Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of α-RuCl_{3}

Magnetodielectric properties in two dimensional magnetic insulators

Magnetodielectric (MD) materials are important for their ability to spin-charge conversion, magnetic field control of electric polarization and vice versa. Among these, two-dimensional (2D) van der Waals (vdW) magnetic materials are of particular interest due to the presence of magnetic anisotropy (MA) originating from the interaction between the magnetic moments and the crystal field. Also, these materials indicate a high degree of stability in the long-range spin order and may be described using suitable spin Hamiltonians of the Heisenberg, XY, or Ising type. Recent reports have suggested effective interactions between magnetization and electric polarization in 2D magnets. However, MD coupling studies on layered magnetic materials are still few. This review covers the fundamentals of MD coupling by explaining related key terms. It includes the necessary conditions for having this coupling and sheds light on the possible microscopic mechanisms behind this coupling starting from phenomenological descriptions. Apart from that, this review classifies 2D magnetic materials into several categories for reaching out each and every class of materials. Additionally, this review summarizes recent advancements of some pioneer 2D MD materials. Last but not the least, the current review provides possible research directions for enhancing MD coupling in those and mentions the possibilities for future developments.

Emergent quantum Majorana metal from a chiral spin liquid

We propose a mechanism to explain the emergence of an intermediate gapless spin liquid phase in the antiferromagnetic Kitaev model in an externally applied magnetic field, sandwiched between the well-known gapped chiral spin liquid and the gapped partially polarized phase. We propose that, in moderate fields, π-fluxes nucleate in the ground state and trap Majorana zero modes. As these fluxes proliferate with increasing field, the Majorana zero modes overlap creating an emergentZ 2 quantum Majorana metallic state with a "Fermi surface" at zero energy. We further show that the Majorana spectral function captures the dynamical spin and dimer correlations obtained by the infinite Projected Entangled Pair States method, thereby validating our variational approach.

ASb3Mn9O19 (A = K or Rb): New Mn-Based 2D Magnetoplumbites with Geometric and Magnetic Frustration

Magnetoplumbites are one of the most broadly studied families of hexagonal ferrites, typically with high magnetic ordering temperatures, making them excellent candidates for permanent magnets. However, magnetic frustration is rarely observed in magnetoplumbites. Herein, the discovery, synthesis, and characterization of the first Mn-based magnetoplumbite, as well as the first magnetoplumbite involving pnictogens (Sb), ASb3Mn9O19 (A = K or Rb) are reported. The Mn3+ (S = 2) cations, further confirmed by DC magnetic susceptibility and X-ray photoelectron spectroscopy, construct three geometrically frustrated sublattices, including Kagome, triangular, and puckered honeycomb lattices. Magnetic properties measurements revealed strong antiferromagnetic spin-spin coupling as well as multiple low-temperature magnetic features. Heat capacity data does not show any prominent λ-anomaly, suggesting minimal associated magnetic entropy. Moreover, neutron powder diffraction (NPD) implied the absence of long-range magnetic ordering in KSb3Mn9O19 down to 3 K. However, several magnetic peaks are observed in RbSb3Mn9O19 at 3 K, corresponding to an incommensurate magnetic structure. Interestingly, strong diffuse scattering is seen in the NPD patterns of both compounds at low angles and is analyzed by reverse Monte Carlo refinements, indicating short-range spin ordering related to frustrated magnetism as well as 2D magnetic correlations in ASb3Mn9O19 (A = K or Rb).

