Generally, a system tends to lower its symmetry to be in the low-energy state. As a consequence, a magnetic material will break certain symmetry according to Landau’s theorem, and the magnetic moments carried by the electron spins will form an ordered pattern at low temperatures.^[1,2] Excitations associated with these ordered spins are conventional magnons with spin S = 1 (Refs. [3]–[9]). However, in systems with small spin and strong quantum fluctuations, such a conventional order can be avoided, leading to a quantum-spin-“liquid” (QSL) state.^[10] Now, it is known that geometrical frustration, a situation where the antiferromagnetic Heisenberg exchange interactions cannot be satisfied simultaneously among different sites in triangular [Fig. 1(a)] or kagomé lattices [Fig. 1(b)], can result in strong quantum fluctuations.^[11–13] In 1973, Anderson proposed the resonant-valence-bond (RVB) model on the triangular lattice for the QSL state. It is a superposition of all possible configurations of the singlets formed by any of the two strongly interacting spins.^[14–17] The elementary excitations in QSLs are fractionalized quasiparticles, e.g., charge-free S = 1/2 spinons, fundamentally different from conventional magnons.^[14–17]

Fig. 1.

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	Fig. 1. (color online) Schematics of two-dimensional (a) triangular, (b) kagomé, and (c) honeycomb structures where quantum spin liquids can be realized. Arrows and question marks represent spins and geometrical frustration, respectively. J is the Heisenberg exchange interaction, and Kx,y,z are Kitaev interactions along three bonds.

The QSL state defined by the RVB model does not have an exact solution. In 2006, Kitaev constructed an exactly-solvable S = 1/2 model on the honeycomb lattice [Fig. 1(c)]. This model, as in Eq. (1), is named the Kitaev model.

Besides the rich and exotic physics of QSLs, they also hold promising application potentials, for example, quantum computation via braiding the non-Abelian anyons in these materials.^[18,20–23] Furthermore, understanding QSLs may help understand the mechanism of high-temperature superconductivity.^[24–28] For these reasons, research on QSL has been surging in the past few decades. There have been many review articles summarizing the progress on QSLs already.^{[11,21,29–42]} In this topical review, we will restrict our discussions to the measurements under magnetic fields only.

In general, an external magnetic field can be detrimental to the QSL phase, as the field may induce symmetry breaking. Ultimately, when the field is strong enough, all the spins will be polarized and the moments will align with the field direction — the system is then a ferromagnet. On the other hand, in the fully polarized state, one can extract the exchange interactions from the spin-wave excitation spectra obtained by inelastic neutron scattering (INS) measurements and understand the magnetic ground state in zero field. Investigating the magnetic excitations under fields with techniques such as nuclear magnetic resonance (NMR), muon spin relaxation (μSR), electron spin resonance, and terahertz spectroscopy also provides key information on the interactions underlying the exotic states of the QSL candidates. Furthermore, studying the field evolution of the thermal transport properties can also provide insights into the QSL physics. Finally, in available systems where the Kitaev physics is relevant, there are other non-Kitaev terms setting in at low temperatures, resulting in an ordered phase instead of the Kitaev QSL. In some of these materials, such as α-RuCl₃, applying a magnetic field suppresses the non-Kitaev interactions and drives the system into a possible QSL state. In this article, we will summarize the results from these magnetic-field experiments on the i) Geometrically-frustrated systems, including triangular-lattice compounds YbMgGaO₄ and YbZnGaO₄, κ-(BEDT-TTF)₂Cu₂(CN)₃, and EtMe₃Sb[Pd(dmit)₂]₂, and the kagomé system ZnCu₃(OH)₆Cl₂; ii) the Kitaev material α-RuCl₃, which has been subject to intensive investigations by ours and other groups recently. We first present experimental evidence for each of these materials, then form discussions based on these results, and raise several questions in the end.

