Muon Spin Relaxation Studies on Quantum Spin Liquid Candidates

Zhu Zi-Hao Shu Lei

1. State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China

2. Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

3. Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

Abstract: Quantum spin liquid is a novel magnetic state without long-range order even at zero temperature due to strong quantum fluctuations. The ground state of quantum spin liquid cannot be described by order parameters, and there is no symmetry breaking in this exotic state, which means that the realization of quantum spin liquid states will break through the paradigm of Landau’s theory. Studies on quantum spin liquid will be helpful for the understanding of the mechanism of high-temperature superconductivity and the application of quantum calculation and quantum information. Although there are many advances in the theory, no material is confirmed to be a real quantum spin liquid so far, making it important to realize a quantum spin liquid material and confirm its properties. Muon spin relaxation,a powerful technique that is extremely sensitive to magnetic fields, has been widely used in the study of quantum spin liquid candidates. Muon spin relaxation can observe whether there is long-range order in the ground state, and measure the fluctuation rate in the system, both of which are the basic properties of quantum spin liquid. This article gives a brief introduction to the quantum spin liquid state and muon spin relaxation technique. Then recent experimental works, especially muon spin relaxation experiments, on different systems, including one-dimensional antiferromagnetic Heisenberg chain (copper benzoate), triangular lattice(YbMgGaO4, NaYbO2 and TbInO3), kagom′e lattice [ZnCu3(OH)6Cl2 and Tm3Sb3Zn2O14],honeycomb lattice (Na2IrO3 and α-RuCl3) and pyrochlore lattice (Tb2Ti2O7, Pr2Ir2O7 and Ce2Zr2O7) are reviewed.

Key words: Strongly-Correlated System; μSR; Quantum Spin Liquid

I. INTRODUCTION TO QUANTUM SPIN LIQUID

FIG.1. Characteristic lattices of two-dimensional QSL candidates. (a). Triangular lattice with geometrical frustration. (b). Kagom′e lattice with geometrical frustration. (c).Frustration on triagular lattice. (d). Honeycomb lattice and Kitaev model.

Quantum spin liquid (QSL) is a novel magnetic state proposed by P.W.Anderson in 1973[1]. Normally,magnetic systems should be in ordered states, for example, ferromagnetic (FM) state or antiferromagnetic(AFM) state, at low temperatures. However, due to geometrical frustration or quantum frustration,there is no long-range order (LRO) but strong fluctuations in QSL down to zero temperature. After the discovery of cuprate high-temperature superconductors, Anderson introduced resonating valence bond (RVB) theory of QSL to understand the mechanism of high-temperature superconductivity[2]. Recently, more and more spin frustrated materials have been discovered[3-6]. Spin frustration enhances quantum fluctuations significantly, which will suppress LRO and derive complex novel quantum phenomena. It is expected that the novel states like QSL will break Landau’s symmetry breaking theory, showing novel behavior like fractional elementary excitations. Study on QSL is of great help to understand the mechanism of high-temperature superconductivity, quantum phase transition and quantum critical phenomena, and may widen the paradigm of theory of condensed matter physics. Besides, QSL has the potential to be used in the realization of topologyprotected quantum information and quantum calculation.

A. Structure and Geometrical Frustration

Although there are many QSL candidates discovered, none of them is convinced to hold a QSL ground state. The elementary excitation of QSL is a fractional excitation calledspinonwhich is believed to be a chargeless, spin-1/2, and highly mobile fermion. These QSL candidates vary from one-dimensional to threedimensional systems.

The simplest model is one-dimensional AFM Heisenberg chain (AFHC) whose ground state has no LRO.Although there is no frustration in this structure,and this is not a true QSL, but one-dimensional Luttinger liquid, the flip of spin which is the excitation in this model, can be seen as spinon.

There are richest frustrated lattices in twodimension, for example, triangular lattice, kagom′e lattice, and several other lattices with AFM interaction(Fig.1.a and b). Ising model on triangular lattice with AFM interaction is the best illustration of geometrical frustration, although the ground state of this model is not a QSL (Fig. 1. c). Consider three spins on a single triangle. All three spins cannot be anti-parallel,indicating that the ground state of this triangle is 6-fold degenerate. On a larger triangle lattice with more atoms, the degeneracy of ground state will be tremendous, which indicates strong fluctuation. This is one of the most important properties of QSL.

FIG. 2. Lattice of magnetic atoms in pyrochlore structure.(a). Ground state, which shows “2 in, 2 out” in both tetrahedrons. (b). Excited state,which shows a pair of magnetic monopoles.

In the three-dimensional cases,there is one famous structure called pyrochlore lattice, or spin ice, since it has similar structure and properties with ice. Hydrogen in the ice composites opposite tetrahedrons, replaced by magnetic atoms,and then a pyrochole lattice is achieved (Fig. 2). The spins at each vertex of the tetrahedron can only point to the center or opposite.Under FM interaction, ground state of a tetrahedron is “2 in, 2 out”. If this system is excited, or one spin flips, then spins on a tetrahedron will be “3 in, 1 out”,while spins on one of the neighbor tetrahedrons will be“1 in, 3 out”. This pair of tetrahedrons is called a pair ofmagnetic monopoles, since the spins are no longer balanced. Spin ice is not generally QSL, but only in several cases, a quantum spin ice is a QSL candidate.

B. Kitaev Spin Liquid

Honeycomb lattice is free of geometrical frustration, however, based on which A. Kitaev proposed a theoretical model which can be exactly solved[7]. The Hamiltonianof this so-called Kitaev model can be written as:

wherex,yandzare coordinates defined locally as shown in Fig. 1. d, andKx,KyandKzare the FM Ising interaction on the respective bond. Based on this strongly anisotropic Kitaev interaction, Kitaev shows that the ground state of Kitaev model is QSL.Depending on the ratio betweenKx,KyandKz, this QSL state could be gapless or gapped,thus giving a possible approach to regulate the system and realize quantum calculation.

