Session B22: Kitaev Quantum Spin Liquids I: Experiments on RuCl3

Stacking orders and magnetic ground state in the Kitaev antiferromagnet α-RuCl3

Quantum spin liquids (QSLs) are of great interest in condensed matter physics due to the strong quantum fluctuations rising from the infinite manifolds of the magnetic ground state. Among various mechanisms that stabilize QSLs, the Kitaev model has garnered significant attention because it is exactly solvable by expressing the Hamiltonian using Majorana fermion operators.  α-RuCl₃ is a recognized Kitaev quantum spin liquid candidate with unique properties such as quantized thermal-Hall effects, a fingerprint of the Majorana fermion modes. However, the sample-dependent thermal transport properties in α-RuCl₃ and the debated spin Hamiltonian calls for a conclusive understanding of its crystal structure and magnetic ground state, which has been elusive due to the complex stacking faults. Here, we investigated the stacking orders of several different α-RuCl₃ crystals, including one untwinned crystal, in their magnetic ordered state using neutron diffraction. While all crystals follow the rhombohedral ABC-stacking extinction rule, the intensities of equivalent rhombohedral-lattice Bragg peaks show systematic deviation from the C₃ symmetry, indicating a lower symmetry than R-3 in the long-ranged average structure. By introducing a degree of stacking disorder, we found that the low-temperature crystal structure of α-RuCl₃ follows the rhombohedral stacking but with stacking faults related to the monoclinic stacking history. The stacking faults present during the structural transition from the high-temperature monoclinic phase in a chiral manner. It also ensures the low-temperature structure phase can be returned to the high-temperature phase with the domain fraction as that before cooling, an interesting “memory effect”. The stacking defects explain the two-fold symmetry in torque, susceptibility, and other anisotropic measurements. Understanding the crystal structure allows precise determination of the ordered magnetic moment size and orientations in the zigzag magnetic ground state. Our result serves as the basis for constructing the spin Hamiltonians and understanding the different thermal-Hall effect observations.   

Re-investigation of Moment Direction in Kitaev Material: α-RuCl3

α-RuCl₃ has been attaining attention due to its possible realization of quantum spin liquid. While a signature of quantum spin liquid was observed in this material [1], the result has been controversial due to discrepancies between samples [2,3]. Here, we report X-ray diffraction and resonant elastic X-ray scattering (REXS) studies on α-RuCl₃ crystals with distinct magnetic transition temperature of T_N = 7.2K and 6.5K. We find the sample with T_N = 6.5K exhibits a high degree of twinning, whereas the T_N = 7.2K sample primarily comprises a single domain of R3. We find that both samples had identical magnetic structure with magnetic moments pointing away from the honeycomb plane by α=31^○ ± 2^○. We argue that the identical moment directions across these samples suggest that intralayer magnetic Hamiltonian remains mostly unchanged regardless of T_N. Moreover, we propose that the observed sample-dependent variations in magnetic transition temperature within α-RuCl₃ can be attributed to the interlayer interaction.   

Physcial properties of high-quality single crystals of the Kitaev spin liquid candidate α-RuCl3 grown by the sublimation method

Kitaev spin liquid is an attractive state which exhibits exotic excitations of fractionalized Majorana fermions. As a promising candidate for the Kitaev spin liquid, α-RuCl₃ has been actively studied. In this material, the half-integer quantization of thermal Hall conductivity has been observed and interpreted as a signature of edge current state of Majorana fermions. Also, specific heat measurements have reported peculiar magnetic field angle dependence that reflects the Majorana gap. These results are considered to be strong evidence for the realization of the Kitaev spin liquid in α-RuCl₃.

However, it has been pointed out that the results of the thermal Hall effect depend on the growth method or crystal quality of samples. Especially, the quantization of thermal Hall conductivity tends to be observed in samples with higher quality and longer mean free paths. Therefore, it is important to synthesize high-quality samples of α-RuCl₃ with various methods and examine the influence of these sample dependences on the Kitaev spin liquid state. 

Here, we synthesized high-quality samples of α-RuCl₃ by the sublimation method. The quality of samples has been confirmed with the larger and sharper specific heat anomaly at antiferromagnetic transition and the largest longitudinal thermal conductivity in the antiferromagnetic phase reported so far. We also performed specific heat measurements in magnetic field and examined Kitaev spin liquid state. We will discuss these results and compare with the reported data in samples synthesized by other methods.   