Different from aforementioned QSL candidates, which have either triangular or kagomé lattice where antiferromagnetic exchange interactions are geometrically frustrated, the Kitaev QSL has a honeycomb lattice where the frustration on a single site arises from the bond-dependent spin anisotropy.^[18] Possible realization of such an exotic state has been suggested in SOC-assisted Mott insulators such as Na₂IrO₃,^{[19,42,87–91]} and Li₂IrO₃.^[91–95] These materials have the honeycomb lattice as shown in Fig. 1(c). Due to the combination of the cubic crystal electric field, strong SOC, and electronic correlations, the ground state is Krammers doublets with an effective spin of 1/2 (Refs. [96]–[98]). Furthermore, because of the strong SOC, the effective spin is expected to be anisotropic due to the spatial anisotropy of the 5d orbitals of Ir⁴⁺. Therefore, the bond-dependent anisotropic Kitaev interaction may be realized on the honeycomb lattice.^[18] However, it is found that they are not QSLs but are instead magnetically ordered at low temperatures.^{[42,91–95,98–101]} Nevertheless, subsequent experimental and theoretical works suggest that the magnetic orders in these materials are unconventional, signifying the presence of notable Kitaev interaction. For instance, it is suggested that the zigzag magnetic order in Na₂IrO₃ can be understood within a Heisenberg–Kitaev model.^[89,102] In the phase diagram constructed using this model, there are regions where the Heisenberg interaction is small and the Kitaev interaction is dominant, leading to the Kitaev QSL phase. Because of this, finding Kitaev QSLs in related materials is still encouraging. In the following, we will discuss another Kitaev material α-RuCl₃, which has been the focus of recent research due to the availability of high-quality single crystals and the feasibility of neutron scattering measurements.

α-RuCl₃ has two-dimensional honeycomb layers formed by the 4d Ru³⁺ ions.^[103–108] In fact, realization of the Kitaev interaction in materials with 4d electrons does not sound very promising in the beginning, because their SOCs are smaller compared to those of the 5d systems. However, although the absolute value of the SOC in RuCl₃ is smaller, the almost −90° bond angles of the Cl–Ru–Cl bonds of the edge-shared RuCl₆ octahedra makes the cubic crystal electric field win and the SOC become a dominant effect.^[103–112] Thus, similar to iridates, α-RuCl₃ is also an SOC-assisted Mott insulator with an effective spin of 1/2, and the strong spatial anisotropy of the 4d orbitals combined with the SOC makes the bond-dependent Kitaev interaction significant.^{[18,103,104,113]} However, similar to Na₂IrO₃, the ground state of α-RuCl₃ is not a Kitaev QSL, but a zigzag magnetic order state instead.^[103–108] It has been proposed that the zigzag order is an indication for the presence of the Kitaev interaction in this system.^{[19,87,105–108]} Moreover, INS results indicate that the ground state of α-RuCl₃ may be proximate to the Kitaev QSL phase,^[114] and both INS^[114–117] and Ramman studies^[118,119] observe broad continuous magnetic excitations that can be associated with fractionalized excitations resulting from the Kitaev QSL phase.

By analyzing the INS spectra, magnetic interactions governing the ground state can be extracted. In Fig. 10(a), we show the spin-wave excitation spectra resulting from the zigzag order state.^[115] Instead of using the widely considered Heisenberg–Kitaev (HK) model^[19,87] to fit the spectra, a K–Γ effective-spin model is used, where K and Γ represent Kitaev and off-diagonal exchange interactions, respectively. This minimal model describing the ground state of α-RuCl₃ is first proposed by Wang et al. in Ref. [120]. They recognize the importance of the off-diagonal interactions, but on the other hand, they find that the Heisenberg interactions are at least an order of magnitude smaller than either K or Γ in the parameter range relevant to this material. Fits to the INS spectra with this model yield a ferromagnetic K of −6.8 meV and a Γ of 9.5 meV,^[115] very close to the calculated results of K = −5.5 meV and Γ = 7.6 meV reported in Ref. [120]. These results unambiguously demonstrate that the Kitaev interaction is large and exists in real material.

Fig. 10.

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	Fig. 10. (color online) (a) Magnetic dispersion along the [100] direction obtained by INS experiments. The solid line is the calculated dispersion as shown in panel (b). (b) Calculated spin-wave spectra along high-symmetry paths. From Ref. [115].

