Abstract
Pursuing the exotic quantum spin liquid (QSL) state in the Kitaev material αRuCl_{3} has intrigued great research interest recently. A fascinating question is on the possible existence of a fieldinduced QSL phase in this compound. Here we perform highfield magnetization measurements of αRuCl_{3} up to 102 T employing the nondestructive and destructive pulsed magnets. Under the outofplane field along the c* axis (i.e., perpendicular to the honeycomb plane), two quantum phase transitions are uncovered at respectively 35 T and about 83 T, between which lies an intermediate phase as the predicted QSL. This is in sharp contrast to the case with inplane fields, where a single transition is found at around 7 T and the intermediate QSL phase is absent instead. By measuring the magnetization data with fields tilted from the c* axis up to 90° (i.e., inplane direction), we obtain the fieldangle phase diagram that contains the zigzag, paramagnetic, and QSL phases. Based on the KJΓ\({{{\Gamma }}}^{{\prime} }\) model for αRuCl_{3} with a large Kitaev term we perform density matrix renormalization group simulations and reproduce the quantum phase diagram in excellent agreement with experiments.
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Introduction
Quantum spin liquid (QSL) constitutes a topological state of matter in frustrated magnets, where the constituent spins remain disordered even down to absolute zero temperature and share longrange quantum entanglement^{1,2,3,4}. Due to the lack of rigorous QSL ground states, such ultra quantum spin states are less wellunderstood in systems in more than one spatial dimension before Alexei Kitaev introduced the renowned honeycomb model with bonddependent exchange^{5}. The ground state of the Kitaev honeycomb model is proven to be a QSL with two types of fractional excitations^{5,6}. Soon after, the Kitaev model was proposed to be materialized in the J_{eff} = 1/2 Mott insulating magnets^{7,8,9,10,11} such as A_{2}IrO_{3} (A = Li and Na)^{12,13}, αRuCl_{3}^{14,15}, etc.
Recently, the 4d spinorbit magnet αRuCl_{3} has been widely accepted as a prime candidate for Kitaev material^{16,17,18,19,20,21,22}. As initially proposed from the firstprinciple analysis^{14,15,23,24,25}, the compound is now believed to be described by the KJΓ\({{{\Gamma }}}^{{\prime} }\) effective model that includes the Heisenberg J_{(1, 3)}, Kitaev exchange K, and the symmetric offdiagonal exchange \({{{\Gamma }}}^{({\prime} )}\) terms. The Kitaev interaction originates from chlorinemediated exchange through edgeshared octahedra arranged on a honeycomb lattice. Similar to the intensively studied honeycomb and hyperhoneycomb iridates^{26}, additional nonKitaev terms \({{{\Gamma }}}^{({\prime} )}\) and/or J_{3}, unfortunately, stabilize a zigzag antiferromagnetic order below T_{N} ≈ 7 K in the compound^{17,18,20,27}. Given that, a natural approach to realizing the Kitaev QSL is to suppress the zigzag order by applying magnetic fields to the compound^{28,29,30,31,32,33,34,35,36,37,38,39,40,41,42}. As shown in certain experiments, a moderate inplane field (about 7 T) can suppress the zigzag order and may induce an intermediate QSL phase before the polarized phase^{34,35,39,40,41}. However, there are also experimental pieces of evidence from, e.g., magnetization^{18,27}, magnetocaloric^{43}, magnetotorque measurements^{44}, etc., that indicate a single transition scenario with no intermediate phase present. Some angledependent experiments, on the other hand, demonstrate the presence of an additional intermediate phase, which however is, due to another zigzag antiferromagnetic order induced by sixlayer periodicity along the outofplane direction^{45}. This leaves an intriguing question to be resolved in the compound αRuCl_{3}.
