Possible gapless quantum spin liquid behavior in the triangular-lattice Ising antiferromagnet PrMgAl$_{11}$O$_{19}$
Zhen Ma Shuhan Zheng Yingqi Chen Ruokai Xu Zhao-Yang Dong Jinghui Wang Hong Du Jan Peter Embs Shuaiwei Li Yao Li Yongjun Zhang Meifeng Liu Ruidan Zhong Jun-Ming Liu Jinsheng Wen







zma@hbnu.edu.cn
Hubei Key Laboratory of Photoelectric Materials and Devices, School of Materials Science and Engineering, Hubei Normal University, Huangshi 435002, China
State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China



Hubei Key Laboratory of Photoelectric Materials and Devices, School of Materials Science and Engineering, Hubei Normal University, Huangshi 435002, China

zhydong@njust.edu.cn
Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China

ShanghaiTech Laboratory for Topological Physics and School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China

Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen, Switzerland




Hubei Key Laboratory of Photoelectric Materials and Devices, School of Materials Science and Engineering, Hubei Normal University, Huangshi 435002, China

Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

jwen@nju.edu.cn
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China




Quantum spin liquids (QSLs) represent a novel state where spins are highly entangled but do not order even at zero temperature due to strong quantum fluctuations. Such a state is mostly studied in Heisenberg models defined on geometrically frustrated lattices. Here, we turn to a new triangular-lattice antiferromagnet , in which the interactions are believed to be of Ising type. Magnetic susceptibility measured with an external field along the c axis is two orders of magnitude larger than that with a field in the ab plane, displaying an ideal easy-axis behavior. Meanwhile, there is no magnetic phase transition or spin freezing observed down to 1.8 K. Ultralow-temperature specific heat measured down to 50 mK does not capture any phase transition either, but a hump at 4.5 K, below which the magnetic specific heat exhibits a quasi-quadratic temperature dependence that is consistent with a Dirac QSL state. Inelastic neutron scattering technique is also employed to elucidate the nature of its ground state. In the magnetic excitation spectra, there is a gapless broad continuum at the base temperature 55 mK, in favor of the realization of a gapless QSL. Our results provide a scarce example for the QSL behaviors observed in an Ising-type magnet, which can serve as a promising platform for future research on QSL physics based on an Ising model.


Possible gapless quantum spin liquid behavior in the triangular-lattice Ising antiferromagnet PrMgAl_11O_19
    Jinsheng Wen
    April 29, 2024
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§ INTRODUCTION


Quantum spin liquids (QSLs) represent a unique quantum disordered state of matter, which can host fractional elementary excitations and emergent gauge structures <cit.>. They have attracted enormous research interest due to the relevance to the understanding of high-temperature superconductivity and promising applications in quantum computation <cit.>. To unveil the nature of QSLs, it is of considerable significance to seek and investigate real QSL materials. One route to this goal is to introduce strong quantum fluctuations in a magnetic system <cit.>, which can be reached via a variety of manners <cit.>, such as geometrical frustration, spin anisotropy, competition between different paths of magnetic interactions, low dimensionality of magnetic correlations, etc. In this context, quite a few QSL candidate materials have been proposed in recent years <cit.>.

Among the currently existing QSL candidates, triangular-lattice rare-earth compounds are a representative branch. In rare-earth magnets, the magnetic correlation is generally strong and 4f electrons are quite localized. The spin-orbit-entangled nature of the local moments often brings highly anisotropic magnetic couplings <cit.>. The combination of geometrical frustration and spin anisotropic greatly improves the possibility to realize a QSL state <cit.>. As such, the triangular-lattice ytterbium-based compound YbMgGaO_4 has become a canonical example <cit.>. In particular, the availability of large single crystals makes it easy to perform various experimental investigations <cit.>. Earlier measurements including specific heat <cit.>, inelastic neutron scattering (INS) <cit.>, muon spin relaxation (μSR) <cit.>, etc, indeed show positive evidences for a QSL. However, subsequent results of thermal conductivity and ac magnetic susceptibility experiments are at odds with the scenario<cit.>. The controversial point lies in the structural disorder that nonmagnetic Mg^2+ and Ga^3+ are randomly distributed in the same Wyckoff position, which results in complicated effect on the ground state <cit.>. Some people think disorder is detrimental to the formation of QSL states and mimics a QSL <cit.>, while others instead believe it is able to enhance quantum fluctuations <cit.>. It has been open to question until now. Anyway, seeking new QSL candidates eliminating the influence from disorder is an imperative way. As a result, the disorder-free AYbCh_ 2 (A = alkali metal and Ch = O, S, Se) compounds are proposed and considered as a perfect investigated regime <cit.>. Nevertheless, given that alkali metal elements are inherently susceptible to site deficiency as well as challenge in growing high-quality single crystals <cit.>, enlarging the QSL candidate family is highly desirable.

More recently, it reports that rare-earth based hexaaluminates ReMAl_11O_19, where Re denotes lanthanides and M is Zn or Mg, as a new family of perfect triangular-lattice compounds were synthesized <cit.>. Due to the significant difference in radii of the constituent ions, there is little structural disorder in this series of compounds <cit.>. Moreover, the larger distance of magnetic layers makes it closer to an ideal two-dimensional structure when compared with YbMgGaO_4. Among them, PrZnAl_11O_19 has firstly caught the attention. It hosts a non-Kramers doublet ground state resulting from strong spin-orbit coupling of 4f^2 Pr^3+ ions and crystal-electric-field effect with D_3h symmetry <cit.>. The T^2 behavior of low-temperature specific heat and diffusive excitations of INS point to a gapless QSL, while μSR experiments also evidence the persistent spin dynamics down to 0.27 K <cit.>. Nonetheless, all the current measurements are confined to its polycrystalline samples, which can only yield powder-average results and thus would miss the crucial information related to crystalline directions, the detailed investigations on its single crystals are still absent at the moment.

Considering the volatile nature upon heating of ZnO in PrZnAl_11O_19 <cit.>, it is actually tough to obtain the single crystals. In this work, we turn to its sister compound , of which the high-quality single crystals have been successfully grown by us. We comprehensively investigate its magnetic properties, specific heat as well as low-energy magnetic excitations via INS at a much lower temperature of several tens of milli-kelvin. It is surprising that the magnetic susceptibility with a field in the ab plane is negligible when compared with that along the c axis, suggesting an ideal Ising model. The fitting to the inverse magnetic susceptibility data gives rise to a Curie-Weiss (Θ_ CW) temperature of -8.1 K, reflecting the dominant antiferromagnetic interactions in this system. Moreover, the absence of bifurcation between zero-field-cooling (ZFC) and field-cooling (FC) data excludes the possibility of spin freezing. Specific heat measurements down to as low as 50 mK reveal no sharp λ-shaped peak indicative of a phase transition, but a hump around 4.5 K, below which the data fit well with a quasi-quadratic temperature dependence. It is reminiscent of a gapless Dirac QSL. INS experiments capture broad spin excitation continuum below 2.5 meV, evidencing the possible realization of a gapless QSL state. Our work provides a promising platform to study QSL physics in a real Ising system.




§ EXPERIMENTAL DETAILS


Polycrystalline samples of were synthesized by solid-state reaction method. The raw materials Pr_2O_3 (99.9%), MgO (99.99%), and Al_2O_3 (99.99%) were weighed with a stoichiometric ratio. Then the precursor powders were mixed and thoroughly ground in an agate mortar. Finally, they were loaded into an alumina crucible and sintered at 1550 ^∘C for 60 hours with an intermediate grinding. High-quality single crystals were grown in air via a two-mirror optical floating-zone furnace (MF-2400, Cyberstar Corp.). During the growth, the seed and feed rods spun in an opposite direction with a speed of 25 r/min, and meanwhile traveled downwards with a speed of 9 mm/h. The obtained single crystals are chartreuse with a size up to 7×3×1.0 mm^3 for a piece, as shown in the inset of Fig. <ref>(c).

Powder x-ray diffraction data were collected at room temperature in an x-ray diffractometer (X^'TRA, ARL) using the Cu-K_α edge with a wavelength of 1.54 Å. In the measurements, the scan range of 2θ is from 10^∘ to 80^∘ with a step of 0.02^∘ and a rate of 10^∘/min, respectively. Rietveld refinements on powder XRD data were performed by GSAS software. X-ray backscattering Laue (Photonic Science) pattern was taken when the beam was perpendicular to the cleavage plane (ab plane) of the single crystal and the exposure time is 60 s. Dc magnetic susceptibility was measured on a piece of single crystal with the typical mass ∼15.9 mg. This procedure was performed within 1.8-300 K in a Quantum Design physical property measurement system (PPMS, Dynacool), equipped with a vibrating sample magnetometer option. Specific heat was measured on a 4.3-mg single crystal in a PPMS Dynacool equipped with a dilution refrigerator, so that the lowest temperature can reach 50 mK. INS experiments were carried out on FOCUS, a time-flight spectrometer located at Paul Scherrer Institute (PSI) at Villigen, Switzerland. About 5-g sample was loaded into a pure copper can in a dilution refrigerator and it was able to cool down to a base temperature of 55 mK. The incident neutron wavelength was fixed as λ∼ 4.7 Å, corresponding an good energy resolution of Δ E = 0.05 meV (half width at half maximum, HWHM). For each temperature, we collect data for about 12 h.