As mentioned above, the ground state of α-RuCl₃ is the zigzag order state instead of a Kitaev QSL. Nevertheless, the magnetic order is rather fragile, with an ordered moment of ∼ 0.4 μ_B and an ordering temperature of ∼ 8 K.^{[103–108,115]} Such a fragile order can be fully suppressed by either an in-plane magnetic field^[22,106,121] or pressure.^[123] How do the magnetic excitations behave in the high-field state? Is the high-field disordered state a QSL? If the high-field state is a QSL, what is the relationship between this phase and the long-sought Kitaev QSL? To answer these questions, measurements utilizing various experimental techniques have been carried out.^{[106,121,122,124–133]} Some of these results are discussed in the following.

By following the magnetic-field dependence of the magnetization and specific heat, it is found that the zigzag order is gradually suppressed, and the system becomes a magnetically disordered state at ∼ 7.5 T,^[124] consistent with earlier reports on the field effect.^{[106,121,122]} NMR spectra on high-quality single crystals also indicate that there is a quantum critical point at B_c ∼ 7.5 T.^[124] Above B_c, the spin-lattice relaxation rate 1/T₁ of ³⁵Cl shows a power-law behavior as 1/T₁ ∼ T^α. In a field range between 8 T and 16 T, α ≈ 3, suggesting a field-induced QSL featuring Dirac nodal-like spin excitations. A phase diagram summarizing these results is shown in Fig. 11(a). Intensive research on the high-field state utilizing various techniques such as magnetization,^{[106,121,122,124,132,135]} specific heat,^{[106,124,126,134,135]} magnetodielectric,^[128] neutron diffraction,^{[122,126,132]} NMR,^[124,134] magnetic torque,^[127] thermal conductivity,^{[127,133,135]} terahertz spectroscopy,^[129,130] and electron spin resonance,^[131] has resulted in many somewhat similar phase diagrams, some of which are shown in Fig. 11.

Fig. 11.

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Fig. 11. (color online) Magnetic phase diagram of α-RuCl3 obtained from various measurements. (a) PM and AFM represent paramagnet and zigzag order, respectively. The low field region of the phase diagram is constructed using magnetization and specific heat data, and the high field region is using NMR data. The magnetic field is applied in the a–b plane. The contour map indicates the exponent of the temperature dependence of the spin-lattice relaxation rate. The inset illustrates the zigzag order in the low-field state. From Ref. [124]. (b) ZZ and QPM represent zigzag order and quantum paramagnet, respectively. The phase boundary between ZZ2 and PM is the transition temperature obtained from specific heat and neutron diffraction measurements. The thick solid line is a fit with the transverse-field Ising model, and the thin solid line is a power-law fit. The dashed line is a power-law fit to the gap size Δ. From Ref. [126]. (c) Transition temperatures and gap values obtained from specific heat measurements. The solid line is a guide to the eye. Dashed lines are the fits of the gap function. The magnetic entropy is shown in a color scale. From Ref. [125]. (d) Phase diagram obtained from specific heat and NMR measurements along with the field dependence of the spin gap Δ extracted from the nuclear-spin relaxation rate (right axis). From Ref. [134]. (e) False-color representation of the T derivative of the ab-plane thermal conductivity (κab) together with the gap values (solid squares) extracted from the phononic fits. The color scale is in units of W/K2·m. From Ref. [133]. (f) Magnetic transition temperature as a function of field obtained from susceptibility measurements with field applied along the [110] direction. The inset shows the susceptibility data. From Ref. [132].