Giant Anisotropic Magnetoresistance in Few-Layer α-RuCl3 Tunnel Junctions

The spin-orbit-assisted Mott insulator α-RuCl3 is proximate to the coveted quantum spin liquid (QSL) predicted by the Kitaev model. In the search for the pure Kitaev QSL, reducing the dimensionality of this frustrated magnet by exfoliation has been proposed as a way to enhance magnetic fluctuations and Kitaev interactions. Here, we perform angle-dependent tunneling magnetoresistance (TMR) measurements on ultrathin α-RuCl3 crystals with various layer numbers to probe their magnetic, electronic, and crystal structures. We observe a giant change in resistance, as large as ∼2500%, when the magnetic field rotates either within or out of the α-RuCl3 plane, a manifestation of the strongly anisotropic spin interactions in this material. In combination with scanning transmission electron microscopy, this tunneling anisotropic magnetoresistance (TAMR) reveals that few-layer α-RuCl3 crystals remain in the high-temperature monoclinic phase at low temperatures. It also shows the presence of a zigzag antiferromagnetic order below the critical temperature TN ≃ 14 K, which is twice the one typically observed in bulk samples with rhombohedral stacking. Our work offers valuable insights into the relation between the stacking order and magnetic properties of this material, which helps lay the groundwork for creating and electrically probing exotic magnetic phases such as QSLs via van der Waals engineering.

Incommensurate charge super-modulation and hidden dipole order in layered kitaev material α-RuCl3

The magnetism of Kitaev materials has been widely studied, but their charge properties and the coupling to other degrees of freedom are less known. Here we investigate the charge states of α-RuCl3, a promising Kitaev quantum spin liquid candidate, in proximity to graphite. We discover that few-layered α-RuCl3 experiences a clear modulation of charge states, where a Mott-insulator to weak charge-transfer-insulator transition in the 2D limit occurs by means of heterointerfacial polarization. More notably, distinct signals of incommensurate charge and lattice super-modulations, regarded as an unconventional charge order, accompanied in the insulator. Our theoretical calculations have reproduced the incommensurate charge order by taking into account the antiferroelectricity of α-RuCl3 that is driven by dipole order in the internal electric fields. The findings imply that there is strong coupling between the charge, spin, and lattice degrees of freedom in layered α-RuCl3 in the heterostructure, which offers an opportunity to electrically access and tune its magnetic interactions inside the Kitaev compounds.

Suppression of antiferromagnetic order by strain-enhanced frustration in honeycomb cobaltate

Layered honeycomb cobaltates are predicted as promising for realizing the Kitaev quantum spin liquid, a many-body quantum entangled ground state characterized by fractional excitations. However, they exhibit antiferromagnetic ordering at low temperatures, hindering the expected quantum state. We demonstrate that controlling the trigonal distortion of CoO6 octahedra is crucial to suppress antiferromagnetic order through enhancing frustration in layered honeycomb cobaltates. Using heterostructure engineering on Cu3Co2SbO6 thin films, we adjust the trigonal distortion of CoO6 octahedra and the resulting trigonal crystal field. The original Néel temperature of 16 kelvin in bulk Cu3Co2SbO6 decreases (increases) to 7.8 kelvin (22.7 kelvin) in strained Cu3Co2SbO6 films by decreasing (increasing) the magnitude of the trigonal crystal fields. The first-principles calculation suggests the enhancement of geometrical frustration as the origin of the suppression of antiferromagnetism. This finding supports the potential of layered honeycomb cobaltate heterostructures and strain engineering in realizing extremely elusive quantum phases of matter.

Possible intermediate quantum spin liquid phase in α-RuCl3 under high magnetic fields up to 100 T

Pursuing the exotic quantum spin liquid (QSL) state in the Kitaev material α-RuCl3 has intrigued great research interest recently. A fascinating question is on the possible existence of a field-induced QSL phase in this compound. Here we perform high-field magnetization measurements of α-RuCl3 up to 102 T employing the non-destructive and destructive pulsed magnets. Under the out-of-plane field along the c* axis (i.e., perpendicular to the honeycomb plane), two quantum phase transitions are uncovered at respectively 35 T and about 83 T, between which lies an intermediate phase as the predicted QSL. This is in sharp contrast to the case with in-plane fields, where a single transition is found at around 7 T and the intermediate QSL phase is absent instead. By measuring the magnetization data with fields tilted from the c* axis up to 90° (i.e., in-plane direction), we obtain the field-angle phase diagram that contains the zigzag, paramagnetic, and QSL phases. Based on the K-J-Γ-[Formula: see text] model for α-RuCl3 with a large Kitaev term we perform density matrix renormalization group simulations and reproduce the quantum phase diagram in excellent agreement with experiments.