§ RESULTS



 §.§ Crystal structure and x-ray diffraction


Figures <ref>(a) and <ref>(b) show the schematics of the crystal structure and a two-dimensional triangular layer of , respectively. Magnetic Pr^3+ ions form edge-shared triangular layers that are well separated by several nonmagnetic layers consisting of AlO_6 octahedra, AlO_5 hexahedra, and Al/MgO_4 tetrahedra, featuring an ideal two-dimensional nature of magnetic couplings <cit.>. Along the c axis, the magnetic triangular layers have an ABAB stacking arrangement. It is emphasized that although the nonmagnetic Al^3+ ions located at the center of tetrahedra suffer from 1/2 occupation mixed with Mg^2+ ions, we believe it should have little impact on the magnetic ground state, since the antisite disorder only exists inside Al/MgO_4 tetrahedra and the ratio of this configuration of polyhedra is relatively less in the structural unit cells. Moreover, the nonmagnetic layers with this disorder problem are not directly adjacent to the magnetic layers, either. We will discuss it more in the discussion part later.



In Fig. <ref>(d), we present the powder XRD patterns as well as the Rietveld refinement results of this compound. It can be seen that the polycrystalline sample is a single-phase material, with no additional impurity peaks observed in the whole measured range. This is vital for the further preparation of its single crystals. The refinements were performed based on a hexagonal structure in the space group P6_3/mmc <cit.>. It gives rise to the lattice parameters a = b = 5.5846(3) Å, c = 21.8730(12) Å, and α = β = 90^∘, γ = 120^∘. From this result, we can further calculate that the separations of the nearest intralayer and interlayer Pr^3+ ions are 5.5846 Å and 11.9582 Å, respectively. It means the strength of magnetic interplay within the magnetic triangular layers is actually ∼10 times larger than that outside the triangular layers when considering the dipolar energy goes as 1/r^3 (Ref. PhysRevLett.84.3430). It justifies the two-dimensional nature of this compound as told above. These parameters are also in good agreement with the previous report <cit.>. The final refinement parameters are R_ p≈ 8.97%, R_ wp≈ 7.14%, and χ^2 ≈ 1.485, respectively, suggesting a high purity of our synthesized samples. Upon these well-prepared polycrystalline samples, we have succeeded in growing the single crystals of . X-ray backscattering Laue pattern was taken to characterize them when the incident beam was perpendicular to the cleavage plane, as shown in Fig. <ref>(c). We can clearly see the sharp diffraction spots with a six-fold symmetry, reflecting the structural characteristic of a triangular framework. The diffraction partten also shows some sporadic intensity near the strong spots. It indicates that there is a small domain of the single crystal with its crystallographic orientation in the ab plane somewhat deviating from the main one. It should have ignorable impact on the following measured results, since such a phenomenon is very weak and almost all the experimental investigations were performed along the crystallographic c axis or upon powder sample.



 §.§ Magnetic susceptibility




We then characterize the magnetic properties of single crystals by measuring dc magnetic susceptibility (χ) at a small field of 0.1 T, and the results are shown in Fig. <ref>(a). The susceptibility with an external field along the c axis is two orders of magnitude larger than that with a field applied in the ab plane. It signals that is an experimental realization of a triangular-lattice Ising model with the c axis as its easy axis of magnetization. More intriguing, there is no any magnetic transition or anomaly captured in the whole measurement process, which indicates the paramagnetic behavior persisting to the lowest temperature of 1.8 K. We speculate that the low-temperature paramagnetic state should be modulated by quantum fluctuations, since the thermal fluctuations are proportional to temperature and it should have little impact on the ground state when down to as low as 1.8 K. Besides that, no signature of splitting between ZFC and FC data can be distinguished in both directions, ruling out the possibility of spin freezing down to the lowest temperature available. Such a behavior is also observed in its polycrystalline sample as well as its sister compound PrZnAl_11O_19 <cit.>. In general, the Curie-Weiss fitting to low-temperature data can reveal the intrinsic physics related to its ground state, while the electrons will be thermally populated to excited crystal-electric-field (CEF) levels at higher temperatures, and thus the inverse magnetic susceptibility deviates from the temperature-linear behavior due to CEF effect <cit.>. In the inset of Fig. <ref>(a), we present the inverse low-temperature susceptibility data along the c axis that can be well fitted by the Curie-Weiss law. It yields a Curie-Weiss temperature Θ_ CW∼ -8.1 K within a typical fitting range of 1.8-50 K, indicating the dominant antiferromagnetic nature of the exchange interactions between the Pr^3+ local moments in this system. This value is close to that of -6.4 K determined upon powder data <cit.>. It is also noteworthy that the intralayer Pr^3+-Pr^3+ distance is ∼5.584 Å from the above structural characterization, which is larger than those of the magnetic ions in other referenced compounds. For instance, it is ∼3.401 Å for Yb^3+-Yb^3+ separation in the heavily-studied YbMgGaO_4, of which the Curie-Weiss temperature is about -4 K for the powder sample <cit.>. However, the main path of the magnetic interactions in is a nearly straight “Pr^3+-O^2--Pr^3+" bridge <cit.>, whose length is ∼5.590 Å determined by us, while it is a polygonal “Yb^3+-O^2--Yb^3+" one with the length of ∼4.456 Å in the latter <cit.>. That means the effective distance of the superexchange paths is actually comparable for these two compounds. On the other hand, the relatively large effective moment of Pr^3+ (∼2.96 μ_eff <cit.>) compared with Yb^3+ (∼ 2.6μ_eff <cit.>) at low temperatures would further strengthen the magnetic interactions in this regime. Thus, it is reasonable to yield such a value of Curie-Weiss temperature even though there is relatively large distance between the intralayer magnetic ions in . For the magnetic susceptibility data perpendicular the c axis, it cannot be fitted by the Curie-weiss law, since the spins in this compound exhibit an ideal easy-axis behavior and there is almost no component of spin moments along the hard-axis direction, as shown in Fig. <ref>(a).

The isothermal field-dependent magnetization up to 9 T with two different directions is depicted in Fig. <ref>(b). The magnetization with the field perpendicular to the c axis is insignificant compared with that along the c axis, which is in line with the Ising-type magnetic behavior determined in magnetic susceptibility. For the case of field along the c axis at 2 K, the magnetization curve progressively increases in a small-field range. Then it turns to deviate from the linear response regime around 1.5 T, which corresponds to the relatively low energy scale of the spin interactions in the ground state. Subsequently, it enters into a smooth transition regime and hardly shows the sign of saturation even though the external field increases up to 9 T, where the maximum magnetization value ∼2.0 μ_ B/Pr^3+ is far away from 3.2 μ_ B/Pr^3+ expected for free Pr^3+ ions <cit.>. It means that higher external field is needed to fully polarize the magnetic moments. With increasing the temperature to 15 K, the magnetic couplings between the Pr^3+ ions are gradually suppressed and the excited crystal field states would be thermally activated, which further keeps the magnetization from saturation and thus makes the magnetization curve remains mostly linear within the current field range.




 §.§ Specific heat




Above magnetic susceptibility and magnetization characterizations on the single crystals provide the basic information of the magnetic moments and the magnetic interactions in . To reveal the properties of low-energy magnetic excitations, we perform the specific heat measurements down to an ultralow temperature of 50 mK. In Fig. <ref>(a), we depict the specific heat results of as well as a nonmagnetic reference compound LaMgAl_11O_19 for comparison. At zero field, there is no sharp λ-shaped peak indicative of a phase transition, but a hump around 4.5 K. Such a feature is commonly observed in quite a few QSL candidates, such as triangular-lattice materials κ-(BEDT-TTF)_2Cu_2(CN)_3 <cit.> and YbMgGaO_4 <cit.>, and the kagome-lattice compound ZnCu_3(OH)_6Cl_2 <cit.>, etc. It may suggest an establishment of short-range correlations while the Schottky anomaly in a two-level system is also proposed <cit.>. Considering there has been no consensus on its exact physical origin so far, we cannot make a solid conclusion at the moment. However, no matter which case it is, the long-range magnetic transition can be excluded. When an external field is applied on this regime, the hump is gradually suppressed and it shifts towards higher temperatures, reflecting the continuous modulation of magnetic ground state via the external fields. To reveal the possible novel excitations in the ground state, it is helpful to perform the quantitative analysis on its magnetic specific heat (C_ m) data. In general, the total specific heat of a crystallized system with gapless elementary excitations can be represented as C_ p = γ T + β T^3 (Refs. np4_459,nc2_275), where the first term can originate from itinerant electrons, spinons or other quasiparticles with fractional spin while the second one may result from phonon and/or magnon excitations. As a result, the magnetic specific heat of can be obtained by subtracting the total specific heat of an isostructural reference compound LaMgAl_11O_19 [also shown in the Fig. <ref>(a)], which can be taken as the pure lattice part of the target compound <cit.>.