On one hand, these phase diagrams all show that the zigzag magnetic order is gradually suppressed by an in-plane magnetic field, and the system reaches a quantum critical point around B_c ≈ 7.5 T. Furthermore, accumulating evidence suggests that the high-field disordered state above B_c is a QSL. In particular, Banerjee et al. have carried out INS measurements to examine the magnetic-field evolution of the magnetic excitations, and some of the results are shown in Fig. 12.^[132] They find that the spin-wave excitations associated with the zigzag order near the M point is suppressed with the field. As shown in Fig. 12(e), above B_c, at μ₀H = 8 T, excitations near the M point are completely gone, and only excitations near the Γ point remain, similar to the results under zero field above the ordering temperature, as shown in Fig. 12(f). By comparing with calculations, they suggest that the excitations under high fields evidence a field-induced Kitaev QSL state.^[132]

Fig. 12.

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	Fig. 12. (color online) (a)–(e) Magnetic-field evolution of the INS spectra along the [100] direction, measured at T = 2 K with an external magnetic field applied in the a–b plane. (f) The zero-field data at 15 K, above the ordering temperature. From Ref. [132].

On the other hand, the nature of the field-induced QSL phase, in particular, whether the low-energy magnetic excitations associated with this state are finite,^{[125,126,130–132,134]} is still under debate. To resolve this issue, Yu et al.^[135] have performed ultralow-temperature thermal conductivity measurements down to 80 mK under magnetic fields, and the results are summarized in Fig. 13. As shown in Fig. 13(a), κ/T increases with field and then decreases to a minimum at the critical field B_c. Above B_c, κ/T increases with field again. These results clearly show that there is a quantum critical point at B_c, consistent with other works.^{[124–127,130,131,134]} By examining the residual κ at zero temperature, it is found that κ₀ is effectively zero in the whole field range probed [Fig. 13(b)]. In the low-field range below B_c, there is a spin-anisotropy gap about 2 meV associated with the zigzag order.^{[114,115,132]} Therefore, absence of the thermal conductivity at zero temperature is expected. If the high-field state is also fully gapped as suggested in Refs. [125], [126], [130]–[132], [134], these thermal conductivity results are easily explainable. However, as shown in Fig. 11(a), a gapless QSL state near B_c is suggested.^[124] We here provide one possible solution to reconcile this discrepancy. According to the report, this gapless state is in fact featured by Dirac-like excitations with gap nodes in the momentum space.^[124] In this case, the magnetic density of states, represented as C_m in κ = 1/3C_mv_Fl is small. At present, the estimated magnetic specific heat C_m has big uncertainties due to the lack of a proper reference sample to subtract the phonon contributions.^{[106,124–126,134,135]} If the real C_m is small, then the gapless state with nodal excitations proposed in Ref. [124] can also be consistent with the thermal conductivity results.^[135]

Fig. 13.

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	Fig. 13. (color online) (a) Field dependence of κ/T at various temperatures. The minimum of κ/T at ∼7.5 T corresponds to Bc, where the zigzag magnetic order disappears. (b) Field dependence of the residual linear term κ0/T. From Ref. [135].

As partially reflected from the discussions above, research on the QSL candidates has been quite dynamic. A lot of progress has been made already in recent years. However, it still lacks an ideal QSL candidate so far. Quite often, the spin-“liquid” behavior may have some other origins than quantum fluctuations. Below, we will show some examples.

There is accumulating evidence suggesting YbMgGaO₄ to be a promising candidate as a gapless QSL.^[43–49] However, the report of no positive contributions from the magnetic excitations to the thermal conductivity is difficult to be reconciled with the gapless QSL picture.^[50] One possibility is that the severe disorder effect caused by the random mixing of Mg²⁺ and Ga³⁺ makes the otherwise itinerant spinons localized and thus not conduct heat.^{[43,44,48,50,52]} However, the disorder is considered to be detrimental to the QSL phase for this compound.^[53] Ma et al.^[51] have carried out measurements on YbZnGaO₄, a sister compound of YbMgGaO₄, utilizing various techniques, including d.c. susceptibility, specific heat, INS, and ultralow-temperature thermal conductivity. They have found that a spin-glass phase can explain the experimental observations in YbZnGaO₄: including no long-range magnetic order, prominent broad excitation continua observed by INS, and absence of magnetic thermal conductivity. By analogy, they suggest the spin-glass phase is also applicable to YbMgGaO₄.