Theoretical progress lately suggests the absence of intermediate QSL under inplane fields, while predicting the presence of an intermediate phase by switching the magnetic fields from inplane to the much less explored outofplane direction. The numerical calculations^{46,47,48,49,50} of the KJΓ\({{{\Gamma }}}^{{\prime} }\) spin model show that the offdiagonal exchanges \({{{\Gamma }}}^{({\prime} )}\) terms dominate the magnetic anisotropy in the compound. Due to the strong magnetic anisotropy in αRuCl_{3}, the critical field increases dramatically from the inplane to the outofplane direction. The authors in ref. ^{47} further point out an interesting twotransition scenario with a fieldinduced intermediate QSL phase, which is later confirmed by other theoretical calculations^{49}, except for subtlety in lattice rotational symmetry breaking (such a socalled nematic order is, however, not directly relevant to our experimental discussion here as the realistic compound αRuCl_{3} does not strictly have a C_{3} symmetry^{15,23,30}). More recently, H. Li et al. proposed a large Kitaevterm spin Hamiltonian^{51} also based on the KJΓ\({{{\Gamma }}}^{{\prime} }\) model. With the precise model parameters determined from fitting the experimental thermodynamics data, they theoretically reproduced the suppression of zigzag order under the 7T inplane field, and find a gapless QSL phase located between two outofplane transition fields that are about 35 T and of 100T class, respectively. Therefore, the previously unsettled debates on the fieldinduced transitions and the concrete theoretical proposal of the intermediate QSL phase strongly motivate a highfield experimental investigation on αRuCl_{3} along the outofplane direction and up to 100 T.
In this work, we report the magnetization (M) process of αRuCl_{3} by applying magnetic fields (H) in various directions within the honeycomb plane and along the c* axis (outofplane) up to 100 T, and find clear experimental evidence supporting the twotransition scenario. Here, the c* axis is the axis perpendicular to the honeycomb plane^{27}. Under fields applied along and close to the c* axis, an intermediate phase is found bounded by two transition fields \({H}_{c}^{l}\) and \({H}_{c}^{h}\). In particular, besides the previously reported \({H}_{c}^{l}\simeq 32.5\) T^{44,52}, remarkably we find a second phase transition at a higher field \({H}_{c}^{h}\simeq 83\) T. Below \({H}_{c}^{h}\) and above \({H}_{c}^{l}\) there exists an intermediate phase — the predicted fieldinduced QSL phase^{47,51}. When the field tilts an angle from the c* axis by 9°, only the transition field H_{c} is observed, indicating the intermediate QSL phase disappears. Accordingly, we also perform the densitymatrix renormalization group (DMRG) calculations based on the previously proposed KJΓ\({{{\Gamma }}}^{{\prime} }\) model of αRuCl_{3}, and find the simulated phase transitions and extended QSL phase are in agreement with experiments. Therefore, we propose a complete fieldangle phase diagram and provide the experimental evidence for the fieldinduced QSL phase in the prominent Kitaev compound αRuCl_{3}.
Results
Experimental results
Figure 1a–c shows the magnetization process and the magnetic field dependence of dM/dH along the c* (outofplane) direction. The magnetization data represented by the dash lines (0 T to 30 T) are very noisy because of the huge switching electromagnetic noise inevitably generated for injection megaampere driving currents at the beginning of the destructive ultrahigh field generation^{53}. The magnetization process and dM/dH are precisely measured from 30 to 95 T, which shows two peaks labeled by \({H}_{c}^{l}\) and \({H}_{c}^{h}\). To be specific, we have conducted three independent measurements (i), (ii), and (iii) in Fig. 1, where \({H}_{c}^{l}\) is found to be about 35 T in three measurements (we also note that the ~ 35 T signal was not observed in the previous magnetization measurement^{18,27}, it maybe caused by the increasing ABAB stacking fault in αRuCl_{3}), and in agreement with the magnetotorque probe result (32.5 T)^{44} (marked with the vertical dashed line in Fig. 1). On the other hand, the measured \({H}_{c}^{h}\) fields are somewhat different in cases (i), (ii), and (iii), with values of 76 T, 83 T, and 87 T, respectively. This difference can be attributed to the small angle ambiguity ( ± 2. 5°) in the three measurements and also to the high sensitivity of the transition field for the field angle near the c* axis of the compound^{46}. Moreover, we average the dM/dH curves from experiments (iiii), show the results in Fig. 1d, and find the averaging process has significantly reduced the electrical noise. This allows us to identify more clearly the two peaks at \({H}_{c}^{l}\) and \({H}_{c}^{h}\), respectively.