Figure <ref>(b) shows the results of magnetic specific heat of at various fields. There is indeed no phase transition observed, confirming the absence of long-range magnetic order down to at least 50 mK, two orders of magnitude lower than its interaction energy scale of |Θ_ CW| ∼ 8.1 K. More importantly, the low-temperature data can be well fitted by a power-law function of C_ m/T = AT^α-1, implying the promising nontrivial excitations in the ground state. At zero field, it gives rise to a α value of 1.98(2) within the fitting range between 0.05 and 1.5 K. The result of α approaching to 2 is unusual for an insulator and it may evidence the gapless Dirac QSL state with Dirac nodes <cit.>. In fact, such a behavior has already been observed in a wider range of QSL candidates, such as the diluted honeycomb-lattice Kitaev regime α-Ru_0.8Ir_0.2Cl_3 <cit.>, kagome-lattice antiferromagnet YCu_3(OH)_6Br_2[Br_x(OH)_1-x] <cit.>, square-kagome compound Na_6Cu_7BiO_4(PO_4)_4 <cit.>, as well as the triangular-lattice compound YbZn_2GaO_5 <cit.>. Nevertheless, it is the first time to capture this feature in an Ising-type QSL candidate. When a magnetic field is applied along the c axis, the magnetic specific heat is quickly suppressed, indicating the effective modulation of the magnetic ground state by the external field. We have also extracted the exponent of α at other fields in this way, and the result is shown in the inset of Fig. <ref>(b). It can be seen that α increases dramatically with the field below around 3 T, followed by a saturation value close to 4 at the moderate fields. As the field continues increasing up to 9 T, it falls back to 3 to some extent. This nonmonotonic response of α to the increasing field is also observed in its sister compound PrZnAl_11O_19 and it was attributed to the inhomogeneous effective magnetic field throughout the polycrystalline sample when considering different orientations of crystallographic axes for each grain inside the polycrystal <cit.>. However, current results are obtained based on high-quality single crystals. Thus, the reason of the polycrystalline sample itself can be excluded. It should reflect more complex intrinsic physics. Considering the significant decrease of magnetic specific heat with the applied magnetic field, the occurrence of field-induced spin gap that annihilates partial low-energy magnetic excitations may be responsible for the behavior. It deserves more detailed investigations on the magnetic-field effect in the future. Figure <ref>(c) shows the temperature dependence of magnetic entropy by integrating the C_ m/T data presented in Fig. <ref>(b). At zero field, the released magnetic entropy from the lowest measured temperature of 50 mK to 30 K is about Rln2, where R is the ideal gas constant. It shows no sign of saturation within the current temperature range, just like the case of its sister compound PrZnAl_11O_19 (Ref. PhysRevB.106.134428). The calculated residual magnetic entropy at 50 mK is less than 0.0001%Rln2. The near-zero value of residual magnetic entropy rules out the possibility of magnetic transition at lower temperatures. Thus, our low-temperature result can reveal the magnetic properties related to the ground state. It is consistent with the feature of a QSL. When applying a magnetic field into this regime, the magnetic entropy is continuously reduced, indicating the suppression of spin fluctuations by the external field.




 §.§ INS spectra






To further elucidate the nature of low-energy magnetic excitations, INS experiments were performed at several characteristic temperatures. In fact, we have observed a low-lying CEF level around 1.5 meV (not shown), which is the same as that measured in its sister compound PrZnAl_11O_19 <cit.>. However, given that this low-lying CEF level may mask the possible magnetic excitation signals, just like the cases in some other rare-earth QSL candidates with non-Kramers ions, such as TbInO_3 and Tm_3Sb_3Mg_2O_14 <cit.>, the high-temperature data of 50 K were used for background subtraction. Although at 50 K the CEF level of 1.5 meV would be occupied by the thermally-populated electrons from the ground state, what we concern most is if there are any underlying low-energy magnetic excitations below the strong CEF excitations. One route to this goal is to subtract a high-temperature constant data to pick it out. Thus, it should be fine whether taking the data of 50 K or a lower temperature, such as 10 K as the background, since they both represent an invariant constant that needs to be subtracted.  The so-obtained results are shown in Fig. <ref>. Aside from the strong elastic scattering signals around zero energy associated with crystal structure, there are diffusive excitations captured below 2.5 meV, which are in sharp contrast to the well-defined spin-wave excitations expected in a magnetically ordered regime. Therefore, INS results rule out the possibility of long-range magnetic order at an ultralow temperature of 55 mK. The broad continuum of spin excitations has been also observed in a few frustrated quantum magnets and it is considered as a hallmark of QSLs <cit.>. It should be true for when taking into account the above result of quasi-quadratic temperature dependence of magnetic specific heat indicative of a Dirac QSL state. These diffusive excitations weaken with the increasing temperatures and nearly disappear at 25 K. The evident temperature dependence of spectral intensities confirms its magnetic nature. We also need to point out that the intenisity in the elastic region is different at different temperatures, just as shown in Fig. <ref>. It results from the small intensity scale displayed in the figure. In fact, there is a huge variation of intensity between the measured elastic and inelastic signals. That means the statistical error of the elastic signal may be even larger than the inelastic intensity itself. Here, what we care about most is the inelastic scattering region and the intensity scale is thus defined to a small window. It would significantly affect the image of the elastic region.

To investigate the low-energy excitations in detail, we integrate the magnetic spectral weight with energy ranging from 0.25 to 2.5 meV and plot the intensities as a function of wave-vector Q, as shown in Fig. <ref>(a). It is clear that there are higher scattering intensities centered around the intermediate Q range for the low-temperature data, while the intensities at both small and large momentum-transfer positions are relatively low. This feature is different from the CEF excitations where the excitation intensity has a monotonic Q dependence <cit.>, proving the origin of low-energy magnetic excitations. When the temperature increases to 25 K, the broad peak becomes indistinguishable, suggesting the complete exhaustion of magnetic excitations due to the predominant role of thermal fluctuations at this temperature. We also integrate the spectral intensities all over Q and obtain the energy E dependence of intensities at various temperatures, as shown in Fig. <ref>(b). The intensities on the neutron-energy-loss (E>0) side have the same temperature dependence as that of Q, affirming the magnetic nature of spectral weight. Moreover, the low-energy magnetic excitations are very close to and difficult to be distinguished from the strong elastic line. In other words, may be a gapless spin system, which is consistent with what we deduce from specific heat measurements. To further clarify whether there is a gap or not, INS experiment with higher energy resolution is required. The magnetic excitations gradually weaken and finally vanish above approximately 2.5 meV. In contrast, the temperature dependence of intensities on the energy-gain (E<0) side are opposite due to detailed balance <cit.>. Similar phenomena are also observed in other QSL candidates, such as Zn_3Cu(OH)_6Cl_2 <cit.>, Ce_2Zr_2O_7 <cit.>, and NaRuO_2 <cit.>. Such a behavior of contrasting temperature dependence of excitation intensities emphasizes the reliability of our experimental data. It should be noted that the integrated intensity in Figs. <ref>(a) and <ref>(b) is affected by the selected area of the INS spectra shown in Fig. <ref>, but the overall behaviors of Q and energy dependence of intensity would not be changed. Based on the original measured magnetic spectra (no background subtraction), the integrated intensities stemmed from the elastic channels are as well extracted and shown in Fig. <ref>(c). The neutron diffraction data of 55 mK and 50 K are overlapped and there is no additional magnetic Bragg peak observed, especially in the low-angle range where magnetic peaks generally show up for a magnetically ordered system. This result agrees well with the absence of magnetic order determined in both magnetic susceptibility and specific heat, pointing to a fascinating QSL ground state in this Ising magnet.



 §.§ CEF analysis


To further facilitate our understanding of its ground state, especially to clarify the issue of singlet or doublet in a low-symmetric CEF environment of the non-Kramers ion Pr^3+, we also perform a CEF analysis based on the method of point-charge approximation. Pr^3+ has an electron configuration of 4f^2, in which the spin and orbital angular momentums (S = 1, L = 5) are entangled into a total angular momentum J = 4 due to the strong spin-orbit coupling. The corresponding 9-fold degenerate states are further split under the CEF effect. The effective CEF Hamiltonian is obtained by the point-charge model according to the D_3h point-group symmetry <cit.>,

    H_ CEF=∑_n,mB_n^m O_n^m,

where O_n^m and B_n^m are the Steven operators <cit.> and CEF parameters, respectively. It turns out that there are two lowest singlets with an extremely small gap of 0.00011 meV, which would compose a nearly degenerate non-Kramers doublet ground state, as shown in Table <ref>. These two singlet levels are |+⟩=-0.3458|-3⟩+0.6168|-1⟩-0.6168|1⟩+0.3458|3⟩ and |-⟩=0.0177|-4⟩-0.4098|-2⟩+0.8145|0⟩-0.4098|2⟩+0.0177|4⟩, respectively.



Unlike Kramers doublets, for this non-Kramers doublet, the longitudinal spin component behaves as the magnetic dipole moment, while the transverse spin component behaves as the magnetic quadrupole moment. This anisotropy of the magnetic moments implies a possible Ising-type magnetic behavior. While the current CEF results merely depending on symmetry analysis cannot match up the excited CEF levels, such as the energy level of 1.5 meV captured by INS, it is able to qualitatively identify the low-lying ground state, since the non-Kramers doublet is mainly determined by the point-group symmetry <cit.>. Thus, it is sufficient to simply do the symmetry analysis despite the lack of experimental CEF data to optimize the parameters for the effective CEF Hamiltonian. It is also worth noting that such a tiny gap of 0.00011 meV between the quasidoublet ground state is only a rough value rather than a quite accurate one due to the accuracy lamination of the point-charge calculations <cit.>. Further fitting upon full CEF levels measured by both low and high-energy INS experiments would give rise to more accurate CEF parameters. Nevertheless, the components of the non-Kramers doublet would not change significantly. In fact, the case that non-Kramers ions in a low-symmetric CEF environment create a quasidoublet ground state is commonly observed in other non-Kramers compounds, such as PrTiNbO_6 <cit.>, Pr_3BWO_9 <cit.>, Tb_3Sb_3Mg_2O_14 and Ho_3Sb_3Mg_2O_14 <cit.>, etc. It is thus reasonable to identify the ground state as a quasidoublet state in .