Many theorists came up with different theories about Kitaev model in detail, however, instead of K-itaev QSL states, several materials that are proved to have Kitaev interaction all hold ordered ground states. For example, Na2IrO3was reported to transit into zigzag AFM state at 15 K[8], andα-RuCl3has AFM transition at 7 K[9,10]. More complicated magnetic structure is realized inα-,β-,andγ-Li2IrO3,all of which have incommensurate counterrotating magnetic order with different magnetic structures[11-13]. Recently, one new Kitaev QSL candidate without magnetic transition, H3LiIr2O6, was discovered[14]. But the existence of impurities in the system restricts the study on the system.

C. Experimental Approach to Quantum Spin Liquid

QSL can be checked by several different measurements. The main purpose is to differentiate QSL from other states, for example, ordered state like FM or AFM, and spin freezing state without fluctuation like spin glass.

QSL is generally expected to be an insulator,which can be determined by resistivity measurements. By measuring the direct current (DC) magnetic susceptibility,one can see whether there is magnetic transition,and get information about the type of magnetic interaction, FM or AFM. Besides, the effective spin and Curie-Weiss temperature are both important parameters for study on QSL.

Specific heat is one of the most important measurements in the study on QSL to see whether there is magnetic phase transition, and whether there is spin freezing or spin glass behavior with temperature decreasing.Additionally, the behavior of magnetic specific heat at extremely low temperature can give information about the density of state of spinon, and the residual entropy at zero temperature can be determined.

Since QSL are generally insulators, thermal transport is the only available transport experiment. Thermal conductivity is the simplest form of thermal transport experiment. All electrons are localized, thus no contribution of electron is expected. Subtracting the contribution of phonon, we expect significant magnetic excitation contribution due to highly mobile spinon.Besides, similar to Hall effect, spinon can transport heat in the direction perpendicular to the external field,called thermal Hall effect.

The most direct method to measure the excitation of spinon is inelastic neutron scattering due to its sen-sitivity to momentum space. Unlike magnon or many other quasi-particles showing a sharp dispersion relation, spinon shows a very indistinct dispersion relation called spinon continuum.

To determine the fluctuation of the system, alternating current(AC)magnetic susceptibility,nuclear moment resonance (NMR), and muon spin relaxation(μSR) are used. These techniques can detect fluctuation at different frequencies. Besides, they can all detect spin glass or spin freezing in different ways.

A large amount of QSL candidates are oxides. It is hard to grow single crystals due to high melting points of raw oxide materials. Techniques like thermal conductivity measurement requires high quality single crystals,and inelastic neutron scattering requires a large amount of single crystals for analysis in detail. When single crystalline samples are not available,μSR is a powerful technique sinceμSR can be performed on single crystals or polycrystals .

No single technique at this moment can identify a QSL state. Exploiting various experimental techniques to identify a true QSL material is the key to the current research.

Although there is no persuasive QSL material at the moment,many theories have been developed in the past years. Theory in detail is beyond this review. Interested readers can follow more specific reviews[6].Based on Hubbard model, QSL can be viewed as a state during the Mott transition. Depending on the gauge symmetry of the system, there are several classes of QSL, for example,U(1) QSL and Z2QSL.

II. INTRODUCTION TO μSR TECHNIQUE

Muon spin relaxation/rotation/resonance(μSR)is a technique using muon as a local magnetic probe to study the physical properties of different systems. Due to the basic properties of muon,μSR can detect very small magnetic field(down to about 0.1 G).Besides,unlike techniques like magnetic susceptibility, which can only detect bulk properties,μSR can observe local magnetic properties. μSR can be used to study magnetic systems, superconductors and semiconductors. It can also be used on study of ionic transport and chemical reaction kinetics.

This section focuses on a brief introduction to theμSR technique. For more details, the reader is referred to several specified reviews[15].

A. Principle of μSR Technique

Muon is a basic particle, or a lepton in the standard model of particles. In 1936, muon was discovered in the cosmic ray by Carl D. Anderson and Seth Neddermeyer[16]. Its mass is 105.658 MeV/c2, about 207 times of an electron, or 1/9 of a proton. Muon or negative muon, is a charged particle with 1 elementary charge. InμSR experiments,the anti-muon,or positive muon, is used instead due to the Coulomb interaction from positive nuclei. Muon is a spin-1/2 particle, and its magnetic moment is about 3.18 times of a proton.The gyro-magnetic ratioγμof muon is 851.615 MHz/T,making muon sensitive to magnetic field. Muon is unstable,with a 2.197μs mean lifetime. It will decay into a positive electron (positron) and two neutrinos:

In μSR experiments, low-energy muon produced from pion decay is used. Pions are produced from the collisions of high-energy protons with the nuclei of an target, usually carbon (graphite). Pion, being very unstable, willβ-decay into a muon and a neutrino:

This process is parity violated, which is, in fact, one of the experiments aroused by T. D. Lee and C. N. Yang to prove their theory is correct[17]. Parity violation guarantees that the spin of muonSμis opposite to its momentummμ. Therefore, 100% polarized muon can be produced by selecting its momentum without any external field.

During μSR experiments, nearly 100% polarized muon is implanted into the sample and then stops there.Muons can be considered as an isotope of hydrogen(proton). Since muon is unstable, it will decay into positron quickly. Lee and Yang found that the momentum of positronmedepends on the direction ofSμ[17].The angular distributionW(θ) takes the form:

FIG.3. Schematic of 2 setups ofμSR experiments. (a). TF setup. (b). LF setup.

whereθis the angle betweenmeandSμ, anda(ε) is called asymmetry parameter, depending on the energy of positronε. If we consider all possibleε,a(ε)will give a mean value of 1/3, then, the positron alongSμwill be more than the opposite. Generally, if the polarization of muon is not 1, the asymmetry, which should be proportional to the polarization, will be smaller than 1/3.