A magnetic continuum in the cobalt-based honeycomb magnet BaCo2(AsO4)2

Quantum spin liquids (QSLs) are topologically ordered states of matter that host fractionalized excitations. A particular route towards a QSL is via strongly bond-dependent interactions on the hexagonal lattice. A number of Ru- and Ir-based candidate Kitaev QSL materials have been pursued, but all have appreciable non-Kitaev interactions. Using time-domain terahertz spectroscopy, we observed a broad magnetic continuum over a wide range of temperatures and fields in the honeycomb cobalt-based magnet BaCo₂(AsO₄)₂, which has been proposed to be a more ideal version of a Kitaev QSL. Applying an in-plane magnetic field of ~0.5 T suppresses the magnetic order, and at higher fields, applying the field gives rise to a spin-polarized state. Under a 4 T magnetic field that was oriented principally out of plane, a broad magnetic continuum was observed that may be consistent with a field-induced QSL. Our results indicate BaCo₂(AsO₄)₂ is a promising QSL candidate.

A Three-Stage Magnetic Phase Transition Revealed in Ultrahigh-Quality van der Waals Bulk Magnet CrSBr

van der Waals (vdW) magnets are receiving ever-growing attention nowadays due to their significance in both fundamental research on low-dimensional magnetism and potential applications in spintronic devices. The high crystalline quality of vdW magnets is the key to maintaining intrinsic magnetic and electronic properties, especially when exfoliated down to the two-dimensional limit. Here, ultrahigh-quality air-stable vdW CrSBr crystals are synthesized using the direct solid-vapor synthesis method. The high single crystallinity and spatial homogeneity have been thoroughly evidenced at length scales from submm to atomic resolution by X-ray diffraction, second harmonic generation, and scanning transmission electron microscopy. More importantly, specific heat measurements of ultrahigh-quality CrSBr crystals show three thermodynamic anomalies at 185, 156, and 132 K, revealing a stage-by-stage development of the magnetic order upon cooling, which is also corroborated with the magnetization and transport results. Our ultrahigh-quality CrSBr can further be exfoliated down to monolayers and bilayers easily, providing the building blocks of heterostructures for spintronic and magneto-optoelectronic applications.

Unveiling the S=3/2 Kitaev honeycomb spin liquids

The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z₂ gauge field. Employing an SO(6) Majorana representation of spin3/2’s, we find an exact representation of the conserved plaquette fluxes in terms of static Z₂ gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by the addition of a [001] single ion anisotropy (SIA) which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. In the presence of a weak [111] SIA, we discuss an emergent chiral QSL within a perturbation theory.

Recently, material realizations of the spin 3/2 Kitaev honeycomb model have been proposed, but the model has not been solved by either analytical or numerical methods. Here the authors report exact results for the spin 3/2 model consistent with numerical simulations, and find gapped and gapless quantum spin liquids.

Work statistics across a quantum critical surface

We study the universality of work statistics of a system quenched through a quantum critical surface. By using the adiabatic perturbation theory, we obtain the general scaling behavior for all cumulants of work. These results extend the studies of Kibble-Zurek mechanism scaling of work statistics from an isolated quantum critical point to a critical surface. As an example, we study the scaling behavior of work statistics in the two-dimensional (2D) Kitaev honeycomb model featured with a critical line. By utilizing the trace formula for quadratic fermionic Hamiltonian, we obtain the exact characteristic function of work of the 2D Kitaev model at zero temperature. The results confirm our prediction.

Honeycomb-Structure RuI3 , A New Quantum Material Related to α-RuCl3

The layered honeycomb lattice material α-RuCl₃ has emerged as a prime candidate for displaying the Kitaev quantum spin liquid state, and as such has attracted much research interest. Here a new layered honeycomb lattice polymorph of RuI₃ , a material that is strongly chemically and structurally related to α-RuCl₃ is described_. The material is synthesized at moderately elevated pressures and is stable under ambient conditions. Preliminary characterization reveals that it is a metallic conductor, with the absence of long-range magnetic order down to 0.35 K and an unusually large T-linear contribution to the heat capacity. It is proposed that this phase, with a layered honeycomb lattice and strong spin-orbit coupling, provides a new route for the characterization of quantum materials.