§ DISCUSSIONS


Currently, the relationship between structural disorder and the QSL state has become a hotly debated issue <cit.>. The prototypical examples include a kagome-lattice material Zn_3Cu(OH)_6Cl_2 and a triangular-lattice compound YbMgGaO_4. Strong disorder effect hinders the direct insight into the intrinsic ground states of these candidate materials. However, unlike the case in Zn_3Cu(OH)_6Cl_2 where antisite disorder directly occurs between magnetic Cu^2+ and nonmagnetic Zn^2+ ions <cit.>, there is no such mixed occupation in , but only weak disorder between the nonmagnetic Al^3+ and Mg^2+ ions <cit.>. Although this problem is quite analogous to the case of YbMgGaO_4 where nonmagnetic Ga^3+ and Mg^2+ ions are randomly distributed at the same Wyckoff position <cit.>, we should stress that the disorder content in this material is much less than that in YbMgGaO_4, since there is no structural disorder in AlO_6 and AlO_5 polyhedra and site mixing only exists in Al/MgO_4 tetrahedra which are in minority in the structure of  <cit.>. On the other hand, the nonmagnetic layers with structural disorder in YbMgGaO_4 are directly adjacent to the magnetic Yb^3+ layers, which has a significant influence on the ground state <cit.>. In contrast, there is an intermediate nonmagnetic layer consisting of disorder-free AlO_6 octahedra between the magnetic Pr^3+ layer and the antisite-disorder Al/MgO_4 layer that can effectively block their correlations. Thus, we argue that weak disorder effect in should have ignorable impact on the ground state and all our measured results can reflect the intrinsic physics. Additionally, while the observed CEF level of 1.5 meV is close to the quasidoublet ground state, as presented by the above CEF analysis, we believe the low-temperature behavior of quasi-quadratic temperature dependence of magnetic specific heat can effectively reveal the magnetic properties associated with its ground state, since it is measured below 1 K (∼0.09 meV) that is far away from the contaminations from this CEF level. To quantitatively study the influence of the CEF effect on milli-kelvin specific heat, one can introduce nonmagnetic ions of La^3+ into to dilute its magnetism, so that the single-ion magnetism related to the CEF splitting would be separated out when the content of La^3+ reaches a very high level, since the ingredient of cooperative magnetism resulting from the inter-site magnetic couplings is excluded at this time, like the cases of Tm_1-xLu_xMgGaO_4 <cit.> and Pr_1-xLa_xTiNbO_6 <cit.>. The low-lying CEF levels will also affect the low-energy magnetic excitations related to the ground state, which is commonly observed in other compounds with non-Kramers ions, such as TbInO_3 <cit.> and Tm_3Sb_3Mg_2O_14 <cit.>. It is thus highly desirable to perform a systematical CEF analysis based on full excited levels measured by low and high-energy INS in the future. In this case, the information of CEF splitting can be accurately learnt, which will help thoroughly unveil the intrinsic magnetic properties of the ground state. Given these, we believe the present results have well fulfilled our goal to propose a scarce example of an Ising magnet that exhibits QSL behaviors in a perfect triangular-lattice compound .

In fact, there is a relevant research on its sister compound PrZnAl_11O_19, which shows similar behaviors to  <cit.>. Given that the experimental results are quite similar for these two isostrutural compounds, we speculate that they should share the same QSL ground state. In other words, PrZnAl_11O_19 may be another Ising system that hosts a QSL state. In the meanwhile, we also notice that there have already been a few other works suggesting possible QSL states in an Ising model, such as the triangular-lattice compounds NdTa_7O_19 and ErMgGaO_4 <cit.>. However, the single-crystal sample of the former is still lacking for now while the latter suffers from the same disorder problem as YbMgGaO_4 <cit.>. Compared with them, may serve as an ideal platform to explore the QSL physics in an Ising magnet. Especially, the availability of large-size and high-quality single crystals of this material will make it more favorable. Besides the experimental investigations, there is a more recent numerical calculation work that also supports the possible realization of a QSL state in a triangular-lattice Ising regime <cit.>. The growing research interests in theory and experiment both advance the QSL investigations from a conventional Heisenberg model to a promising Ising one.




§ SUMMARY


In summary, we have succeeded in growing single crystals for a perfect triangular-lattice magnet . The magnetic susceptibility characterizations reveal an ideal Ising system with the c axis as its easy axis of magnetization. Meanwhile, there is no long-range magnetic order nor spin freezing observed down to 1.8 K. Specific heat measurements confirm the absence of magnetic order at a much too low temperature of 50 mK. More importantly, the behavior of a quasi-quadratic temperature dependence of magnetic specific points to a Dirac QSL state, which can be further modulated by the external field. Moreover, the broad continuum of low-energy magnetic excitation spectra is captured by INS experiments, which is also consistent with a gapless QSL state. Our work provides a rare example of a real Ising system that exhibits QSL behaviors, which serves as a fertile ground to explore the nontrivial quantum states in an Ising system.



§ ACKNOWLEDGMENTS


We thank Tong Chen for helpful discussions. The work was supported by the National Key Projects for Research and Development of China with Grant No. 2021YFA1400400, National Natural Science Foundation of China with Grants No. 12204160, No. 12225407, No. 12074174, No. 12074111, and No. 52272108, Hubei Provincial Natural Science Foundation of China with Grant No. 2023AFA105, and Fundamental Research Funds for the Central Universities. We thank the neutron beam time from PSI with Proposal No. 20202284 and the support from the Sample Environment Group for setting up the dilution refrigerator on FOCUS.