Figure 2 shows the measured dM/dH results for various tilting angles ranging from θ ≃ 0° (i.e., outofplane fields) to 90° (inplane). For θ ≃ 0° and 9°, the data are obtained by the destructive method, while the dM/dH curves with θ ≃ 20°, 30° and 90° are obtained by the nondestructive magnet and up to about 30 T.
The three θ ≃ 0° cases are also plotted in Fig. 2. Here only the highquality data above 30 T are shown, which exhibit double peaks at \({H}_{c}^{l}\) and \({H}_{c}^{h}\). With the singleturn coil technique reaching the ultrahigh magnetic field of 100 T class, here we are able to reach the higher transition field near \({H}_{c}^{h}\simeq 83\) T that has not been reached before. It is noteworthy that although the downsweep data in the fielddecreasing measurements are unavailable to be integrated due to the field inhomogeneity^{54,55}, nevertheless the signals at \({H}_{c}^{l}\) and \({H}_{c}^{h}\) in the upsweep and downsweep processes are consistent (c.f., Supplementary Fig. 4). This indicates unambiguously that these two anomalies are not artifacts due to noise but genuine features of phase transitions in αRuCl_{3}, and the possibility that the sample becomes degraded by applying the ultrahigh field can be excluded.
At θ ≃ 9° and 20°, the signals in dM/dH curve becomes rather weak (see also Fig. 3) although we measure the data at 9° by employing the more sensitive pickup coil with 1.4 mm diameter. The highfield downturn feature of the curve at 9° is thought to reflect the saturation of the magnetization as field increases. To see the transition for clarity, we show the averaged dM/dH curves measured by the nondestructive magnet in the two middle insets of Fig. 2, where roundpeak signals are observed near 25 and 20 T for θ ≃ 9° and 20°. These round peaks in the middleinset of Fig. 2 are thought to be the phase transitions. The two dome structures of averaged dM/dH curves at 9° leads to an uncertainty in θ of ± 2. 5°. We note that the two transition fields (\({H}_{c}^{l}\) and \({H}_{c}^{h}\)) for θ ≃ 0° seem to merge into one, and as this two curves are averaged results with θ ≃ 9 ± 2. 5° and 20 ± 2. 5°, the peaks are very broad. Therefore, we define a large error bar, i.e., ± 5 T for θ ≃ 9°, and ± 2 T for θ ≃ 20°.
The results at larger angles θ ≃ 30° and 90° are also shown in Fig. 2^{44,52}. The dM/dH curve at 90° shows two peaks and one shoulder structures. The peaks at 6.2 T and 7.2 T correspond respectively to the transition boundaries of the magnetic zigzag order (zigzag1) and another zigzag order (zigzag2), in agreement to previous studies^{42,45}. The shoulder structure seen at around 8.5 T is likely due to another antiferromagnetic (AFM) order^{56}. Because this feature is insensitive to the field angle as we show in the latter part, such AFM order is deemed to be caused by the ABAB stacking components and the transition field is denoted as \({H}_{c}^{AB}\). For large angles, the critical field H_{c}, e.g., H_{c} ≃ 7.2 T for θ ≃ 90° (i.e., inplane), labels the upper boundary between the zigzag and paramagnetic phases. Such a transition has been widely recognized for the inplane case as observed by neutron scattering experiments^{18,20,33}, and for tilted angles based on the magnetotorque measurements^{44}. Besides, the additional peak at 6.2 T is generally believed to reflect the transition between two different zigzag antiferromagnetic phases, with period3 and period6 spin structures along the c* direction in the ABC stacking, respectively, (see, e.g., ref. ^{45}). Here we dub this transition field as \({H}_{c}^{{\prime} }\).