Defect-induced low-energy Majorana excitations in the Kitaev magnet α-RuCl3

Recently, the layered honeycomb material α-RuCl₃ exhibits several anomalous features that are consistent with expectations of the Kitaev quantum spin liquid (KQSL) under an in-plane magnetic field. Most remarkably, finite planar thermal Hall conductivity has been observed, whose magnitude is close to the half-integer quantization value expected for the chiral edge currents of Majorana fermions. However, it has been reported that the thermal Hall conductivity shows strong sample dependence. Therefore, in order to understand the nature of the high-field state, it is crucial to elucidate the effects of disorder, which inevitably exists in real materials. Here, we artificially introduce point defects by electron irradiation and compare the low-energy excitations in the pristine and irradiated sample by high-resolution specific heat measurements under in-plane magnetic field rotation[1]. We find a slight suppression of AFM order and the field-dependent Majorana gap similar to the pristine sample in the high-field state. In addition, we observe a field-dependent additional T-linear term in C/T, which indicates that low-energy excitations with a linear density of states emerge. The C/T data show the scaling behaviors with the Majorana gap, which is consistent with recent theoretical calculations.

 

[1] K. Imamura et al., arXiv:2306.17380 (2023).   

Presence/absence of thermal conductivity oscillations and quantized thermal Hall effect in ultra-clean crystals of Kitaev magnet α-RuCl3

Kitaev quantum spin liquid (QSL) harbors exotic excitations such as Majorana fermions and non-Abelian anyons as a result of the fractionalization of quantum spins. One of the prime candidates for Kitaev magnets is a spin-orbit assisted Mott insulator α-RuCl₃ with layered honeycomb structure. In a field induced quantum disordered (FIQD) state of α-RuCl₃, half-integer quantized thermal Hall (HIQTH) effect, a signature of the chiral Majorana edge currents and non-Abelian anyons, has been reported. However, despite intensive research efforts, the exact nature of the FIQD state has been highly controvertial. For example, instead of the HIQTH effect, the longitudinal thermal conductivity κ_xx shows oscillatory feasures as a function of in-plane magnetic field both in zigzag antiferromagnetic and FIQD states, suggesting further exotic quantum spin states. Moreover, the HIQTH effect is observed only in the crystals with larger κ_xx in zero field below T_N, depending on growth method. These results call for more systematic studies using high-quality crystals grown by the different methods. Here we study the thermal transport properties in the ultra-clean α-RuCl₃ crystals grown by the sublimation method. In zero field, the magnitude of thermal conductivity peak below T_N is the largest reported so far, supporting the high quality of the crystals. In the field dependence of κ_xx, oscillatory features are not clearly observed, in contrast to expectation for the conventional quantum oscillations. We will also discuss the results of κ_xy in the ultra-clean crystals for the nature of the HIQTH effect in α-RuCl₃.    

Anomalous spatial oscillations in a monolayer Kitaev quantum spin liquid candidate α-RuCl3

Quantum spin liquids (QSLs) are exotic states of matter with fractionalized excitations and topological order. Kitaev QSL is distinct from other QSLs as the fractionalized excitations are Majorana fermions, yielding non-abelian anyons critical for topological quantum computations. It is thus highly desirable to detect a sign of the fractionalized particles in a spatially resolved manner toward locally controlling them. Here, using spectroscopic-imaging scanning tunneling microscopy, we report anomalous spatial oscillations of the local density of states around defects of monolayer α-RuCl₃, one of the most promising Kitaev QSL candidates. The oscillation decays away from the defects, is incommensurate, and exhibits a characteristic bias dependence. It therefore is distinguished from known oscillatory phenomena, such as charge density waves, Friedel oscillations, and quasiparticle interference. The oscillation characteristics indicate the presence of itinerant fermionic quasiparticles, implying that the oscillation manifests an unexpected feature of Kitaev QSL.   

Possible Kitaev behaviour in the angle-resolved torque magnetometry of α-RuCl3

We have measured the magnetic torque of T_N = 7 K, and T_N = 14 K single crystal samples of α-RuCl₃, as a function of the field angle in the ab-plane, focusing on temperatures between 2 and 20 K and fields from 0 to 9 T. There is a 60^◦ periodic (i.e. six-fold periodicity) sawtooth pattern found only within the zigzag ordered phase, which we ascribe to zigzag domain reorientation, and which was only observed in a large single crystal 7 K sample. Further, in both the large and small T_N = 7 K samples there is a sinusoidal six-fold signal at the temperature-field boundary of the zigzag order, and at low temperatures and higher fields. In addition to other contributions, our key findings are features that appear at low temperature along the two non-monoclinic b-axes. We find a sharp step-like feature upon applying a field strong enough to suppress the zigzag order. This feature resembles theoretical predictions for hypothesized field-induced Kitaev Ising Topological Order in the intermediate field phase.   