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@bib@innerbibempty

[Anderson(1973)]Anderson1973153
  author author P.W. Anderson, title title Resonating
  valence bonds: A new kind of insulator? @noop  journal journal Mater. Res. Bull. volume 8, pages 153 (year
  1973)NoStop
[Balents(2010)]nature464_199
  author author Leon Balents, title title Spin liquids
  in frustrated magnets, 10.1038/nature08917 journal journal Nature volume 464,  pages 199 (year 2010)NoStop
[Zhou et al.(2017)Zhou,
  Kanoda, and Ng]RevModPhys.89.025003
  author author Yi Zhou, author Kazushi Kanoda,
   and author Tai-Kai Ng,  title title Quantum spin liquid
  states, 10.1103/RevModPhys.89.025003 journal journal Rev. Mod. Phys. volume
  89, pages 025003 (year 2017)NoStop
[Broholm et al.(2020)Broholm, Cava, Kivelson, Nocera, Norman, and Senthil]Broholmeaay0668
  author author C. Broholm, author R. J. Cava,
  author S. A. Kivelson, author D. G. Nocera, author
  M. R. Norman,  and author
  T. Senthil, title title Quantum spin liquids, 10.1126/science.aay0668 journal journal
  Science volume 367, pages eaay0668
  (year 2020)NoStop
[Anderson(1987)]anderson1
  author author P. W. Anderson, title title The
  Resonating Valence Bond State in La_2CuO_4 and Superconductivity,  @noop  journal journal Science volume 235, pages 1196 (year
  1987)NoStop
[Kitaev(2003)]Kitaev20032
  author author A. Yu. Kitaev, title title Fault-tolerant
  quantum computation by anyons, http://dx.doi.org/10.1016/S0003-4916(02)00018-0 journal
  journal Ann. Phys. volume 303,  pages 2 (year 2003)NoStop
[Nayak et al.(2008)Nayak,
  Simon, Stern, Freedman, and  Das Sarma]RevModPhys.80.1083
  author author Chetan Nayak, author Steven H. Simon, author Ady Stern,
  author Michael Freedman,  and  author Sankar Das Sarma,  title title Non-Abelian anyons and
  topological quantum computation, 10.1103/RevModPhys.80.1083 journal journal Rev.
  Mod. Phys. volume 80, pages
  1083 (year 2008)NoStop
[Clark and Abdeldaim(2021)]doi:10.1146/annurev-matsci-080819-011453
  author author Lucy Clark and author Aly H. Abdeldaim, title title Quantum Spin
  Liquids from a Materials Perspective, 10.1146/annurev-matsci-080819-011453 journal journal Annu. Rev. Mater. Res. volume
  51, pages 495 (year 2021)NoStop
[Chamorro et al.(2021)Chamorro, McQueen, and Tran]doi:10.1021/acs.chemrev.0c00641
  author author Juan R. Chamorro, author Tyrel M. McQueen,  and author Thao T. Tran, title title Chemistry of
  Quantum Spin Liquids, 10.1021/acs.chemrev.0c00641
  journal journal Chem. Rev. volume 121, pages 2898. (year
  2021)NoStop
[Gardner et al.(2010)Gardner, Gingras, and Greedan]RevModPhys.82.53
  author author Jason S.  Gardner, author Michel J. P.  Gingras,  and author John E.  Greedan, title title
  Magnetic pyrochlore oxides, 10.1103/RevModPhys.82.53
  journal journal Rev. Mod. Phys. volume 82, pages 53 (year
  2010)NoStop
[Li et al.(2016a)Li, Wang, and Chen]PhysRevB.94.035107
  author author Yao-Dong  Li, author Xiaoqun Wang,  and author Gang Chen, title title Anisotropic spin
  model of strong spin-orbit-coupled triangular antiferromagnets, 10.1103/PhysRevB.94.035107 journal journal Phys. Rev. B volume 94, pages 035107 (year 2016a)NoStop
[Liu et al.(2018a)Liu, Zhang, Ji, Liu,
  Li, Wang, Lei, Chen, and Zhang]Liu_2018
  author author Weiwei Liu, author Zheng Zhang,
  author Jianting Ji, author Yixuan Liu, author
  Jianshu Li, author
  Xiaoqun Wang, author
  Hechang Lei, author
  Gang Chen,  and author
  Qingming Zhang, title
  title Rare-Earth Chalcogenides: A Large Family of
  Triangular Lattice Spin Liquid Candidates, 10.1088/0256-307x/35/11/117501 journal journal
  Chin. Phys. Lett. volume 35, pages
  117501 (year 2018a)NoStop
[Li et al.(2015a)Li, Liao, Zhang, Li,
  Jin, Ling, Zhang,
  Zou, Pi, Yang, Wang, Wu, and Zhang]sr5_16419
  author author Yuesheng  Li, author Haijun Liao, author Zhen Zhang,
  author Shiyan Li, author Feng Jin, author
  Langsheng Ling, author
  Lei Zhang, author Youming Zou, author Li Pi, author Zhaorong  Yang, author Junfeng  Wang, author Zhonghua  Wu,  and author Qingming  Zhang, title title
  Gapless quantum spin liquid ground state in the two-dimensional spin-1/2
  triangular antiferromagnet YbMgGaO_4, @noop  journal journal Sci. Rep. volume
  5, pages 16419 (year
  2015a)NoStop
[Li et al.(2015b)Li, Chen, Tong, Pi,
  Liu, Yang, Wang, and  Zhang]prl115_167203
  author author Yuesheng  Li, author Gang Chen, author Wei Tong,
  author Li Pi, author
  Juanjuan Liu, author
  Zhaorong Yang, author
  Xiaoqun Wang,  and author
  Qingming Zhang, title
  title Rare-Earth Triangular Lattice Spin Liquid: A
  Single-Crystal Study of YbMgGaO_4, @noop 
  journal journal Phys. Rev. Lett. volume 115, pages 167203 (year
  2015b)NoStop
[Shen et al.(2016)Shen,
  Li, Wo, Li, Shen, Pan, Wang, Walker,
  Steffens, Boehm, Hao,
  Quintero-Castro, Harriger, Frontzek, Hao, Meng, Zhang,
  Chen, and Zhao]nature540_559
  author author Yao Shen, author Yao-Dong Li,
  author Hongliang Wo, author Yuesheng Li, author
  Shoudong Shen, author
  Bingying Pan, author
  Qisi Wang, author H. C. Walker, author P. Steffens, author M. Boehm, author Yiqing  Hao, author D. L. Quintero-Castro, author L. W. Harriger, author M. D. Frontzek, author Lijie Hao,
  author Siqin Meng, author Qingming Zhang, author
  Gang Chen,  and author
  Jun Zhao, title title Evidence for a spinon Fermi surface in a
  triangular-lattice quantum-spin-liquid candidate, @noop 
  journal journal Nature volume 540, pages 559 (year
  2016)NoStop
[Paddison et al.(2017)Paddison, Daum, Dun, Ehlers, Liu, Stone, Zhou,  and Mourigal]np13_117
  author author Joseph A. M.  Paddison, author Marcus  Daum, author Zhiling  Dun, author Georg Ehlers, author Yaohua Liu,
  author Matthew B. Stone,
  author Haidong Zhou,  and  author Martin Mourigal,  title title Continuous excitations of
  the triangular-lattice quantum spin liquid YbMgGaO_4, @noop 
  journal journal Nat. Phys. volume 13, pages 117 (year
  2017)NoStop
[Li et al.(2016b)Li, Adroja, Biswas, Baker,
  Zhang, Liu, Tsirlin,
  Gegenwart, and Zhang]PhysRevLett.117.097201
  author author Yuesheng  Li, author Devashibhai  Adroja, author Pabitra K.  Biswas, author Peter J.  Baker, author Qian Zhang, author Juanjuan Liu,
  author Alexander A. Tsirlin,
  author Philipp Gegenwart,  and  author Qingming Zhang,  title title Muon Spin Relaxation
  Evidence for the U(1) Quantum Spin-Liquid Ground State in the Triangular
  Antiferromagnet YbMgGaO_4, 10.1103/PhysRevLett.117.097201 journal journal
  Phys. Rev. Lett. volume 117, pages
  097201 (year 2016b)NoStop
[Li et al.(2017a)Li, Adroja, Bewley, Voneshen, Tsirlin, Gegenwart, and  Zhang]PhysRevLett.118.107202
  author author Yuesheng  Li, author Devashibhai  Adroja, author Robert I.  Bewley, author David  Voneshen, author Alexander A.  Tsirlin, author Philipp  Gegenwart,  and author Qingming Zhang, title title Crystalline Electric-Field Randomness in the Triangular Lattice
  Spin-Liquid YbMgGaO_4, 10.1103/PhysRevLett.118.107202 journal journal
  Phys. Rev. Lett. volume 118, pages
  107202 (year 2017a)NoStop
[Li et al.(2017b)Li, Adroja, Voneshen, Bewley, Zhang, Tsirlin, and Gegenwart]nc8_15814
  author author Yuesheng  Li, author Devashibhai  Adroja, author David  Voneshen, author Robert I.  Bewley, author Qingming  Zhang, author Alexander A.  Tsirlin,  and author Philipp  Gegenwart, title title
  Nearest-neighbour resonating valence bonds in YbMgGaO_4, @noop
   journal journal Nat. Commun. volume 8, pages 15814 (year
  2017b)NoStop
[Li et al.(2019)Li,
  Bachus, Liu, Radelytskyi,
  Bertin, Schneidewind, Tokiwa,
  Tsirlin, and Gegenwart]PhysRevLett.122.137201
  author author Yuesheng  Li, author Sebastian  Bachus, author Benqiong  Liu, author Igor Radelytskyi, author Alexandre Bertin, author Astrid Schneidewind, author Yoshifumi Tokiwa, author Alexander A. Tsirlin,  and author Philipp Gegenwart, title title