H_{c} and \({H}_{c}^{{\prime} }\) are found to monotonously increase with decreasing the field angle. In contrast, \({H}_{c}^{AB}\) is independent of the field angle, suggesting that \({H}_{c}^{AB}\) at 8.5 T comes from a magnetically isotropic origin which is different from the transitions at H_{c} and \({H}_{c}^{{\prime} }\). According to the previous study^{18,56}, the field location of 8.5 T indicates that the transition occurs in the stacking fault ABAB layers in the sample. Therefore, the phase transition due to the suppression of the antiferromagnetic order in the ABAB stacking component is found to be isotropic, suggesting a 3dimensional order which is different from the 2dimensional zigzag orders.
Here, we should note that the presence of ABAB stacking fault is almost inevitable for αRuCl_{3} in the outofplane high magnetic field experiment. This is because the stress caused by the strong magnetic anisotropy under the magnetic field along the c* axis would more or less deform the sample^{57}. We can even damage αRuCl_{3} by deforming the sample and produce lots of ABAB stacking faults, which now exhibits ordering temperature at about 14 K (c.f., Supplementary Fig. 5). Then we perform highfield experiments up to 100 T along the c* axis on this sample, and find only \({H}_{c}^{{{{{{{{\rm{AB}}}}}}}}}\) peak at around 14( ± 4) T. Based on the experimental results, we conclude that the \({H}_{c}^{l}\) and \({H}_{c}^{h}\) signals should belong to the ABC stacking component. Furthermore, we also note that the pulse time of the destructive magnet is only a few microseconds^{55}, much shorter as compared to the nondestructive magnet. This allows the samples to withstand less stress impulse during the measurement, rendering some advantages in measuring fragile and strong anisotropic samples such as αRuCl_{3}.
In Fig. 2, by comparing the dM/dH results at different θ angles from 90° to 0°, we find strong magnetic anisotropy consistent with previous measurements^{18,27}. We measured the magnetization process for θ ≃ 90° (within the abplane) up to 90 T using the singleturn coil techniques. The results are shown in Fig. 1e, which demonstrate that only the 7 T transition is present for θ ≃ 90° and our measurements reproduce excellently the results in ref. ^{27} [c.f., Fig. 1 e]. It is found that H_{c} monotonically increases with decreasing angle from 90° to 0°, which is consistent with the results of Modic et al.^{44}.
As we described in Fig. 1, the dM/dH at 0° is significantly different from that at large angles (θ ≥ 9°) and exhibits two phase transitions. The two phase transitions indicate that an intermediate phase emerges between \({H}_{c}^{l}\) and \({H}_{c}^{h}\). Because Modic et al.^{44} have claimed that the zigzag order is suppressed for H > H_{c} or \({H}_{c}^{l}\), the intermediate phase between \({H}_{c}^{l}\) and \({H}_{c}^{h}\) should be disordered and counts as the experimental evidence of the recently proposed QSL phase in αRuCl_{3} with fields applied along outofplane c* axis^{47,51}. We also note that there is another scenario that \({H}_{c}^{h}\) corresponds to the transition field that suppresses the AFM order, and \({H}_{c}^{l}\) just separates two different AFM phases. However, based on the experimental results here, the reported data of Modic et al.^{44}, and calculated results as shown in the following section, we find strong evidence that the transition at \({H}_{c}^{l}\) is an intrinsic characteristic of the ABC stacking component, and consider it is more reasonable that \({H}_{c}^{l}\) suppresses the AFM order of the ABC stacking sample.