Structure transition and zigzag magnetic order in Ir/Rh-substituted honeycomb lattice α-RuCl3

The effect of moderate nonmagnetic chemical substitution of Ir and Rh in α-RuCl₃ is investigated. Single crystals are grown with nominal 10% Ir and 10/20% Rh substituted at the magnetic Ru site. The atomic radii of the nonmagnetic substituted atoms are similar in size to Ru. Neutron diffraction reveals little structural change of the local RuCl₆ environment. In both variants, a room temperature C2/m to low temperature R-3 phase transition is observed with a large recoverable hysteresis like the parent. At the base temperature of 5K, a zigzag magnetic spin order is observed with the same characteristic wavevector (0,0.5,1) present it the parent α-RuCl₃. Magnetic structure refinement of the 10% Ir-substituted sample reveals an ordered moment of 0.32(5) µB/Ru ion and canting angle with upper boundary of 15(4)° which is different from previous reports in the parent α-RuCl₃. The undistorted octahedra environment, reduced magnetic moment size and canting angle indicate the potential to investigate quantum spin liquid behavior through non-magnetic ion doping and presents an attractive approach for tailoring the magnetic properties of materials proximate to a quantum spin liquid state.   

(H,Li)6Ru2O6 (Ru3+, Jeff=1/2) : a novel Kitaev Quantum Spin Liquid alternative to α-RuCl3

Kitaev Quantum Spin Liquids (KQSL) host novel ground state and excited state properties.  A prominent example is having Ru³⁺ (J_eff=1/2) on a honeycomb lattice.  In zero applied field this compound is magnetically ordered and a field of 80 kOe is required to suppress the order and reveal the KQSL state. Herein we report the synthesis of (H, Li)₆Ru₂O₆ with Ru³⁺ (J_eff=1/2) on a honeycomb lattice.  Our heat capacity measurements suggest no ordering down to 400 mK in spite of a large Curie-Weiss temperature (- 44 K) as extracted from our susceptibility data. All techniques suggest a crossover to a liquid-like state below about 40 K. A two-step entropy release in heat capacity indicative of Z₂ flux (low-T) and itinerant Majorana fermions (high-T) is observed. The ⁷Li NMR shift shows non-zero T-independent spin susceptibility at low T. NMR 1/T₁ power law variation and data collapse/scaling with T/B are signatures of gapless excitations and (perhaps) a field-dependent Density of States DOS (E, B). The flattening/saturation of muon spin relaxation rate, λ (persistent spin dynamics) sets in at 10 times the temperature in H₃LiIr₂O₆ which is a KQSL. μSR asymmetry at low-T with time shows data-collapse when plotted vs t/B². The scaling behavior seen in C_m, 1/T_1, and also μSR asymmetry is indicative of underlying topological symmetry that emerges in the ground state. Based on these findings, we propose a possible KQSL in (H,Li)₆Ru₂O₆ with Ru³⁺ and no evidence of magnetic ordering, unlike α-RuCl₃.   

Spin Seebeck effect in the Kitaev spin liquid

Quantum spin liquids (QSLs) have nontrivial elementary excitations due to quantum entanglement. However, these elementary excitations are not easy to detect, and this is one of the reasons why the identification of QSLs is difficult. In one-dimensional QSLs, the spinon-mediated spin Seebeck effect (SSE) has been discovered [1], indicating that the SSE is useful for characterizing QSLs. However, the SSE for QSLs in more than one dimension remains unexplored both theoretically and experimentally. In this study, we focus on two-dimensional Kitaev QSLs and analyze the SSE based on the tunnel spin-current theory using the density matrix renormalization group method and the time-dependent variational principle that accurately incorporate quantum many-body effects. In addition, we examine the low-field and the high-field behaviors by the perturbation theory and the spin-wave theory, respectively. As a result, we found that the SSE changes its sign depending on the sign of the Kitaev interaction. This indicates that the lowest energy excitation in the Kitaev QSL, the Majorana fermions, contribute to the spin current with (up-)down-spin like nature when the Kitaev interaction is (anti-)ferromagnetic. Our results suggest that the Majorana fermions excitations contribute not only to the heat flow but also to the thermal spin current transport, and that their signatures are useful for detecting QSLs.

[1] D. Hirobe et al., Nat. Phys. 13, 30 (2017).