  Rearrangement of Uncorrelated Valence Bonds Evidenced by Low-Energy Spin
  Excitations in YbMgGaO_4, 10.1103/PhysRevLett.122.137201 journal journal
  Phys. Rev. Lett. volume 122, pages
  137201 (year 2019)NoStop
[Xu et al.(2016)Xu,
  Zhang, Li, Yu, Hong, Zhang, and Li]PhysRevLett.117.267202
  author author Y. Xu, author J. Zhang, author Y. S. Li, author
  Y. J. Yu, author X. C. Hong, author Q. M.  Zhang,  and author S. Y.  Li, title title
  Absence of Magnetic Thermal Conductivity in the Quantum Spin-Liquid
  Candidate YbMgGaO_4, 10.1103/PhysRevLett.117.267202 journal journal
  Phys. Rev. Lett. volume 117, pages
  267202 (year 2016)NoStop
[Ma et al.(2018a)Ma, Wang, Dong, Zhang,
  Li, Zheng, Yu, Wang, Che, Ran, Bao,
  Cai,  ČČermák, Schneidewind, Yano,
  Gardner, Lu, Yu,
  Liu, Li, Li, and  Wen]PhysRevLett.120.087201
  author author Zhen Ma, author Jinghui Wang,
  author Zhao-Yang Dong, author Jun Zhang, author
  Shichao Li, author
  Shu-Han Zheng, author
  Yunjie Yu, author Wei Wang, author Liqiang  Che, author Kejing Ran, author Song Bao,
  author Zhengwei Cai, author P.  ČČermák, author A. Schneidewind, author S. Yano,
  author J. S. Gardner, author Xin Lu, author
  Shun-Li Yu, author
  Jun-Ming Liu, author
  Shiyan Li, author Jian-Xin Li,  and author Jinsheng Wen, title title Spin-Glass Ground State in a Triangular-Lattice Compound
  YbZnGaO_4, 10.1103/PhysRevLett.120.087201 journal journal
  Phys. Rev. Lett. volume 120, pages
  087201 (year 2018a)NoStop
[Ma et al.(2018b)Ma, Ran, Wang, Bao,
  Cai, Li, and Wen]ZhenMa:106101
  author author Zhen Ma, author Kejing Ran,
  author Jinghui Wang, author Song Bao, author
  Zhengwei Cai, author
  Shichao Li,  and author
  Jinsheng Wen, title
  title Recent progress on magnetic-field studies on
  quantum-spin-liquid candidates, 10.1088/1674-1056/27/10/106101 journal journal
  Chin. Phys. B volume 27, eid
  106101 (year 2018b)NoStop
[Kimchi et al.(2018)Kimchi,
  Nahum, and Senthil]PhysRevX.8.031028
  author author Itamar Kimchi, author Adam Nahum,   and author T. Senthil,  title title Valence Bonds in Random
  Quantum Magnets: Theory and Application to YbMgGaO_4,  10.1103/PhysRevX.8.031028 journal journal Phys. Rev. X volume 8, pages
  031028 (year 2018)NoStop
[Wen et al.(2019)Wen,
  Yu, Li, Yu, and  Li]npjqm4_12
  author author Jinsheng  Wen, author Shun-Li Yu, author Shiyan Li,
  author Weiqiang Yu,  and  author Jian-Xin Li, title title Experimental identification of quantum
  spin liquids, @noop  journal journal
  npj Quant. Mater. volume 4, pages 12
  (year 2019)NoStop
[Zhu et al.(2017)Zhu,
  Maksimov, White, and Chernyshev]PhysRevLett.119.157201
  author author Zhenyue Zhu, author P. A. Maksimov,
  author Steven R. White,  and  author A. L. Chernyshev,  title title Disorder-Induced Mimicry of
  a Spin Liquid in YbMgGaO_4, 10.1103/PhysRevLett.119.157201 journal journal
  Phys. Rev. Lett. volume 119, pages
  157201 (year 2017)NoStop
[Ma et al.(2020)Ma,
  Dong, Wu, Zhu, Bao, Cai, Wang, Shangguan,
  Wang, Ran, Yu, Deng, Mole, Li, Yu,
  Li, and Wen]PhysRevB.102.224415
  author author Zhen Ma, author Zhao-Yang Dong,
  author Si Wu, author
  Yinghao Zhu, author
  Song Bao, author Zhengwei Cai, author Wei  Wang, author Yanyan Shangguan, author Jinghui Wang, author Kejing Ran,
  author Dehong Yu, author Guochu Deng, author
  Richard A. Mole, author
  Hai-Feng Li, author
  Shun-Li Yu, author
  Jian-Xin Li,  and author
  Jinsheng Wen, title
  title Disorder-induced spin-liquid-like behavior in
  kagome-lattice compounds, 10.1103/PhysRevB.102.224415
  journal journal Phys. Rev. B volume 102, pages 224415 (year
  2020)NoStop
[Li(2019)]https://doi.org/10.1002/qute.201900089
  author author Yuesheng  Li, title title
  YbMgGaO_4: A Triangular-Lattice Quantum Spin Liquid Candidate, https://doi.org/10.1002/qute.201900089 journal
  journal Adv. Quant. Technol. volume 2, pages 1900089 (year 2019)NoStop
[Liu et al.(2022a)Liu, Yuan, Li, Li,
  Zhao, Liao, and Li]PhysRevB.105.024418
  author author Jiabin Liu, author Long Yuan,
  author Xuan Li, author Boqiang Li, author
  Kan Zhao, author Haijun Liao,  and author Yuesheng Li, title title Gapless spin liquid behavior in a kagome Heisenberg antiferromagnet
  with randomly distributed hexagons of alternate bonds, 10.1103/PhysRevB.105.024418 journal journal
  Phys. Rev. B volume 105, pages
  024418 (year 2022a)NoStop
[Furukawa et al.(2015)Furukawa, Miyagawa, Itou, Ito, Taniguchi, Saito, Iguchi, Sasaki, and Kanoda]PhysRevLett.115.077001
  author author T. Furukawa, author K. Miyagawa,
  author T. Itou, author
  M. Ito, author H. Taniguchi, author M. Saito, author S. Iguchi, author T. Sasaki,  and author K. Kanoda, title title
  Quantum Spin Liquid Emerging from Antiferromagnetic Order by Introducing
  Disorder, 10.1103/PhysRevLett.115.077001 journal journal Phys. Rev. Lett. volume 115, pages 077001 (year
  2015)NoStop
[Hu et al.(2021)Hu,
  Pajerowski, Zhang, Podlesnyak, Qiu, Huang, Zhou, Klich, Kolesnikov, Stone, and Lee]PhysRevLett.127.017201
  author author Xiao Hu, author Daniel M. Pajerowski, author Depei Zhang, author Andrey A. Podlesnyak, author Yiming Qiu, author Qing Huang,
  author Haidong Zhou, author Israel Klich, author
  Alexander I. Kolesnikov, author
  Matthew B. Stone,  and author
  Seung-Hun Lee, title
  title Freezing of a Disorder Induced Spin Liquid with
  Strong Quantum Fluctuations, 10.1103/PhysRevLett.127.017201 journal journal
  Phys. Rev. Lett. volume 127, pages
  017201 (year 2021)NoStop
[M. Bordelon et al.(2019)M. Bordelon, Kenney, Liu, Hogan, Posthuma, Kavand, Lyu, Sherwin, Butch, Brown,
  Graf, Balents, and Wilson]np15_1058
  author author Mitchell  M. Bordelon, author Eric  Kenney, author Chunxiao  Liu, author Tom Hogan, author Lorenzo Posthuma, author Marzieh Kavand, author Yuanqi Lyu,
  author Mark Sherwin, author N. P. Butch, author
  Craig Brown, author
  M. J. Graf, author
  Leon Balents,  and author
  Stephen D. Wilson, title
  title Field-tunable quantum disordered ground state
  in the triangular lattice antiferromagnet NaYbO_2, @noop 
  journal journal Nat. Phys. volume 15, pages 1058 (year
  2019)NoStop
[Yan et al.(2019)Yan,
  Okamoto, Wu, Zheng,
  Zhou, Cao, and McGuire]PhysRevMaterials.3.074405
  author author J.-Q. Yan, author S. Okamoto,
  author Y. Wu, author
  Q. Zheng, author H. D. Zhou, author H. B.  Cao,  and author M. A.  McGuire, title title
  Magnetic order in single crystals of
  Na_3Co_2SbO_6 with a honeycomb
  arrangement of 3d^7 Co^2+ ions, 10.1103/PhysRevMaterials.3.074405 journal journal
  Phys. Rev. Mater. volume 3, pages
  074405 (year 2019)NoStop
[Dai et al.(2021)Dai,
  Zhang, Xie, Duan,
  Gao, Zhu, Feng, Tao, Huang, Cao, Podlesnyak, Granroth, Everett,
  Neuefeind, Voneshen, Wang,
  Tan, Morosan, Wang,
  Lin, Shu, Chen, Guo, Lu, and Dai]PhysRevX.11.021044
  author author Peng-Ling  Dai, author Gaoning Zhang, author Yaofeng Xie,
  author Chunruo Duan, author Yonghao Gao, author
  Zihao Zhu, author Erxi Feng, author Zhen  Tao, author Chien-Lung  Huang, author Huibo Cao, author Andrey Podlesnyak, author Garrett E. Granroth, author Michelle S. Everett, author Joerg C. Neuefeind, author David Voneshen, author Shun Wang,
  author Guotai Tan, author Emilia Morosan, author
  Xia Wang, author Hai-Qing Lin, author Lei  Shu, author Gang Chen, author Yanfeng Guo,
  author Xingye Lu,  and  author Pengcheng Dai,  title title Spinon Fermi Surface Spin
  Liquid in a Triangular Lattice Antiferromagnet NaYbSe_2,  10.1103/PhysRevX.11.021044 journal journal Phys. Rev. X volume 11, pages 021044 (year 2021)NoStop
[Dufault et al.(2023)Dufault, Bahrami, Streeter, Yao, Gonzalez, Zhang, and Tafti]PhysRevB.108.064405
  author author Emilie Dufault, author Faranak Bahrami, author Alenna Streeter, author Xiaohan Yao,
  author Enrique Gonzalez,
  author Qiang Zhang,  and  author Fazel Tafti, title title Introducing the monoclinic polymorph of