To measure the asymmetry, we need to count positrons on opposite directions. Assuming that the counts on the two positron detectors at timetareN1(t)andN2(t), the time dependent asymmetry asy(t) can be defined as:

Thus, the factor of muon decay can be neutralized. S-ince the mean lifetime of muon is 2.197 μs, the time scale of the spectrum is about 15 μs.

Depending on the direction of initial muon spinSμ(0) and the positron detectors, there are 2 main setups of μSR experiments as shown in Fig. 3:transversal-field (TF) setup, and longitudinal-field(LF) setup. For TF setup, an external magnetic fieldBperpendicular toSμ(0) is applied, makingSμprecess aroundB, and the asymmetry will be cosine-like.For LF setup,an external magnetic fieldBor zero field(ZF) alongSμ(0) is applied. If the field is ideally uniform, there will be no relaxation of asymmetry, which means that the asymmetry is a constant. However,in samples, muons stopping at different muon sites will precess along different directions with different frequencies under local field,which means that the asymmetry will decay. By fitting the time spectrum with different relaxation functions, we can get the properties of different systems.

B. Relaxation Functions

Depending on the properties of different samples,the relaxation functions can be very different. Herein,we will only cover several most important conclusions which are generally used in the study of QSL in this review. For readers interested in the proof process,please follow other reviews[15].

Single crystals or polycrystals may be used in the study of QSL. ZF-μSR can be used to check the presence of magnetic order, and LF-μSR under different fields can be used to see if the system is static or dynamic, and get the information about the fluctuation rate.In real samples, the fitting function could be combination or product of the theoretical relaxation functions.The former one indicates that there are multiple muon sites, while the latter one suggests that muon is affected by different effects at one site. Since the relaxation functions could have similar behavior, we need to differentiate between these confusing functions carefully.

1. ZF-μSR

Spin of muon implanted into non-magnetic material will not depolarize,which means that the asymmetry is a constant. Hence, we can use non-magnetic element like silver to make a sample holder. Muon might stop in the holder, but it will not change the relaxation process. It will only add a constant background to the total asymmetry.

For system with no magnetic order,but with Gaussian distribution of static field, the relaxation functionP(t) is Kubo-Toyabe (KT) function:

whereσ=γμ?,γμis the gyro-magnetic ratio of muon,and ?is the standard deviation of field distribution[18].The most significant property of KT function is the tail that recovers to 1/3 as shown in Fig. 4.

For system with no magnetic order, but with dynamic field distribution, provided that the correlation frequency isνc, ifνct ?1, which is called motional narrowing limit, the relaxation functionP(t) will be:

For system with LRO,we expect cosine oscillation of polarization. In the perfect single crystal with magnetic moment perpendicular toSμ(0),it should be only a cosine. But for polycrystal with random moment direction, the relaxation functionP(t) will be:

More often, the magnitude of field could be slightly different, which will give:

This is often called internal field relaxation function.

2. LF-μSR

For system that can be described by ZF-KT function, if we apply a longitudinal fieldB=ω0/γμ, the relaxation functionP(t) will be:

where theσhere is the equal to theσin the ZF-KT function[18]. As shown in Fig.4,withBincreasing,the tail increases,and the relaxation slows down. This phenomenon is called decouple. This indicates that muon spin is decoupled from the internal field of the sample,but affected by the external field. Typically, the external field needs to be about 10 times of the internal field to be fully decoupled.

FIG. 4. Plot of characteristic KT functions under zero field and selective longitudinal field.

For dynamic system with correlation frequencyνc,if we apply a longitudinal fieldB=ω0/γμ, the relaxation functionP(t) will still be:

Supposing that the time correlation of spins in the systemS(t) takes the form

theλhere along with ZFλsatisfies

whereBlocis the standard deviation of local field. This equation is called Redfield formula. More generally,supposing the time correlationS(t) takes the form

based on which one can derive a more general relation

C. Why μSR?

To verify a true QSL, one must make sure that it has strong fluctuations down to zero temperature. μSR is one of the optimal techniques for several reasons.

? ZF-μSR can detect magnetic order, either longrange or short-range,which is a decisive signature for a QSL candidate.

? As the temperature goes down,the spins are usually frozen at some point, which will give a static spin-glass-like behavior. In ZF-μSR experiments,spin glass behaves like a perfect realization of KT function,which can be clearly observed due to its unique 1/3 tail.

? As mentioned in section II.B.2,muon spin will be decoupled from the static field distribution like nuclear moments or spin glass by strong external field. For spin glass, whose internal field is typically about 10 G, an external 100 G field is large enough to see the decouple effect.

? Unlike static systems, there is no decouple effect in a strongly dynamic system. The relaxation shows a robust behavior under strong fields (typically 3000 G).

? To figure out how dynamic the system is,we need to know its fluctuation rateνc. We can getνcby fitting the field dependence of the relaxation rate with Redfield formula.

In combination with other techniques, μSR is a powerful tool to determine whether a QSL candidate is real QSL or not.

III. ONE-DIMENSIONAL CASES

A. Antiferromagnetic Heisenberg Chain

1. Copper Benzoate

Copper benzoate,Cu(C6H5COO)2·3H2O,is a rare ideal realization of spin-1/2 AFHC[19]. As shown in Fig. 5, it has monoclinic crystal structure, with very long interchain distance. The intrachain couplingJ/kB≈18.6 K, while the interchain coupling is very weak. The specific heat measurement shows that there is no magnetic phase transition,which is also confirmed by neutron scattering[20]. The magnetic susceptibility measurements show a very broad peak around 10 K under high fields,which can be interpreted by sine-Gorden theory[21]. All these experimental evidences indicate that copper benzoate is an ideal spin-1/2 AFHC. Besides,inelastic neutron scattering also finds evidence of spinon continuum[20].