Antiferromagnetic Kitaev interaction inJeff1/2 cobalt honeycomb materials Na3Co2SbO6and Na2Co2TeO6

Finding new materials with antiferromagnetic (AFM) Kitaev interaction is an urgent issue for quantum magnetism research. We conclude that Na₃Co₂SbO₆and Na₂Co₂TeO₆are new honeycomb cobalt-based systems with AFM Kitaev interaction by carrying out inelastic neutron scattering experiments and subsequent analysis. The spin-orbit excitons observed at 20-28 meV in both compounds strongly support the idea that Co²⁺ions of both compounds have a spin-orbital entangledJ_eff= 1/2 state. Furthermore, we found that a generalized Kitaev-Heisenberg Hamiltonian can describe the spin-wave excitations of both compounds with additional 3rd nearest-neighbor interaction. Our best-fit parameters show significant AFM Kitaev terms and off-diagonal symmetric anisotropy terms of a similar magnitude in both compounds. We also found a strong magnon-damping effect at the higher energy part of the spin waves, entirely consistent with observations in other Kitaev magnets. Our work suggests Na₃Co₂SbO₆and Na₂Co₂TeO₆as rare examples of the AFM Kitaev magnets based on the systematic studies of the spin waves and analysis.

Field-induced quantum spin disordered state in spin-1/2 honeycomb magnet Na2Co2TeO6

Spin-orbit coupled honeycomb magnets with the Kitaev interaction have received a lot of attention due to their potential of hosting exotic quantum states including quantum spin liquids. Thus far, the most studied Kitaev systems are 4d/5d-based honeycomb magnets. Recent theoretical studies predicted that 3d-based honeycomb magnets, including Na₂Co₂TeO₆ (NCTO), could also be a potential Kitaev system. Here, we have used a combination of heat capacity, magnetization, electron spin resonance measurements alongside inelastic neutron scattering (INS) to study NCTO's quantum magnetism, and we have found a field-induced spin disordered state in an applied magnetic field range of 7.5 T < B (⊥ b-axis) < 10.5 T. The INS spectra were also simulated to tentatively extract the exchange interactions. As a 3d-magnet with a field-induced disordered state on an effective spin-1/2 honeycomb lattice, NCTO expands the Kitaev model to 3d compounds, promoting further interests on the spin-orbital effect in quantum magnets.

Identification of magnetic interactions and high-field quantum spin liquid in α-RuCl3

The frustrated magnet α-RuCl3 constitutes a fascinating quantum material platform that harbors the intriguing Kitaev physics. However, a consensus on its intricate spin interactions and field-induced quantum phases has not been reached yet. Here we exploit multiple state-of-the-art many-body methods and determine the microscopic spin model that quantitatively explains major observations in α-RuCl3, including the zigzag order, double-peak specific heat, magnetic anisotropy, and the characteristic M-star dynamical spin structure, etc. According to our model simulations, the in-plane field drives the system into the polarized phase at about 7 T and a thermal fractionalization occurs at finite temperature, reconciling observations in different experiments. Under out-of-plane fields, the zigzag order is suppressed at 35 T, above which, and below a polarization field of 100 T level, there emerges a field-induced quantum spin liquid. The fractional entropy and algebraic low-temperature specific heat unveil the nature of a gapless spin liquid, which can be explored in high-field measurements on α-RuCl3.

Spin Hamiltonians in Magnets: Theories and Computations

The effective spin Hamiltonian method has drawn considerable attention for its power to explain and predict magnetic properties in various intriguing materials. In this review, we summarize different types of interactions between spins (hereafter, spin interactions, for short) that may be used in effective spin Hamiltonians as well as the various methods of computing the interaction parameters. A detailed discussion about the merits and possible pitfalls of each technique of computing interaction parameters is provided.