  the honeycomb magnet
  Na_2Co_2TeO_6, 10.1103/PhysRevB.108.064405 journal journal
  Phys. Rev. B volume 108, pages
  064405 (year 2023)NoStop
[Ashtar et al.(2019a)Ashtar, Gao,
  Wang, Qiu, Tong,
  Zou, Zhang, Marwat,
  Yuan, and Tian]ASHTAR2019146
  author author M. Ashtar, author Y.X. Gao,
  author C.L. Wang, author Y. Qiu, author
  W. Tong, author Y.M. Zou, author X.W.  Zhang, author M.A. Marwat, author S.L. Yuan,   and author Z.M. Tian,  title title Synthesis, structure and
  magnetic properties of rare-earth REMgAl_11O_19 (RE = Pr, Nd)
  compounds with two-dimensional triangular lattice, https://doi.org/10.1016/j.jallcom.2019.06.177 journal
  journal J. Alloy. Compd. volume 802,  pages 146 (year
  2019a)NoStop
[Ashtar et al.(2019b)Ashtar, Marwat,
  Gao, Zhang, Pi, Yuan, and Tian]C9TC02643F
  author author M. Ashtar, author M. A. Marwat,
  author Y. X. Gao, author Z. T. Zhang, author
  L. Pi, author S. L. Yuan,  and author Z. M. Tian, title title REZnAl_11O_19 (Re = Pr, Nd, Sm-Tb): a new family of ideal 2D
  triangular lattice frustrated magnets, 10.1039/C9TC02643F journal journal J. Mater.
  Chem. C volume 7, pages 10073
  (year 2019b)NoStop
[Bu et al.(2022)Bu,
  Ashtar, Shiroka, Walker,
  Fu, Zhao, Gardner,
  Chen, Tian, and Guo]PhysRevB.106.134428
  author author Huanpeng  Bu, author Malik Ashtar, author Toni Shiroka,
  author Helen C. Walker, author Zhendong Fu, author
  Jinkui Zhao, author
  Jason S. Gardner, author
  Gang Chen, author Zhaoming Tian,  and author Hanjie Guo, title title Gapless triangular-lattice spin-liquid candidate
  PrZnAl_11O_19, 10.1103/PhysRevB.106.134428 journal journal
  Phys. Rev. B volume 106, pages
  134428 (year 2022)NoStop
[den Hertog and Gingras(2000)]PhysRevLett.84.3430
  author author Byron C.  den Hertog and author Michel
  J. P. Gingras, title title Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore
  Magnets, 10.1103/PhysRevLett.84.3430 journal journal Phys. Rev. Lett. volume 84, pages 3430 (year
  2000)NoStop
[Ranjith et al.(2019)Ranjith, Dmytriieva, Khim, Sichelschmidt, Luther, Ehlers,
  Yasuoka, Wosnitza, Tsirlin,
  Kühne, and Baenitz]PhysRevB.99.180401
  author author K. M. Ranjith, author D. Dmytriieva,
  author S. Khim, author
  J. Sichelschmidt, author
  S. Luther, author D. Ehlers, author H. Yasuoka, author J. Wosnitza, author A. A.  Tsirlin, author H. Kühne,  and author M. Baenitz, title title Field-induced
  instability of the quantum spin liquid ground state in the
  J_eff=1/2 triangular-lattice compound
  NaYbO_2, 10.1103/PhysRevB.99.180401
  journal journal Phys. Rev. B volume 99, pages 180401(R) (year
  2019)NoStop
[Yamashita et al.(2008)Yamashita, Nakazawa, Oguni, Oshima, Nojiri, Shimizu, Miyagawa, and Kanoda]np4_459
  author author Satoshi Yamashita, author Yasuhiro Nakazawa, author Masaharu Oguni, author Yugo Oshima,
  author Hiroyuki Nojiri, author Yasuhiro Shimizu, author Kazuya Miyagawa,  and author Kazushi Kanoda, title title Thermodynamic properties of a spin-1/2
  spin-liquid state in a κ-type organic salt, @noop 
  journal journal Nat. Phys. volume 4, pages 459 (year
  2008)NoStop
[Helton et al.(2007)Helton,
  Matan, Shores, Nytko,
  Bartlett, Yoshida, Takano,
  Suslov, Qiu, Chung,
  Nocera, and Lee]prl98_107204
  author author J. S. Helton, author K. Matan,
  author M. P. Shores, author E. A. Nytko, author
  B. M. Bartlett, author
  Y. Yoshida, author Y. Takano, author A. Suslov, author Y. Qiu, author J.-H.  Chung, author D. G. Nocera,  and author Y. S. Lee, title title Spin Dynamics of
  the Spin-1/2 Kagome Lattice Antiferromagnet
  ZnCu_3(OH)_6Cl_2, @noop 
  journal journal Phys. Rev. Lett. volume 98, pages 107204 (year
  2007)NoStop
[Yamashita et al.(2011)Yamashita, Yamamoto, Nakazawa,
  Tamura, and Kato]nc2_275
  author author Satoshi Yamashita, author Takashi Yamamoto, author Yasuhiro Nakazawa, author Masafumi Tamura,  and author Reizo Kato, title title Gapless spin
  liquid of an organic triangular compound evidenced by thermodynamic
  measurements, @noop  journal journal
  Nat. Commun. volume 2, pages 275
  (year 2011)NoStop
[Ran et al.(2007)Ran,
  Hermele, Lee, and Wen]PhysRevLett.98.117205
  author author Ying Ran, author Michael Hermele,
  author Patrick A. Lee,  and  author Xiao-Gang Wen,  title title Projected-Wave-Function
  Study of the Spin-1/2 Heisenberg Model on the Kagomé Lattice, 10.1103/PhysRevLett.98.117205 journal journal Phys. Rev. Lett. volume 98, pages 117205 (year 2007)NoStop
[Hu et al.(2019)Hu,
  Zhu, Eggert, and He]PhysRevLett.123.207203
  author author Shijie Hu, author W. Zhu, author Sebastian Eggert,  and author Yin-Chen He, title
  title Dirac Spin Liquid on the Spin-1/2 Triangular
  Heisenberg Antiferromagnet, 10.1103/PhysRevLett.123.207203 journal journal
  Phys. Rev. Lett. volume 123, pages
  207203 (year 2019)NoStop
[Do et al.(2020)Do,
  Lee, Kihara, Choi,
  Yoon, Kim, Cheong,
  Chen, Chou, Nojiri, and  Choi]PhysRevLett.124.047204
  author author Seung-Hwan  Do, author C. H. Lee, author T. Kihara,
  author Y. S. Choi, author Sungwon Yoon, author
  Kangwon Kim, author
  Hyeonsik Cheong, author
  Wei-Tin Chen, author
  Fangcheng Chou, author
  H. Nojiri,  and author
  Kwang-Yong Choi, title
  title Randomly Hopping Majorana Fermions in the
  Diluted Kitaev System
  -Ru_0.8Ir_0.2Cl_3,  10.1103/PhysRevLett.124.047204 journal
  journal Phys. Rev. Lett. volume 124,  pages 047204 (year 2020)NoStop
[Zeng et al.(2022)Zeng,
  Ma, Wu, Li, Tao, Lu, Chen, Mi,
  Song, Cao, Che, Li, Li, Luo, Meng, and  Li]PhysRevB.105.L121109
  author author Zhenyuan  Zeng, author Xiaoyan  Ma, author Si Wu,
  author Hai-Feng Li, author Zhen Tao, author
  Xingye Lu, author Xiao-hui Chen, author Jin-Xiao Mi, author Shi-Jie  Song, author Guang-Han  Cao, author Guangwei  Che, author Kuo Li, author Gang Li, author Huiqian Luo, author
  Zi Yang Meng,  and author
  Shiliang Li, title title Possible Dirac quantum spin liquid in the kagome quantum
  antiferromagnet
  YCu_3(OH)_6Br_2[Br_x(OH)_1x],  10.1103/PhysRevB.105.L121109 journal
  journal Phys. Rev. B volume 105,  pages L121109 (year 2022)NoStop
[Liu et al.(2022b)Liu, Zeng, Xu, Sun,
  Yakubovich, Shvanskaya, Li,  and Vasiliev]PhysRevB.105.155153
  author author Bo Liu, author Zhenyuan Zeng,
  author Aini Xu, author Yili Sun, author
  Olga Yakubovich, author
  Larisa Shvanskaya, author
  Shiliang Li,  and author
  Alexander Vasiliev, title
  title Low-temperature specific-heat studies on two
  square-kagome antiferromagnets, 10.1103/PhysRevB.105.155153 journal journal
  Phys. Rev. B volume 105, pages
  155153 (year 2022b)NoStop
[Xu et al.()Xu,
  Bag, Sherman, Yadav,
  Kolesnikov, Podlesnyak, Moore, and Haravifard]arXiv_2305.20040
  author author Sijie Xu, author Rabindranath Bag,
  author Nicholas E. Sherman,
  author Lalit Yadav, author Alexander I. Kolesnikov, author Andrey A. Podlesnyak, author Joel E. Moore,  and author Sara Haravifard, title title Realization of U(1) Dirac Quantum Spin
  Liquid in YbZn_2GaO_5, @noop  journal
  arXiv:2305.20040 NoStop
[Clark et al.(2019)Clark,
  Sala, Maharaj, Stone,
  Knight, Telling, Wang,
  Xu, Kim, Li, Cheong, and Gaulin]np15_262
  journal   author author Lucy Clark, author Gabriele  Sala, author Dalini D.  Maharaj, author Matthew B.  Stone, author Kevin S.  Knight, author Mark T. F.  Telling, author Xueyun  Wang, author Xianghan  Xu, author Jaewook Kim, author Yanbin Li,
  author Sang-Wook Cheong,  and  author Bruce D. Gaulin,  title title Two-dimensional spin liquid
  behaviour in the triangular-honeycomb antiferromagnet TbInO_3, @noop
   journal journal Nat. Phys. volume 15, pages 262 (year
  2019)NoStop
[Scheie et al.(2018)Scheie,
  Sanders, Krizan, Christianson, Garlea, Cava, and  Broholm]PhysRevB.98.134401
  author author A. Scheie, author M. Sanders,
  author J. Krizan, author A. D. Christianson, author V. O. Garlea, author
  R. J. Cava,  and author
  C. Broholm, title title Crystal field levels and magnetic anisotropy in the
