \(({\mathrm{ln}}\,2)/2\) [cf. 45.5 T and 91 T lines in Fig. 6b]. This, together with the lowT (below \({T}_{{\rm{l}}}^{^{\prime\prime} }\)) algebraic specific heat behavior reflecting the gapless excitations, can be probed through highfield calorimetry^{46}.
Lastly, it is interesting to note that the emergent highfield QSL under outofplane fields may be closely related to the offdiagonal Γ term (see, e.g., Refs. ^{49,50}) in the compound αRuCl_{3}. The Γ term has relatively small influences in ab plane, while introduces strong effects along the c^{*} axis — from which the magnetic anisotropy in αRuCl_{3} mainly originates. The zigzag order can be suppressed by relatively small inplane fields and the system enters the polarized phase, as the Γ term does not provide a strong “protection” of both zigzag and QSL phases under inplane fields (recall the QSL phase in pure FM Kitaev model is fragile under external fields^{59,67,72}). The situation is very different for outofplane fields, where the Kitaev (and also Γ) interactions survive the QSL after the zigzag order is suppressed by high fields. Intuitively, the emergence of this QSL phase can therefore be ascribed to the strong competition between the Γ interaction and magnetic field along the hard axis c^{*} of αRuCl_{3}. With such insight, we expect a smaller critical field for compounds with a less significant Γ interaction. In the fastmoving Kitaev materials studies, such compounds with relatively weaker magnetic anisotropy, e.g., the recent Na_{2}Co_{2}TeO_{6} and Na_{2}Co_{2}SbO_{6}^{73,74,75,76}, have been found, which may also host QSL induced by outofplane fields at lower field strengths.
Methods
Exponential tensor renormalization group
The thermodynamic quantities including the specific heat, magnetic susceptibility, Z_{2} flux, and the spin correlations can be computed with the exponential tensor renormalization group (XTRG) method^{44,45} on the Ytype cylinders with width W = 4 and length up to L = 6 (i.e., YC4 × 6 × 2). We retain up to D = 800 states in the XTRG calculations, with truncation errors ϵ ≲ 2 × 10^{−5}, which guarantees a high accuracy of computed thermal data down to the lowest temperature T ≃ 1.9 K. Note the truncation errors in XTRG, different from that in DMRG, directly reflects the relative errors in the free energy and other thermodynamics quantities. The lowT data are shown to approach the T = 0 DMRG results (see Fig. 5f). In the thermodynamics simulations of the αRuCl_{3} model, one needs to cover a rather wide range of temperatures as the high and lowT scales are different by more than one order of magnitude (100 K vs. 7 K in αRuCl_{3} under zero field). In the XTRG cooling procedure, we represent the initial density matrix ρ_{0}(τ) at a very high temperature T ≡ 1/τ (with τ ≪ 1) as a matrix product operator (MPO), and the series of lower temperature density matrices ρ_{n}(2^{n}τ) (n ≥ 1) are obtained by successively multiplying and compressing ρ_{n} = ρ_{n−1} ⋅ ρ_{n−1} via the tensornetwork techniques. Thus XTRG is very suitable to deal with such Kitaev model problems, as it cools down the system exponentially fast in temperature^{59}.
Density matrix renormalization group
The ground state properties are computed by the density matrix renormalization group (DMRG) method, which can be considered as a variational algorithm based on the matrix product state (MPS) ansatz. We keep up to D = 2048 states to reduce the truncation errors ϵ ≲ 1 × 10^{−8} with a very good convergence. The simulations are based on the highperformance MPS algorithm library GraceQ/MPS2^{77}.
Variational Monte Carlo
The ground state of αRuCl_{3} model are evaluated by the variational Monte Carlo (VMC) method based on the fermionic spinon representation. The spin operators are written in quadratic forms of fermionic spinons \({S}_{i}^{m}=\frac{1}{2}{C}_{i}^{\dagger }{\sigma }^{m}{C}_{i},m=x,y,z\) under the local constraint \({\hat{N}}_{i}={c}_{i\uparrow }^{\dagger }{c}_{i\uparrow }+{c}_{i\downarrow }^{\dagger }{c}_{i\downarrow }=1\), where \({C}_{i}^{\dagger }=({c}_{i\uparrow }^{\dagger },{c}_{i\downarrow }^{\dagger })\) and σ^{m} are Pauli matrices. Through this mapping, the spin interactions are expressed in terms of fermionic operators and are further decoupled into a noninteracting meanfield Hamiltonian H_{mf}(R), where R denotes a set of parameters (see Supplementary Note 5). Then we perform Gutzwiller projection to the meanfield ground state \(\left{{{\Phi }}}_{{\rm{mf}}}({\boldsymbol{R}})\right\rangle\) to enforce the particle number constraint (\({\hat{N}}_{i}=1\)). The projected states \(\left{{\Psi }}({\boldsymbol{R}})\right\rangle ={P}_{G}\left{{{\Phi }}}_{{\rm{mf}}}({\boldsymbol{R}})\right\rangle ={\sum }_{\alpha }f(\alpha )\left\alpha \right\rangle\) (here α stands for the Ising bases in the manybody Hilbert space, same for β and γ below) provide a series of trial wave functions, depending on the specific choice of the meanfield Hamiltonian H_{mf}(R). Owing to the huge size of the manybody Hilbert space, the energy of the trial state \(E({\boldsymbol{R}})=\left\langle {{\Psi }}({\boldsymbol{R}})\rightH\left{{\Psi }}({\boldsymbol{R}})\right\rangle /\langle {{\Psi }}({\boldsymbol{R}}) {{\Psi }}({\boldsymbol{R}})\rangle ={\sum }_{\alpha }\frac{ f(\alpha ){ }^{2}}{{\sum }_{\gamma } f(\gamma ){ }^{2}}\left(\right.{\sum }_{\beta }\langle \beta  H \alpha \rangle \frac{f{(\beta )}^{* }}{f{(\alpha )}^{* }}\left)\right.\) is computed using Monte Carlo sampling. The optimal parameters R are determined by minimizing the energy E(R). While the VMC calculations are performed on a relatively small size (up to 128 sites), once the optimal parameters are determined we can plot the spinon dispersion of a QSL state by diagonalizing the meanfield Hamiltonian on a larger lattice size, e.g., 120 × 120 unit cells in practice.