FIG. 5. Experimental results of copper benzoate. (a) Crystal structure of copper benzoate [19]. (b)Magnetic susceptibility of single crystals under 2 T field‖c [21]. (c)Asymmetry spectra of ZF-μSR on single crystals at selective temperatures. Lines are the fitting results. [22]. (d) Asymmetry spectrum of ZF-μSR and LF-μSR under 100 G on single crystals at 20 mK. Lines are the fitting results. [22].

ZF-μSR experiments were carried out on single crystals down to 20 mK, with initial muon spin parallel to thec-axis[22]. No magnetic order is observed down to 20 mK. There is no significant differences in the asymmetry spectra at different temperatures. Subtracting the constant background, the spectra can be well described by

whereGKT(σ,t) is the ZF-KT function (Eq. 6). The fitting result gives temperature independentλ=0.15 μs?1,and internal field distributionσ/γμ=3.4 G,which indicates a static ground state. Furthermore,LF-μSR shows that the internal field can be easily decoupled by 100 G field, which is consistent with the ZF-μSR results.

In summary, μSR experiments show that there is no LRO in copper benzoate down to 20 mK, and no strong dynamics is observed in this system.

IV. TWO-DIMENSIONAL CASES

A. Triangular Lattice

1. YbMgGaO4

Once YbMgGaO4was discovered in 2015, it aroused experimentalists and theorists’ attention[23].The Yb3+ions in this system form triangular lattice with AFM interaction, and its effective spinSeffis 1/2[23]. Compared with the in-plane interaction, the inter-plane interaction is very weak, resulting in very good two-dimensional properties. The magnetic susceptibility and specific heat measurements show that there is no phase transition or spin glass behavior down to 60 mK[23-25]. Inelastic neutron scattering experiments show the presence of spinon excitations[24,26,27].However, thermal conductivity measurements indicate a spin freezing ground state[25], and a frequency dependent cusp, suggesting a spin-glass-like transition, was observed around 0.1 K in AC susceptibility[28]. YbMgGaO4is believed to be a gaplessU(1)QSL, and detailed theories are developed by many theorists [29-31].

FIG. 6. Basic properties of YbMgGaO4. (a) Crystal structure of Yb3+ in YbMgGaO4[24]. (b)Magnetic contribution of specific heat of single crystals under selective fields [26].(c) Magnetic susceptibility χ of single crystals under zerofield cooling (ZFC) and field cooling (FC) under 1 T [24].Inset: Inverse of magnetic susceptibility χ?1 at low temperature, fitted by Curie-Weiss law (dashed line). (d) Field dependence of magnetization of single crystals at 1.7 K [26].

FIG. 7. μSR results of YbMgGaO4[32]. (a) Asymmetry spectra of ZF-μSR at selective temperatures with initial muon spin along c-axis. Lines are the fitting results. (b)Temperature dependence of fitting parameter β. (c) Temperature dependence of ZF relaxation rate λZF and magnetic susceptibility χ, which shows similar behavior. (d)Asymmetry spectra of LF-μSR at 0.07 K under selective magnetic fields. Lines are the fitting results. (e) Field dependence of LF relaxation rate λLF at 0.07 K. The line is the fitting result.

The μSR results are shown in Fig. 7. ZF-μSR experiments were carried out on single crystals in 2 directions,with initial muon spin perpendicular and parallel to thec-axis. The results of both sets of experiments are very similar[32]. There is no signature of LRO down to 70 mK. Subtracting the constant background,the spectra can be well described by

The stretched exponential function could indicate dynamic spins. With decreasing temperature, the exponentβdecreases from 1 to 0.6,suggesting the absence of spin glass, in whichβis expected to drop to 1/3[33,34].Instead of continuous increase, relaxation rateλZFsaturates at 0.4 K, which is believed to be a signal of persistent spin dynamics.

LF-μSR supports the conclusion above. The spectra at 0.07 K show a robust relaxation under up to 0.18 T field, which can also be well described by the same function as ZF spectra (Eq. 17). The field dependence of LF relaxation rateλLFcannot be fitted by Redfield formula (Eq. 13), but can be described by the general form(Eq.15),indicating the long-time spin correlations. The fitting result givesx= 0.66(5) andν=9.4(6) MHz.

However, another μSR work on YbMgGaO4was reported recently[35]. By using weak transverse fieldμSR (wTF-μSR), it is proved that there are 2 muon stopping sites by the analysis on Knight shift. Normally, an integration of two exponential terms is very similar to a stretched exponential term, which is very confusing to the researchers. Based on the conclusion made by wTF-μSR, the researchers believed that the real fitting function for LF-μSR is an integration of two exponential terms due to two stopping sites

Besides, due to the existence of nuclear moments, a static KT relaxation was observed. Except for these differences,they both agree that there is no spin freezing but persistent spin dynamics down to the base temperature[32,35].

In summary, most of the experiments show that YbMgGaO4could have a QSL ground state. However,later on, it turns out that the persistent spin dynamics could arise from the structural disorder due to exchange between Mg2+and Ga3+ions,which makes it not a true QSL[31,36-38].

2. NaYbO2

After YbMgGaO4was studied thoroughly, researchers wanted to find new compounds with similar structure but without the fatal exchange disorder. In 2018, a brand new large family of QSL candidates was reported to have a much simpler structure but keeps the frustrated triangular lattice[39,40]. One of them is NaYbO2, which is so close to YbMgGaO4that similar properties were expected.