Orbital Effects in Solids: Basics, Recent Progress, and Opportunities

The properties of transition metal compounds are largely determined by nontrivial interplay of different degrees of freedom: charge, spin, lattice, and also orbital ones. Especially rich and interesting effects occur in systems with orbital degeneracy. For example, they result in the famous Jahn-Teller effect, leading to a plethora of consequences for static and dynamic properties, including nontrivial quantum effects. In the present review, we discuss the main phenomena in the physics of such systems, paying central attention to the novel manifestations of those. After shortly summarizing the basic phenomena and their descriptions, we concentrate on several specific directions in this field. One of them is the reduction of effective dimensionality in many systems with orbital degrees of freedom due to the directional character of orbitals, with the concomitant appearance of some instabilities that lead in particular to the formation of dimers, trimers, and similar clusters in a material. The properties of such cluster systems, which are largely determined by their orbital structure, are discussed in detail, and many specific examples of those in different materials are presented. Another big field that has acquired special significance relatively recently is the role of the relativistic spin-orbit interaction. The mutual influence of this interaction and the more traditional Jahn-Teller physics is treated in detail in the second part of the review. In discussing all of these questions, special attention is paid to novel quantum effects.

Tuning Topological Orders by a Conical Magnetic Field in the Kitaev Model

We show that a conical magnetic field H=(1,1,1)H can be used to tune the topological order and hence, anyon excitations of the Z_{2} quantum spin liquid in the isotropic antiferromagnetic Kitaev model. A novel topological order, featured with Chern number C=4 and Abelian anyon excitations, is induced in a narrow range of intermediate fields H_{c1}≤H≤H_{c2}. On the other hand, the C=1 Ising-topological order with non-Abelian anyon excitations, as previously known to be present at small fields, is found here to survive up to H_{c1}. The results are obtained by developing and applying a Z_{2} mean field theory that works at finite fields and is asymptotically exact in the zero field limit and the associated variational quantum Monte Carlo.

Phase Diagram of the Spin-1/2 Kitaev-Gamma Chain and Emergent SU(2) Symmetry

We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.

Ab Initio Mismatched Interface Theory of Graphene on α-RuCl_{3}: Doping and Magnetism

Recent developments in twisted and lattice-mismatched bilayers have revealed a rich phase space of van der Waals systems and generated excitement. Among these systems are heterobilayers, which can offer new opportunities to control van der Waals systems with strong in plane correlations such as spin-orbit-assisted Mott insulator α-RuCl_{3}. Nevertheless, a theoretical ab initio framework for mismatched heterobilayers without even approximate periodicity is sorely lacking. We propose a general strategy for calculating electronic properties of such systems, mismatched interface theory (MINT), and apply it to the graphene/α-RuCl_{3} (GR/α-RuCl_{3}) heterostructure. Using MINT, we predict uniform doping of 4.77% from graphene to α-RuCl_{3} and magnetic interactions in α-RuCl_{3} to shift the system toward the Kitaev point. Hence, we demonstrate that MINT can guide targeted materialization of desired model systems and discuss recent experiments on GR/α-RuCl_{3} heterostructures.

Possible Kitaev Quantum Spin Liquid State in 2D Materials with S=3/2

Quantum spin liquids (QSLs) form an extremely unusual magnetic state in which the spins are highly correlated and fluctuate coherently down to the lowest temperatures, but without symmetry breaking and without the formation of any static long-range-ordered magnetism. Such intriguing phenomena are not only of great fundamental relevance in themselves, but also hold promise for quantum computing and quantum information. Among different types of QSLs, the exactly solvable Kitaev model is attracting much attention, with most proposed candidate materials, e.g., RuCl_{3} and Na_{2}IrO_{3}, having an effective S=1/2 spin value. Here, via extensive first-principles-based simulations, we report the investigation of the Kitaev physics and possible Kitaev QSL state in epitaxially strained Cr-based monolayers, such as CrSiTe_{3}, that rather possess a S=3/2 spin value. Our study thus extends the playground of Kitaev physics and QSLs to 3d transition metal compounds.