Comparison between experimental and calculated results
The recently proposed realistic microscopic spin model with large Kitaev coupling might support our experimental results. We consider the KJΓ\({{{\Gamma }}}^{{\prime} }\) model \({{{{{{{{\mathcal{H}}}}}}}}}_{0}={\sum }_{{\langle i,j\rangle }_{\gamma }}[K{S}_{i}^{\gamma }{S}_{j}^{\gamma }+J\,{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}+{{\Gamma }}({S}_{i}^{\alpha }{S}_{j}^{\beta }+{S}_{i}^{\beta }{S}_{j}^{\alpha })+{{{\Gamma }}}^{{\prime} }({S}_{i}^{\gamma }{S}_{j}^{\alpha }+{S}_{i}^{\gamma }{S}_{j}^{\beta }+{S}_{i}^{\alpha }{S}_{j}^{\gamma }+{S}_{i}^{\beta }{S}_{j}^{\gamma })]\) (α, β, γ ∈ {x, y, z}) with parameters K = − 25 meV, J = − 0.1∣K∣, Γ = 0.3∣K∣, and \({{{\Gamma }}}^{{\prime} }=0.02 K\)^{51}.
In Fig. 3, we compare the experimental and calculated dM/dH results as well as the integrated MH results. For the experimental data, only the critical fields associated with the pristine ABC stacking component, i.e., H_{c}, \({H}_{c}^{l}\), and \({H}_{c}^{h}\), are marked.
From Fig. 3a, b, we find semiquantitative agreement between the experimental and calculated dM/dH results. Similarly, the experimental and calculated MH results also show consistency to each other as shown in Fig. 3c, d. In Fig. 3b, for small angles θ = 0°, 0.8°, and 2.0° located within the angle range θ ≃ 0° ± 2. 5°, the calculated curves exhibit two transition fields as indicated by the solid black triangles and circles, and we find the upper transition fields \({H}_{c}^{h}\) are rather sensitive to the small change of θ near 0°. Therefore it explains the visible difference in \({H}_{c}^{h}\) among the three θ ≃ 0° measurements. On the contrary, the lower transition field \({H}_{c}^{l}\) is found rather stable in Fig. 3b, also in agreement with experiments. As the angle θ further increases, e.g., θ = 10°, there exists a single transition field, in agreement with the experimental result of 9° in Fig. 3a. The calculated transition fields H_{c}, from our DMRG simulations based on the 2D spin model, of θ ≃ 20°, 30°, and 90° cases in Fig. 3b show quantitative agreement to measurements in Fig. 3a. We note that there are still certain differences between the DMRG and experimental results, such as the height of peaks, which are understandable. The difference might be ascribed to the finitesize effects in the model calculations (c.f., Supplementary Fig. 7) or other possible terms/factors not considered in the present model study, e.g., the next and thirdnearest neighbor Heisenberg couplings, the interlayer interactions, and the inhomogeneous external field in the highfield measurements. In particular, as the DMRG calculations are performed on an effective twodimensional spin model, the interlayer stacking effects in αRuCl_{3} compounds are not considered.