FIG. 8. Basic properties of NaYbO2[41]. (a) Crystal structure of Yb3+ in NaYbO2. (b) Temperature dependence of inverse of magnetic susceptibility χ?1. The red line is Curie-Weiss fit in low temperature range, which gives θCW=-10.3(8) K.(c)Specific heat of NaYbO2 down to 80 mK under zero field. The result of NaLuO2 is used as the phonon contribution of specific heat. Almost all of the magnetic entropy are released during freezing. (d) Specific heat under different fields. The peak under 5 T is likely due to second order transition. Inset: Fitting of low temperature specific heat under zero field with power law.

The basic properties of NaYbO2are shown in Fig.8. No phase transition is observed under low fields down to 80 mK, but a sharp peak in specific heat is observed under 5 T,which is due to field-induced phase transition[41,42]. The AC magnetic susceptibility measurements also suggest that there is no spin freezing in this compound[39,41]. However, the single crystalline NaYbO2has not been synthesized successfully yet, resulting in the lack of several key measurements like inelastic neutron scattering and thermal conductivity.

The ZF-μSR results are shown in Fig. 9[43]. No LRO is observed down to 0.1 K. The spectra can be well described by

whereb0is the constant background, the two exponential terms indicate two muon stopping sites with respective relaxation rateλ1andλ2, andfis the fraction of the first site. This fitting function is very close to a stretched exponential,which needs careful analysis to distinguish.fis temperature independent with its value close to 0.5, which suggests the same population at this two sites.λ1(about 0.9 μs?1) is significantly larger thanλ2(about 0.6 μs?1). Bothλ1andλ2shows a significant increase around 5 K, and saturates below 1 K, which suggests persistent spin dynamics. They performed LF-μSR at 1.5 K under various field, which also shows a robust relaxation up to 0.1 T field.

FIG.9. μSR results of NaYbO2[43]. (a)Asymmetry spectra of ZF-μSR at selective temperatures with initial muon spin along c-axis. Lines are the fitting results. (b) Temperature dependence of fitting parameter λ1.

In summary,the studies on NaYbO2show that it is a good QSL candidate. However, the failure in synthesis of single crystals results in the lack of measurements with powerful techniques like inelastic neutron scattering and thermal conductivity. Single crystals of several other compounds from the same family, i.e. NaYbS2,NaYbSe2, and CsYbSe2, are much easier for synthesis,making them more convenient to study with various techniques[40,44].

3. TbInO3

Recently, another QSL candidate TbInO3was reported[45-47]. Instead of an ideal 2D triangular lattice,the Tb3+ions at 2 different sites lie in 2 layers close to each other. Tb3+ions at one site form an ideal honeycomb lattice, and the ions at the other site occupy the center of the resulting hexagon(Fig. 10.a). No magnetic transition is observed by magnetic susceptibility and specific measurements down to 1.5 K[45,46]. The broad peak of specific heat around 10 K is arised from crystalline electric field (CEF) effect, as confirmed by inelastic neutron scattering measurements[46]. Furthermore, inelastic neutron scattering measurements also find evidence of spinon continuum[47].

The μSR results are shown in Fig. 11[46]. Neither LRO nor spin freezing is observed, and the authors believed that the initial asymmetry loss is due to the large internal magnetic field generated by Tb3+. The spectra can be well described by

The increase of relaxation rateλindicates the slowing down of spin fluctuations,and the saturation below 2 K is an evidence of persistent spin dynamics. Additionally,λbelow 10 K can be fitted by

FIG. 10. Basic properties of TbInO3. (a) Structure of Tb3+ ions [45]. Red and purple spheres are Tb3+ at different sites, which forms honeycomb lattice each, while form distorted triangular lattice together. (b) Magnetic susceptibility χ of single and polycrystals, and Curie-Weiss fit to χ?1 [46]. (c) Temperature dependence of specific heat subtracting phonon contribution [45]. Inset: Results under different fields. (d) Powder inelastic neutron scattering with initial energy Ei = 11 meV at 5 K [46]. The relatively dispersionless scattering at 2 meV can be attributed to CEF excitations.

The LF-μSR measurements also suggest a dynamically fluctuating state. In summary, μSR measurements support that TbInO3is a good QSL candidate.

FIG. 11. μSR results of TbInO3[46]. (a) Asymmetry spectra of ZF-μSR at selective temperatures. Lines are the fitting results. Inset: Asymmetry spectra of LF-μSR at 100 mK under selective fields. (b)Temperature dependence of fitting parameter λ. The solid line is the fitting results by Eq. 21.

B. Kagom′e Lattice

1. ZnCu3(OH)6Cl2

Cu4(OH)6Cl2is a crystal with pyrochlore structure. Doped by non-magnetic Zn2+, this compound shows rich physical properties. One doping is ZnCu3(OH)6Cl2, where Cu2+ions form twodimensional kagom′e lattice (Fig. 12. a). Compared with triangular lattice, kagom′e lattice has stronger quantum fluctuations due to larger geometrical frustration and smaller coordination number.

FIG. 12. Experimental results of ZnCu3(OH)6Cl2. (a)Transformation from pyrochlore structure (left) to kagom′e lattice (right) [49]. (b) Specific heat results under various fields [49]. Inset: Specific heat results with wider temperature range. (c) Magnetic susceptibility χ results [48]. Inset:χ?1 results with a Curie-Weiss fit. (d) ZF- and LF-μSR spectra [53]. Lines are the fitting results. (e) Temperature dependence of relaxation rate λOH = 1/T1[53]. The line is based on their theoretical calculations.

The experimental results are shown in Fig.12. No magnetic transition is observed by magnetic susceptibility measurements down to 100 mK[48,49], which is also supported by specific heat and AC magnetic susceptibility measurements[49]. The inelastic neutron scattering on ZnCu3(OD)6Cl2shows a continuous excitation spectrum[50], and NMR experiments also indicate QSL ground state[51]. However, the specific heat and neutron scattering experiments suggest a gapless ground state, while NMR results indicate a gapped ground state. Later on, another neutron scattering experiment shows that the gapless excitation could arise from the exchange disorder between Zn2+and Cu2+[52]. It remains controversial whether the ground state of this system is gapped or gapless.