The spin-glass phase, with frozen, short-range correlations below the freezing temperature T_f,^[136–138] can be identified from the a.c. susceptibility. Ma et al. have performed such measurements on both YbMgGaO₄ and YbZnGaO₄ with temperatures spanning about 3 decades, ranging from 0.05 K to 4 K. Some of the results are shown in Fig. 14. In both compounds, they observe strong frequency-dependent peaks below 0.1 K, evidencing a broad distribution of the spin relaxation times around T_f, typical for a spin glass.^[136–142] They consider disorder and frustration to give rise to the spin-glass phase.

Fig. 14.

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	Fig. 14. (color online) (a) and (b) Temperature dependence of the a.c. susceptibility (χ′) for YbMgGaO4 and YbZnGaO4, respectively. In the insets, χ′ in an extended temperature range up to 4 K are plotted. Dashed lines indicate the Curie–Weiss fits for the 100-Hz data. From Ref. [51].

However, a μSR study on YbMgGaO₄^[47] shows that there is no signature of spin freezing down to 0.07 K, which is already below the T_f reported in the a.c. susceptibility measurements.^[51] One possible origin of this discrepancy is that these two techniques cover different time scales: the former and latter probes are sensitive to fluctuations with frequencies larger and smaller than 10⁴ Hz, respectively.^[143] Furthermore, as estimated from the INS results, the portions of the spectral weight in the elastic channel over the total weight, are 16% and 13% for YbMgGaO₄ and YbZnGaO₄, respectively.^[45,51] These roughly represent the portions of moments that have been frozen. All these together may cause difficulties to detect the spin freezing by μSR.

Another important feature for a spin glass is that in the d.c. susceptibility measurements, there should be a cusp at T_f, where the zero-field-cooling and field-cooling susceptibility begin to separate from each other. At present, there are no such data available for YbMgGaO₄ and YbZnGaO₄, so performing d.c. susceptibility down to temperatures below 0.1 K will be useful to further clarify the ground state of these compounds.

For an organic compound such as κ-(BEDT-TTF)₂Cu₂(CN)₃ and EtMe₃Sb[Pd(dmit)₂]₂, disorder effect is expected to be significant, and it is unclear whether the QSL phase can survive in the presence of strong disorder.^{[11,21,29–32,144–146]} Moreover, as we discuss in Subsection 2.2. for κ-(BEDT-TTF)₂Cu₂(CN)₃, the specific heat indicates a gapless ground state. On the other hand, thermal conductivity measurements reveal no contributions from the magnetic excitations, inconsistent with the gapless QSL state. How to reconcile these contradicting results? Is it because the disorder effect makes the spinons localized and thus not conduct heat? Or the ground state is not a QSL at all? At the current stage, we believe these are open questions, calling for further investigations.

For the most heavily studied kagomé compound, ZnCu₃(OH)₆Cl₂, disorder also plays an important role. In particular, there are 5%–15% excess Cu²⁺ replacing the nonmagnetic Zn²⁺, which induces randomness in the magnetic exchange coupling.^[76,147,148] It is believed that such disorder can be accountable for many experimental observations.^{[76,84,147,149–155]} As an example, by considering the Cu impurities, Han et al.^[85] estimate a spin gap of 0.7 meV in the kagomé layer, close to that obtained from the NMR results.^[80]

For QSL candidates, frustration is strong. In the presence of strong disorder, the spin-glass phase is often observed, as disorder and frustration are two important ingredients for a spin glass.^{[136–138,156]} A spin glass mimics a QSL in many aspects — it maintains short-range spin–spin correlations, so in the susceptibility, specific heat, and neutron diffraction measurements, it lacks the signature of a long-range magnetic order; moreover, as demonstrated in Ref. [51], a spin-glass phase can also produce the continuous INS spectra, which is arguably the strongest evidence for a QSL so far. Therefore, in the quest for QSLs, the spin-glass phase which can give rise to spin-liquid-like features must be excluded first before labeling the candidate as a QSL.