Discussion
From both experimental and calculated magnetization data, we see intrinsic angle dependence of the quantum spin states in αRuCl_{3} under magnetic fields. Therefore, by collecting the transition fields \({H}_{c}^{l}\) and \({H}_{c}^{h}\) marked in Fig. 3, we summarize the results in a fieldangle phase diagram shown in Fig. 4. In previous theoretical studies, an intermediate QSL phase was predicted between the upper boundary of zigzag phase \({H}_{c}^{l}\) and the lower boundary of paramagnetic phase \({H}_{c}^{h}\)^{47,51}. Nevertheless, the fate of the intermediate QSL phase under tilted angles has not been studied before. Here we show clearly that the QSL states indeed constitute an extended phase in the fieldangle phase diagram in Fig. 4, as further supported by additional DMRG calculations of the spin structure factors here (c.f. the Supplementary Section B). Moreover, when θ becomes greater than about 9°, there exists only one transition field H_{c} in Fig. 4, which decreases monotonically as θ further increases. The previously proposed magnetic transition points determined by the magnetotorque measurements^{44} are also plotted in Fig. 4 and found to agree with our H_{c} for θ from 9° to 90°. In addition, the two transitions (\({H}_{c}^{l}\) and \({H}_{c}^{h}\)) experimentally obtained at θ ≃ 0° are semiquantitatively reproduced by the theoretical simulation, which indicates the existence of an intermediate QSL phase. The transition field of the magnetotorque measurements^{44} at θ = 0° is also found to be in agreement with our results. For 0° < θ ≲ 9°, there is a discrepancy between the theoretical simulation and the results of the torque measurement. Although the reason of the difference is not completely clear at present, the quantum fluctuations in the vicinity of the potential tricritical point where the \({H}_{c}^{l}\) and \({H}_{c}^{h}\) merge disturbs the precise evaluation of the transition field experimentally as well as numerically. Nevertheless, the theoretical proposition of the extension of the QSL phase to the finite small θ is likely to be supported by different experimental \({H}_{c}^{h}\) at θ ≃ 0° with ± 2. 5° uncertainty.
In summary, we find experimentally an interesting twotransition scenario in the prime Kitaev material αRuCl_{3} under high outofplane fields up to 100T class and reveal the existence of a fieldinduced intermediate phase in the fieldangle phase diagram. Such a magnetic disordered phase is separated from the trivial polarized state by a quantum phase transition, suggesting the existence of the longsought QSL phase as predicted in previous model studies^{47,51}. Regarding the nature of the intermediate QSL phase, previous theoretical work^{47} concludes the intermediate QSL phase can be adiabatically connected to the Kitaev spin liquid (KSL) phase. On the other hand, ref. ^{51} draws a different conclusion of gapless QSL in the intermediate regime based on results with multiple manybody approaches. Here we further uncover that the intermediate phase also extends to a finiteangle regime, whose precise nature calls for further theoretical studies. While the phase diagram in Fig. 4 excludes the presence of an inplane QSL phase like certain other recent studies^{43,44}, our work nevertheless opens the avenue for the exploration of the outofplane QSL phase in the Kitaev materials. Moreover, further experimental characteristics of the intermediate QSL phase can be started from here. For example, nuclear magnetic resonance and electron spin resonance spectroscopy under high fields^{58,59} are promising approaches for probing lowenergy excitations in the intermediate QSL phase discovered here.
Methods
Experimental details
A single crystal of αRuCl_{3} was used for the present experiment^{27}. The verticaltype singleturn coil field generator was employed to provide a pulse magnetic field up to 102 T. Things inside of the coil including the sample are generally not damaged by the generation of a magnetic field, although the field generation is destructive^{55}. The magnetization processes under the outofplane fields (Fig. 1) and those with small tilting angles (9° lines in Fig. 2) were measured using a doublelayer pickup coil that consists of two small coils compensating for each other^{53,55}. The sample is cut to 0.9 × 0.9 mm^{2} square. Several sample with ~ 0.2 mm thickness are stacking together to obtain enough thickness to measure the magnetization process in the singleturn coil experiments. The angle between the magnetic field and the c* axis is denoted as θ (c.f. upper inset of Fig. 2). In order to have good control on the angle θ, two glass rods with a section inclination angle θ are employed to clamp the sample in a Kapton tube. The singleturn coil, pickup coil, and the Kapton tube with the sample are placed in parallel visually. As the αRuCl_{3} sample is very soft and has strong anisotropy, it needs to be carefully fixed. Silicone grease instead of cryogenic glue is used to hold the sample, in order to reduce the dislocation of stacking caused by pressure (For more information of the setup around the sample, see in Supplementary Fig. 3). Nevertheless, such an experimental setting inevitably affects the precise control of θ with errors estimated to be ± 2. 5°.