The ZF- and LF-μSR spectra are shown in Fig. 12. d[53],which is same as their former results[54].Due to the existence of OH?and Cl?, the ZF spectra can be well described by

which consists of two terms due to two muon stopping sites with only one adjustable parameterλOH, andfis the fraction of the site close to the OH?.GKT(t)is KT function (Eq. 6), andGOH(t) takes the form

which is based on Lord’s theory[55]. Hence the oscillation in the ZF spectrum is not due to magnetic order,but the interaction between muon and OH?. The LF spectra can be well described by

where the influence of OH?and Cl?are covered by the external field. The fast decay at 20 mK indicates that there is no spin freezing. Temperature dependence of ZF relaxation rateλOHis shown in Fig. 12. e.The rapid increase ofλOHshows spin slowing down with temperature decreasing, and the saturation below 0.4 K suggests persistent spin dynamics. In summary, μSR experiments support the conclusion that ZnCu3(OH)6Cl2has a gapless QSL ground state.

2. Tm3Sb3Zn2O14

In 2016, a new family of pyrochlore derivativesRE3Sb3M2O14(REis various rare-earth elements,andMis Mg or Zn) with kagom′e lattice was reported[56]. This system shows rich physical properties[57-60]. The group of kagom′e lattices formed by magnetic rare-earth atoms were separated by another group of kagom′e lattice formed by non-magnetic atoms, which provides large inter-plane distance and good two-dimensional properties, including the emergent kagom′e Ising order[57], non-magnetic singlet state[60],spin glass[60]and even QSL[60].

FIG. 13. Basic properties of Tm3Sb3Zn2O14[43]. (a) K-agom′e lattice composed of Tm3+ ions in Tm3Sb3Zn2O14.The in-plane Zn2+ ions are also shown. (b) Temperature dependence of inverse of magnetic susceptibility χ?1. The blue line is Curie-Weiss fit between 10 K and 20 K, which gives θCW=-18.6 K. (c) Specific heat coefficient (Cp/T) per mole Tm down to 0.3 K under zero field with phonon contribution subtracted. The green dashed curve indicates the fitting result of Schottky anomaly. The blue line is the experimental magnetic entropy. (d) Low temperature specific heat coefficient (Cp/T) as a function of T2 under various fields. The red line is a linear fit to the zero field data, indicating a finite interception γ(0) = 26.6(1) mJ mol-Tm?1 K?2.

FIG. 14. ZF-μSR results of Tm3Sb3Zn2O14[43]. (a) Asymmetry spectra at selective temperatures. Lines are the fitting results. (b) and (c) Temperature dependence of fitting parameters λ and β. The curve in (b) is a guide to eyes,indicating the persistent spin dynamics below 2 K.

One QSL candidate in this family is Tm3Sb3Zn2O14, where Tm3+can be regarded as an ion with effective spin-1/2[61]. The basic properties are shown in Fig. 13. The specific heat and magnetic susceptibility measurements show that there is no magnetic transition or spin freezing down to 0.3 K,as confirmed by AC magnetic susceptibility results.Besides, the finiteγ(0) suggests a constant density of states at low energies (Fig. 13. d), indicating that Tm3Sb3Zn2O14has a Z2QSL ground state.

ZF-μSR was carried out down to 20 mK on polycrystalline samples. No magnetic order or spin freezing is observed(Fig. 14). The spectra can be well described by

The increase of relaxation rateλindicates the slowing down of spin fluctuations,and the saturation below 2 K suggests the existence of persistent spin dynamics. The stretched exponentβis close to 0 and almost temperature independent, suggesting the absence of impurity induced relaxation[15]. In summary, Tm3Sb3Zn2O14has a possible gapless Z2QSL ground state.

C. Kitaev Spin Liquid

1. Na2IrO3

Na2IrO3,a Mott insulator with strong spin-orbital coupling interaction, is supposed to be the first system which can realize Kitaev spin liquid[8]. Magnetic susceptibility measurements show that this system has effective spin-1/2, which satisfies Kitaev model. Unfortunately, specific heat result finds magnetic transition aroundTN ～15 K. Additionally, neutron scattering shows that it has LRO with zigzag structure(Fig. 15.a)[62].This magnetic structure can be explained by inducing a Heisenberg interaction term to Kitaev model, which shows a rich phase diagram depending on the energy scales[63]. Although the ground state of Na2IrO3is not Kitaev spin liquid, it still is an important system which satisfies Kitaev physics.

ZF-μSR experiments confirm the existence of LRO[62,64]. There is no oscillation but exponential relaxation aboveTN, while oscillations are observed vividly belowTN, indicating the transition into LRO(Fig. 16). The spectra belowTNcan be well described by

FIG. 15. Basic properties of Na2IrO3. (a) Magnetic structure of zigzag order [62]. (b)Specific heat coefficient(Cp/T)under zero field and phonon contribution [8]. (c) Temperature dependence of the inverse of magnetic susceptibility χ?1 [8]. The red curve is Curie-Weiss fitting result. (d)Temperature dependence of magnetic susceptibility on single crystals in different directions and polycrystals [8].

where the first two terms account for two different muon stopping sites with ratio of about 9:1, the third term accounts for the muons polarized parallel to the local magnetic fields, and the last term is background from the silver sample holder. The fitting results show that there is a typical second-order transition, which is consistent with other experiments. The local field at two muon sites have different temperature dependence,which could come from muon sites near the stacking fault planes.