Based on discussions above, we now give several perspectives:

(i) Although great progress has been made in theory,^{[11,21,29–40]} it still lacks a proposal for the defining feature of a QSL that can be detected directly from experiments. At present, observations of the continuous magnetic excitation spectra in INS measurements have often been taken to be the most reliable evidence for a QSL.^[45,46,82] However, this is necessary but not sufficient evidence for the fractionalized excitations.^[51] A feasible direct proposal to identify a QSL should greatly boost this field.

(ii) As we discussed above, there appear to be no ideal QSLs so far. Materials wise, does there exist a QSL candidate with large magnetic exchange interactions, little disorder, and minimal extra interactions that produce the static magnetic order? In the past, most attention had been paid to materials with a triangular or kagomé lattice where strong geometrical frustration is present.^[11] Now, studying the SOC-assisted Mott insulators with anisotropic bond-dependent Kitaev interactions on the honeycomb lattice may offer new possibilities.^[18,19,105] For instance, very recently, H₃LiIr₂O₆ has been suggested to be a Kitaev QSL.^[157]

(iii) According to Anderson’s proposal, high-temperature superconductivity can emerge from QSLs.^[25–27] There have been some successes in making QSL candidates superconducting by applying pressures to some organic compounds.^{[33,61,69,158,159]} However, another more common route to achieve superconductivity — via chemical doping, has not been successful so far.^[160] Is it because there is no ideal QSL candidate so far? Will chemically doping an ideal QSL eventually lead to high-temperature superconductivity as predicted? In this aspect, recent advances in doping using electric-field gating may offer some assistance.^[161–166]

To summarize, we review the recent progress on QSLs, especially on the magnetic-field measurements on several QSL candidates, including the geometrically-frustrated triangular and kagomé compounds, including YbMgGaO₄, YbZnGaO₄, κ-(ET)₂Cu₂(CN)₃, EtMe₃Sb[Pd(dmit)₂]₂, and ZnCu₃(OH)₆Cl₂, and the Kitaev material α-RuCl₃ with the honeycomb lattice. While there are many experimental evidences showing that they are promising candidates for QSLs, there are also some evidences that may be used to argue against the QSL picture. As such, we provide several perspectives hoping to stimulate further investigations. We anticipate that continuous efforts will be paid off by the discovery of more fascinating physics and ideal candidate materials.

Before ending this review, we note that there are many other materials that have been proposed to be QSLs. We list a few examples below:

(I) Na₄Ir₃O₈ is a widely studied QSL candidate with the hyperkagomé lattice.^[167–170] In the initial report^[167] that suggested it to a QSL, spin freezing indicative of a spin-glass phase at T_f = 6 K was observed. The frozen moments were estimated to be less than 10% of the total moments and were thus ignored. Later on, both μSR^[169] and NMR measurements^[171] showed that the spins are frozen and maintain short-range correlations in the ground state.

(II) Kagomé compounds ZnCu₃(OH)₆SO₄^[172] and Zn-substituted barlowite Cu₃Zn(OH)₆FBr,^[173,174] and a hyperkagomé material PbCuTe₂O₆.^[175,176] In Cu₃Zn(OH)₆FBr, it has been shown that the magnetic field dependence of the gap extracted from the NMR data is consistent with that given by fractionalized spin-1/2 spinon excitations.^[173]

(III) A triangular spin-1 material Ba₃NiSb₂O₉.^[177]

(IV) Ca₁₀Cr₇O₂₈, a system with complex structure, and more interestingly, with ferromagnetic interactions.^[178–180] In this compound, although Balz et al. found that there is no static magnetic order in the μSR measurements, they observed frequency dependent peaks in the a.c. susceptibility, which is characteristic of a spin glass.^[178] However, they argued that the spin-glass phase could be ruled out by doing the Cole–Cole analysis for the a.c. susceptibility data.^[178]

(V) Very recently, a protypical charge-density-wave compound 1T-TaS₂ with the David-star structure has attracted a lot of attention due the possibility of realizing the QSL state.^[181–185]

In this review, we did not discuss these materials in detail due to the limited space. Readers who are interested in them can refer to the above references and the references therein.