Two types of doublelayer pickup coils are employed in the measurements; one is the standard type with 1 mm diameter^{53}, and the other is a recently developed one with a larger diameter of 1.4 mm that helps to enhance the signal by nearly three times. The magnetization signal is obtained by subtraction of the background signal from the sample signal, which are obtained by two successive destructivefield measurements^{53,54,55} without and with the sample (see Supplementary Fig. 3), respectively. Magnetization measurements at certain large angles like θ ≃ 9°, 20°, 30°, and 90° are performed by a similar induction method employing nondestructive pulse magnets^{60}. In the nondestructive pulse field experiment, the diameter of the sample is about 2 mm. All of our experiments are performed at a low temperature of 4.2 K.
Density matrix renormalization group calculation
We simulate the system on the cylindrical geometry up to width 6 (c.f. Supplementary Sec. B), and retain D = 512 bond states that lead to accurate results (truncation errors less than ϵ ≃ 1 × 10^{−6}). The direction of the magnetic field H is represented by [lmn] in the spin space (S^{x}, S^{y}, S^{z}), and the Zeeman term reads \({{{{{{{{\mathcal{H}}}}}}}}}_{H}=g{\mu }_{B}{\mu }_{0}{H}_{[lmn]}\frac{l{S}^{x}+m{S}^{y}+n{S}^{z}}{\sqrt{{l}^{2}+{m}^{2}+{n}^{2}}}\) with H_{[l=1, m=1, n]} tilting an angle \(\theta=\arccos (\frac{2+n}{\sqrt{6+3{n}^{2}}})\cdot \frac{18{0}^{\circ }}{\pi }\) to the c* axis within the ac*plane, and the Landé gfactor is fixed as g ≃ 2.3. The magnetization curves shown in Fig. 3b are obtained by computing \(M=g{\mu }_{B}\frac{l\langle {S}^{x}\rangle+m\langle {S}^{y}\rangle+n\langle {S}^{z}\rangle }{\sqrt{{l}^{2}+{m}^{2}+{n}^{2}}}\).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
X.G.Z. thank Yuan Yao for fruitful discussions, and acknowledge Yuto Ishi, Hironobu Sawabe, and Akihiko Ikeda for experimental supports. W.L. and H.L. are indebted to ShunYao Yu, ShouShu Gong, ZhengXin Liu, and Jinsheng Wen for helpful discussions. X.G.Z was supported by a JSPS fellowship. X.G.Z. and Y.M.H. was funded by JSPS KAKENHI No. 22F22332. Y.H.M. was funded by JSPS KAKENHI, GrantinAid for Transformative Research Areas (A) Nos.23H04859 and 23H04860, GrantinAid for Scientific Research (B) No. 23H01117, and GrantinAid for Challenging Research (Pioneering) No.20K20521. H.L. and W.L. were supported by the National Natural Science Foundation of China (Grant Nos. 12222412, 11834014, 11974036, and 12047503), CAS Project for Young Scientists in Basic Research (Grant No. YSBR057), and China National Postdoctoral Program for Innovative Talents (Grant No. BX20220291).
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Y.H.M and W.L supervised the project. X.G.Z and Y.H.M performed the destructive magnetic field experiment. X.G.Z, A.M and K.K performed the nondestructive field experiment. N.K and H.T provided the sample αRuCl_{3}. X.G.Z and Y.H.M analyzed the experimental data. H.L, W.L and G.S performed the model calculations and analyzed the numerical results. X.G.Z, H.L, Y.H.M and W.L contributed to the preparation of the draft.
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Zhou, XG., Li, H., Matsuda, Y.H. et al. Possible intermediate quantum spin liquid phase in αRuCl_{3} under high magnetic fields up to 100 T. Nat Commun 14, 5613 (2023). https://doi.org/10.1038/s41467023412327
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DOI: https://doi.org/10.1038/s41467023412327
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