2.α-RuCl3

α-RuCl3is another material believed to be a candidate that can realize Kitaev physics[65,66],where Ru3+ions haveJeff= 1/2[67]. There is a structure transition from monoclinic lattice to rhombohedral lattice aroundTS ～150K[68,69].α-RuCl3turns into ordered state belowTN, whereTN ～14 K for powder sample, while 2 critical temperaturesTN1～7 K andTN2～14 K are found for single crystalline samples[70].Neutron scattering experiments reveal that this is due to the stacking fault, whereTN1comes from ABC stacking, andTN2comes from AB stacking[71]. Additionally, similar to Na2IrO3, zigzag magnetic order is also found by inelastic neutron scattering on single crystalline sample[72].

FIG. 16. μSR results of Na2IrO3[62]. (a) ZF-μSR spectra above and below TN. (b) and (c) Temperature dependence of fitting parameter λ and ν, respectively.

FIG.17. Basic properties of α-RuCl3. (a)Crystal structure of α-RuCl3[71]. (b) Specific heat Cp under zero field and magnetic susceptibility results [73]. (c)and(d)Temperature dependence of magnetic susceptibility and its inverse in different directions [72]. The powder average susceptibility is defined as χave =(2χab+ χc)/3.

Although the ground state ofα-RuCl3under zero field is ordered, it is possible that there is a QSL phase at higher field shown by heat capacity[68,73], NMR[74,75], thermal conductivity[73],neutron scattering[72,76]and many other techniques. It is shown that the LRO is suppressed by in-plane field,and disappears at 8 T. The phase at high field is controversial, especially whether it is a QSL phase, and whether it is gapped or gapless.

FIG. 18. ZF-μSR results of α-RuCl3[77]. (a) Asymmetry spectra at selective temperatures. The solid line is the fitting result with one (11 K T 14 K) or two (T 11 K)oscillating terms. (b)Fourier transform spectra of the asymmetry spectra.

μSR experiments were carried out by different groups[77,78]. Asymmetry spectra at low temperatures show clear oscillations,suggesting the existence of magnetic order. Two peaks are found in Fourier transform spectra at low temperatures, suggesting multiple muon stopping sites. The spectra can be well described by

whereλ2can be described by a second order transition[77].Further density functional theory(DFT)calculations find the most possible stopping sites, corresponding to the experiment results.

V. THREE-DIMENSIONAL CASES

A. Pyrochlore Lattice

One most important family of pyrochlore lattice can be formulated asA2B2O7, whereAis often a rareearth ion (usually magnetic), andBis often a nonmagnetic ion, both of which form pyrochlore sublattices. Since the choice ofAandBions are various,this family show rich physical phenomena, including metallic state, ordered state, spin glass state, spin ice state and spin liquid state. Interested readers can follow some other relevant reviews[79].

1. Tb2Ti2O7

Tb2Ti2O7, first reported in 1999, was well-studied in the following 20 years[80]. Due to the sensitivity to stoichiometry, strong sample dependence makes it confusing to figure out the properties of Tb2Ti2O7[81]. The temperature dependence of zero-field cooling (ZFC)and field cooling (FC) magnetization shows an obvious split, thus suggesting spin-glass-like behavior[82].The magnetic susceptibility measurements can be well described by Curie-Weiss law, indicating the AFM interaction in this system[80,83]. Further analysis shows that Tb2Ti2O7can be described by Ising model, which should be ordered at low temperatures[84]. However,no evidence of ordered state is observed. Additionally, AC magnetic susceptibility measurements show frequency dependent peaks around 0.3 K,which is another signature of spin-glass-like behavior[85-88].

FIG. 19. Basic properties of Tb2Ti2O7. (a) Temperature dependence of magnetization under 10 mT field in 3 directions [82]. (b)Temperature dependence of inverse of magnetic susceptibility χ?1 [80]. The solid line is the Curie-Weiss fitting result. (c) Temperature dependence of AC magnetic susceptibility with different frequencies [87]. (d) Specific heat of magnetic contribution and magnetic entropy under zero field [90].

FIG. 20. Phase diagram of Tb2+xTi2?xO7+y based on specific heat, magnetic susceptibility and neutron scattering measurements [89].

The specific heat results show strong sample dependence[89]. In addition, the complex CEF and hyperfine interaction makes it difficult to discuss the residual entropy, yet similar results ofRln 4 are archived[86,90,91]. Extremely low thermal conductivity is observed, suggesting that phonons are strongly scattered by spin fluctuations[92].

The CEF energy levels are measured by inelastic neutron scattering and tetrahertz spectroscopy,which shows multiple excitation routes[93,94]. Thorough inelastic neutron scattering studies interpret the observed excitation spectra as magnetoelastic modes recently[95,96]. Based on the specific heat and other techniques results on samples with different stoichiometry,possible quantum critical point between LRO phase and QSL phase is raised[89]. Besides, exotic properties are observed under strong field or high pressure[97,98],and thermal Hall effect is also observed[99].

In summary, short-range correlations, spin freezing, and spin fluctuations are observed in the QSL phase Tb2Ti2O7, but the physics is not clear yet.Strong sample dependence makes the study difficult and confusing.

Two different groups carried out wTF-μSR experiments. The data show strong frequency shift, indicating paramagnetic state[80,91]. However, Yaouancet al.found a kink in the temperature dependence of frequency shift[91].