  kagome compounds
  Nd_3Sb_3Mg_2O_14,
  Nd_3Sb_3Zn_2O_14, and
  Pr_3Sb_3Mg_2O_14,  10.1103/PhysRevB.98.134401 journal journal Phys. Rev. B volume 98, pages 134401 (year 2018)NoStop
[Gaudet et al.(2019)Gaudet,
  Smith, Dudemaine, Beare,
  Buhariwalla, Butch, Stone,
  Kolesnikov, Xu, Yahne,
  Ross, Marjerrison, Garrett,
  Luke, Bianchi, and Gaulin]PhysRevLett.122.187201
  author author J. Gaudet, author E. M. Smith,
  author J. Dudemaine, author J. Beare, author
  C. R. C. Buhariwalla, author
  N. P. Butch, author
  M. B. Stone, author
  A. I. Kolesnikov, author
  Guangyong Xu, author
  D. R. Yahne, author
  K. A. Ross, author
  C. A. Marjerrison, author
  J. D. Garrett, author
  G. M. Luke, author
  A. D. Bianchi,  and author
  B. D. Gaulin, title
  title Quantum Spin Ice Dynamics in the
  Dipole-Octupole Pyrochlore Magnet
  Ce_2Zr_2O_7, 10.1103/PhysRevLett.122.187201 journal journal
  Phys. Rev. Lett. volume 122, pages
  187201 (year 2019)NoStop
[Ortiz et al.(2023)Ortiz,
  Sarte, Avidor, Hay,
  Kenney, Kolesnikov, Pajerowski, Aczel, Taddei, Brown, Wang, Graf, Seshadri, Balents, and Wilson]np19_949
  author author Brenden R.  Ortiz, author Paul M.  Sarte, author Alon Hendler  Avidor, author Aurland  Hay, author Eric Kenney, author Alexander I. Kolesnikov, author Daniel M. Pajerowski, author Adam A. Aczel, author Keith M. Taddei, author Craig M. Brown, author Chennan Wang,
  author Michael J. Graf, author Ram Seshadri, author
  Leon Balents,  and author
  Stephen D. Wilson, title
  title Quantum disordered ground state in the
  triangular-lattice magnet NaRuO_2, @noop  journal
  journal Nat. Phys. volume 19,  pages 943 (year 2023)NoStop
[Hutchings(1964)]HUTCHINGS1964227
  author author M.T. Hutchings, title title Point-Charge
  Calculations of Energy levels of Magnetic Ions in Crystalline Electric
  Fields,   (publisher Academic Press, New York, year 1964) pp.  pages 227–273NoStop
[Stevens(1952)]Stevens_1952
  author author K. W. H. Stevens, title title Matrix Elements
  and Operator Equivalents Connected with the Magnetic Properties of Rare Earth
  Ions, 10.1088/0370-1298/65/3/308 journal
  journal Proc. Phys. Soc. Sec. A  volume 65, pages 209 (year 1952)NoStop
[Edvardsson and Klintenberg(1998)]EDVARDSSON1998230
  author author Sverker Edvardsson and author Mattias Klintenberg, title title Role of
  the electrostatic model in calculating rare-earth crystal-field
  parameters, https://doi.org/10.1016/S0925-8388(98)00309-0 journal
  journal J. Alloy. Compd. volume 275-277, pages 230 (year
  1998)NoStop
[Liu et al.(2018b)Liu, Li, and Chen]PhysRevB.98.045119
  author author Changle Liu, author Yao-Dong Li,   and author Gang Chen,  title title Selective measurements of
  intertwined multipolar orders: Non-Kramers doublets on a triangular
  lattice, 10.1103/PhysRevB.98.045119 journal journal Phys. Rev. B volume
  98, pages 045119 (year
  2018b)NoStop
[Dun et al.(2021)Dun,
  Bai, Stone, Zhou, and  Mourigal]PhysRevResearch.3.023012
  author author Zhiling Dun, author Xiaojian Bai,
  author Matthew B. Stone,
  author Haidong Zhou,  and  author Martin Mourigal,  title title Effective point-charge
  analysis of crystal fields: Application to rare-earth pyrochlores and tripod
  kagome magnets
  R_3Mg_2Sb_3O_14, 10.1103/PhysRevResearch.3.023012 journal journal Phys. Rev. Res. volume 3, pages 023012 (year 2021)NoStop
[Li et al.(2018)Li,
  Bachus, Tokiwa, Tsirlin,  and Gegenwart]PhysRevB.97.184434
  author author Yuesheng  Li, author Sebastian  Bachus, author Yoshifumi  Tokiwa, author Alexander A.  Tsirlin,  and author Philipp  Gegenwart, title title
  Gapped ground state in the zigzag pseudospin-1/2 quantum antiferromagnetic
  chain compound PrTiNbO_6, 10.1103/PhysRevB.97.184434 journal journal Phys.
  Rev. B volume 97, pages 184434
  (year 2018)NoStop
[Nagl et al.(2024)Nagl,
  Flavián, Hayashida, Povarov,
  Yan, Murai, Ohira-Kawamura,
  Simutis, Hicken, Luetkens,
  Baines, Hauspurg, Schwarze,
  Husstedt, Pomjakushin, Fennell, Yan, Gvasaliya, and  Zheludev]nagl2024excitation
  author author J. Nagl, author D. Flavián,
  author S. Hayashida, author K. Yu. Povarov, author
  M. Yan, author N. Murai, author S. Ohira-Kawamura, author G. Simutis, author T. J.  Hicken, author H. Luetkens, author C. Baines,
  author A. Hauspurg, author B. V. Schwarze, author
  F. Husstedt, author
  V. Pomjakushin, author
  T. Fennell, author Z. Yan, author S. Gvasaliya,  and author A. Zheludev, title title Excitation
  Spectrum and Spin Hamiltonian of the Frustrated Quantum Ising Magnet
  Pr_3BWO_9, @noop  journal arXiv:2402.14107 NoStop
[Huang et al.(2021)Huang,
  Xu, Wang, Zhao, Tu, Ni, Wang, Pan,
  Fu, Hao, Liu, Mei, and Li]PhysRevLett.127.267202
  author author Y. Y. Huang, author Y. Xu, author Le Wang, author
  C. C. Zhao, author
  C. P. Tu, author J. M. Ni, author L. S.  Wang, author B. L. Pan, author Ying Fu, author Zhanyang Hao, author
  Cai Liu, author Jia-Wei Mei,  and author S. Y. Li, title title
  Heat Transport in Herbertsmithite: Can a Quantum Spin Liquid Survive
  Disorder? 10.1103/PhysRevLett.127.267202 journal journal Phys. Rev. Lett. volume 127, pages 267202 (year
  2021)NoStop
[Norman(2016)]RevModPhys.88.041002
  author author M. R. Norman, title title
  Colloquium : Herbertsmithite and the search for the quantum spin
  liquid, 10.1103/RevModPhys.88.041002 journal journal Rev. Mod. Phys. volume
  88, pages 041002 (year 2016)NoStop
[Li et al.(2020)Li,
  Bachus, Deng, Schmidt,
  Thoma, Hutanu, Tokiwa,
  Tsirlin, and Gegenwart]PhysRevX.10.011007
  author author Yuesheng  Li, author Sebastian  Bachus, author Hao Deng, author Wolfgang Schmidt, author Henrik Thoma,
  author Vladimir Hutanu, author Yoshifumi Tokiwa, author Alexander A. Tsirlin,  and  author Philipp Gegenwart,  title title Partial Up-Up-Down Order
  with the Continuously Distributed Order Parameter in the Triangular
  Antiferromagnet TmMgGaO_4, 10.1103/PhysRevX.10.011007 journal journal Phys.
  Rev. X volume 10, pages 011007
  (year 2020)NoStop
[Arh et al.(2022)Arh,
  Sana, Pregelj, Khuntia,
  Jagličić, Le, Biswas,
  Manuel, Mangin-Thro, Ozarowski, and Zorko]nm21_416
  author author T. Arh, author B. Sana, author M. Pregelj, author
  P. Khuntia, author Z. Jagličić, author M. D. Le, author P. K.  Biswas, author P. Manuel, author L. Mangin-Thro,
  author A. Ozarowski,  and  author A. Zorko, title title The Ising triangular-lattice
  antiferromagnet neodymium heptatantalate as a quantum spin liquid
  candidate, @noop  journal journal
  Nat. Mater. volume 21, pages
  416 (year 2022)NoStop
[Cai et al.(2020)Cai,
  Lygouras, Thomas, Wilson,
  Beare, Sharma, Marjerrison,
  Yahne, Ross, Gong,
  Uemura, Dabkowska, and Luke]PhysRevB.101.094432
  author author Y. Cai, author C. Lygouras,
  author G. Thomas, author M. N. Wilson, author
  J. Beare, author S. Sharma, author C. A.  Marjerrison, author D. R.  Yahne, author K. A. Ross, author Z. Gong, author Y. J. Uemura, author
  H. A. Dabkowska,  and author
  G. M. Luke, title title SR study of the triangular
  Ising antiferromagnet ErMgGaO_4, 10.1103/PhysRevB.101.094432 journal journal
  Phys. Rev. B volume 101, pages
  094432 (year 2020)NoStop
[Cevallos et al.(2018)Cevallos, Stolze, and Cava]CEVALLOS20185
  author author F. Alex Cevallos, author Karoline Stolze,  and author Robert J. Cava, title title Structural
  disorder and elementary magnetic properties of triangular lattice ErMgGaO_4
  single crystals, https://doi.org/10.1016/j.ssc.2018.03.015 journal journal Solid State Commun. volume 276,  pages 5 (year 2018)NoStop
[Ulaga et al.()Ulaga,
  Kokalj, Wietek, Zorko, and  Prelovsek]arXiv_2307.03545
  author author M. Ulaga, author J. Kokalj,
  author A. Wietek, author A. Zorko,  and author
  P. Prelovsek, title title Quantum spin liquid in the easy-axis Heisenberg model on
  frustrated lattices, @noop  journal
  arXiv:2307.03545 NoStop