Exact diagonalization
The 24site exact diagonalization (ED) is employed to compute the zerotemperature dynamical correlations and energy spectra. The clusters with periodic boundary conditions are depicted in the inset of Fig. 2d and the Supplementary Information, and the αRuCl_{3} model under inplane fields (\({H}_{[11\bar{2}]}\parallel {\bf{a}}\) and \({H}_{[1\bar{1}0]}\parallel {\bf{b}}\)) as well as outofplane fields H_{[111]}∥c^{*} have been calculated, as shown in Fig. 5 and Supplementary Note 3. Regarding the dynamical results—the neutron scattering intensity—is defined as
where f(k) is the atomic form factor of Ru^{3+}, which can be fitted by an analytical function as reported in Ref. ^{78}. \({S}^{\mu \nu }({\bf{k}},\omega )={\sum }_{i,j}\langle {S}_{i}^{\mu }(t){S}_{j}^{\nu }(0)\rangle {e}^{i{\bf{k}}\cdot ({{\bf{r}}}_{j}{{\bf{r}}}_{i})}{e}^{i\omega t}\) is the dynamical spin structure factor, which can be expressed by the continued fraction expansion in the tridiagonal basis of the Hamiltonian using Lanczos iterative method. For the diagonal part,
where z = ω + E_{0} + iη, E_{0} is the ground state energy, \(\left{\psi }_{0}\right\rangle\) is the ground state wave function, η is the Lorentzian broadening factor (here we take η = 0.5 meV in the calculations, i.e., 0.02 times the Kitaev interaction strength ∣K∣ = 25 meV), and a_{i} (b_{i+1}) is the diagonal (subdiagonal) matrix element of the tridiagonal Hamiltonian. On the other hand, for the offdiagonal part, we define a Hermitian operator \({\hat{S}}_{{\bf{k}}}^{\mu }+{\hat{S}}_{{\bf{k}}}^{\nu }\) to do the continued fraction expansion,
Then the offdiagonal S^{μν}(k, ω) can be computed by S^{μν}(k, ω) + S^{νμ}(k, ω) = S^{(μ+ν) (μ+ν)}(k, ω) − S^{μμ}(k, ω) − S^{νν}(k, ω). Following the INS experiments^{14}, the shown scattering intensities in Fig. 2 are integrated over perpendicular momenta k_{z} ∈ [ − 5π, 5π], assuming perfect twodimensionality of αRuCl_{3} in the ED calculations.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
All numerical codes in this paper are available upon request to the authors.
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Acknowledgements
We are indebted to Yang Qi, XuGuang Zhou, Yasuhiro Matsuda, Wentao Jin, Weiqiang Yu, Jinsheng Wen, Kejing Ran, Yanyan Shangguan and Hong Yao for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11834014, 11974036, 11974421, 11804401), Ministry of Science and Technology of China (Grant No. 2016YFA0300504), and the Fundamental Research Funds for the Central Universities (BeihangUZG216S2113, SYSU2021qntd27). We thank the HighPerformance Computing Cluster of Institute of Theoretical PhysicsCAS for the technical support and generous allocation of CPU time.
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W.L., S.S.G. and Z.X.L initiated this work. H.L. and D.W.Q. performed the thermal tensor network calculations, H.L. obtained the Hamiltonian parameters by fitting experiments, Y.G. confirmed the model parameters with Bayesian searching, H.K.Z. (DMRG) and J.C.W. (VMC) performed the ground state simulations, and H.Q.W. computed the dynamical correlations by ED. All authors contributed to the analysis of the results and the preparation of the draft. S.S.G., Z.X.L., and W.L. supervised the project.
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Li, H., Zhang, HK., Wang, J. et al. Identification of magnetic interactions and highfield quantum spin liquid in αRuCl_{3}. Nat Commun 12, 4007 (2021). https://doi.org/10.1038/s41467021242578
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DOI: https://doi.org/10.1038/s41467021242578
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