Four different groups performed LF-μSR experiments in the past 20 years. Their results are slightly different[80,91,100,101]. No magnetic order is observed in these results. The spectra can be well described by

FIG. 21. LF-μSR results of Tb2Ti2O7. (a) Temperature dependence of LF relaxation rate λ=1/T1 under 50 G [80].Inset: Time spectra of LF-μSR at selective temperatures under 50 G. (b) Time spectra of LF-μSR at 50 mK under selective temperatures [101]. (c) Field dependence of inverse of relaxation rate 1/λ at different temperatures [101]. The data are offset for clarity.

where the stretched exponentβis 1 for most of the results, while Yaouancet al. argued that there is a slight increase ofβat low temperatures due to spin freezing[91]. The relaxation rateλsaturates below 0.3 K suggests possible persistent spin dynamics[80,91].Redfield formula (Eq. 13) can not describe the field dependence of relaxation rate at low temperatures over the whole field range, which could arise from the field dependence ofBlocorνc[101]. The fitting on a narrower range givesτc=1/νc ～10 ns, although the origin of field dependentBlocorνcremains unclear.

2. Pr2Ir2O7

In the series of rare-earth iridate pyrochloresR2Ir2O7, Pr2Ir2O7is the only compound that remains metallic down to low temperatures[102]. Such Kondo lattice with geometrically frustration could derive a novel metallic spin liquid state[103]. No evidence of LRO is observed down toTf= 0.12 K by specific heat and magnetic susceptibility measurements, but spin freezing is observed belowTf[103]. Additionally,with surplus Pr,a well-defined phase transition at 0.8 K is observed on Pr2+xIr2?xO7+δ[104].

In the μSR results on Pr2Ir2O7, there is no evidence of LRO[105]. However, due to the atomic hyperfine coupling between141Pr nuclear moment and non-Kramers nonmagnetic 4fCEF ground state, a K-T behavior over a wide range can be observed. The asymmetry spectra can be well described by

FIG. 22. μSR results of Pr2Ir2O7[105]. (a) Time spectra of LF-μSR under 13 G on Pr2Ir2O7 at selective temperatures showing full and early time spectra. (b) Temperature dependence of static and dynamic relaxation rate ? and λ,respectively.

whereλand ? is dynamic and static relaxation rate,respectively, andHis the external field. The dynamic relaxation rateλroughly saturates below 1 K, which suggests persistent spin dynamics.

For μSR experiments on Pr2+xIr2?xO7+δ, similar results were archived[104]. However, unlike the results by specific heat, no significant evidence of magnetic order is observed, which could arise from long-range muon-induced moment suppressing, or nanosecondscale Pr-moment fluctuations. The field dependence of dynamic relaxation rateλshows a power-law relation,or the muon polarization has a time-field scaling

which can be interpreted as evidence of a zero-frequency divergence of the noise power spectrum associated with“glassy” spin dynamics,or proximity to a quantum critical point (QCP).

3. Ce2Zr2O7

FIG. 23. Experimental results of Ce2Zr2O7[107]. (a) Magnetic susceptibility χ?1 results. The dashed lines are the Curie-Weiss fitting results. Inset: crystal electric field splitting measured by inelastic neutron scattering. The line is the fitting result. (b) Magnetic contribution of zero specific heat results, and magnetic entropy Smag derived from specific heat. Inset: Total specific heat results with fitting of phonon contribution. (c). Time spectra of ZF-μSR at different temperatures. (d). Time spectra of LF-μSR at selective temperatures under different fields.

Recently,a Ce-based pyrochlore Ce2Sn2O7was reported to be a QSL candidate by magnetic susceptibility and μSR measurements[106]. However, the lack of single crystalline samples restricts deeper study on the system. Later, another Ce-Based pyrochlore,Ce2Zr2O7, was reported. Single crystals of Ce2Zr2O7have been grown successfully[107,108]. No magnetic transition is observed by magnetic susceptibility and specific heat measurements down to 60 mK.For a classical spin ice, instead ofRln 2, whereRis the ideal gas constant, the magnetic entropySmagsaturates toat high temperatures[109]. However,Smagof Ce2Zr2O7saturates toRln 2 (Fig. 23. b),which suggests that Ce2Zr2O7is not a classical spin ice. Additionally, AC magnetic susceptibility results indicate that there is no spin freezing down to 0.1 K,and inelastic neutron scattering measurements find evidence of spinon continuum[107,108].

ZF-μSR experiments were carried out down to 20 mK. No magnetic order or spin freezing was observed[107]. The time spectra can be well described by

The increase of relaxation rateλsuggests the spin slowing down below 200 mK.LF-μSR study up to 1.4 T was also carried out to show that the relaxation arises from the dynamic spin fluctuations. To summarize, μSR experiments show that the absence of magnetic order and spin freezing, and the presence of spin dynamics in the ground state of Ce2Zr2O7.

VI. SUMMARY

In this review, we give a brief introduction to QSL and μSR technique. Experimental results using various techniques on different materials, including onedimensional antiferromagnetic Heisenberg chain (copper benzoate),triangular lattice(YbMgGaO4,NaYbO2and TbInO3), kagom′e lattice [ZnCu3(OH)6Cl2and Tm3Sb3Zn2O14], honeycomb lattice (Na2IrO3andα-RuCl3) and pyrochlore lattice (Tb2Ti2O7, Pr2Ir2O7and Ce2Zr2O7) are reviewed.

With the cooperation of various techniques, the study on QSL is developed quickly in the past few years.However, there is still no confirmed realization of QSL. Due to the existence of exchange disorder in many different QSL candidates,and similar phenomenon arising from other states like spin glass[28], it is challenging to identify a true QSL ground state. Fortunately,many new QSL candidates were discovered recently,including H3LiIr2O6[14], 1T-TaS2[110,111], NaYbSe2[39],Ce2Zr2O7[107,108],and many other materials. QSL is a wide open field for both theorists and experimentalists.Hopefully, fruitful new physics will appear during our understanding of QSL.

ACKNOWLEDGMENTS

This work is supported by the National Key Research and Development Program of China Nos.2016YFA0300503 and the National Natural Science Foundation of China No. 11774061, and the Shanghai Municipal Science and Technology (Major Project Grant No. 2019SHZDZX01 and No. 20ZR1405300).