Randomly Hopping Majorana Fermions in the Diluted Kitaev System α-Ru_{0.8}Ir_{0.2}Cl_{3}

dc and ac magnetic susceptibility, magnetization, specific heat, and Raman scattering measurements are combined to probe low-lying spin excitations in α-Ru_{1-x}Ir_{x}Cl_{3} (x≈0.2), which realizes a disordered spin liquid. At intermediate energies (ℏω>3  meV), Raman spectroscopy evidences linearly ω-dependent Majorana-like excitations, obeying Fermi statistics. This points to robustness of a Kitaev paramagnetic state under spin vacancies. At low energies below 3 meV, we observe power-law dependences and quantum-critical-like scalings of the thermodynamic quantities, implying the presence of a weakly divergent low-energy density of states. This scaling phenomenology is interpreted in terms of the random hoppings of Majorana fermions. Our results demonstrate an emergent hierarchy of spin excitations in a diluted Kitaev honeycomb system subject to spin vacancies and bond randomness.

Weak-field induced nonmagnetic state in a Co-based honeycomb

Layered honeycomb magnets are of interest as potential realizations of the Kitaev quantum spin liquid (KQSL), a quantum state with long-range spin entanglement and an exactly solvable Hamiltonian. Conventional magnetically ordered states are present for all currently known candidate materials, however, because non-Kitaev terms in the Hamiltonians obscure the Kitaev physics. Current experimental studies of the KQSL are focused on 4d or 5d transition metal-based honeycombs, in which strong spin-orbit coupling can be expected, yielding Kitaev interaction that dominates in an applied magnetic field. In contrast, for 3d-based layered honeycomb magnets, spin-orbit coupling is weak, and thus, Kitaev physics should be substantially less accessible. Here, we report our studies on BaCo₂(AsO₄)₂, for which we find that the magnetic order associated with the non-Kitaev interactions can be fully suppressed by a relatively low magnetic field, yielding a nonmagnetic material and implying the presence of strong magnetic frustration and weak non-Kitaev interactions.

One Proximate Kitaev Spin Liquid in the K-J-Γ Model on the Honeycomb Lattice

In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.

Heisenberg-Kitaev physics in magnetic fields

Magnetic insulators in the regime of strong spin-orbit coupling exhibit intriguing behaviors in external magnetic fields, reflecting the frustrated nature of their effective interactions. We review the recent advances in understanding the field responses of materials that are described by models with strongly bond-dependent spin exchange interactions, such as Kitaev's celebrated honeycomb model and its extensions. We discuss the field-induced phases and the complex magnetization processes found in these theories and compare with experimental results in the layered Mott insulators [Formula: see text]-RuCl₃ and Na₂IrO₃, which are believed to realize this fascinating physics.

Magnetic properties of the BaNi2(As1-x V x )2O8 honeycomb system

The Kitaev model for magnetic interactions on a honeycomb lattice, which underlies the exotic quantum spin liquid (QSL) state in such systems, has long been established and experimentally confirmed. Current studies on Kitaev QSLs have focused on heavy-metal-based Mott insulators where strong spin-orbit coupling (SOC) leads to bond-dependent spin anisotropy. Here we examine a 3d transition-metal-based honeycomb lattice material BaNi₂(As_1-x V _x )₂O₈, where the SOC is weak but spin exchange coupling is strongly frustrated due to the crystal field. We present an investigation of the crystal structure and magnetization of this series of Ni-based honeycomb materials, providing an alternative perspective in the search for the QSL in 3d-element-based honeycomb systems.

Microscopic Mechanism for a Higher-Spin Kitaev Model

The spin S=1/2 Kitaev honeycomb model has attracted significant attention since emerging candidate materials have provided a playground to test non-Abelian anyons. The Kitaev model with higher spins has also been theoretically studied, as it may offer another path to a quantum spin liquid. However, a microscopic route to achieve higher spin Kitaev models in solid state materials has not been rigorously derived. Here we present a theory of the spin S=1 Kitaev interaction in two-dimensional edge-shared octahedral systems. Essential ingredients are strong spin-orbit coupling in anions and strong Hund's coupling in transition metal cations. The S=1 Kitaev and ferromagnetic Heisenberg interactions are generated from superexchange paths. Taking into account the antiferromagnetic Heisenberg term from direct-exchange paths, the Kitaev interaction dominates the physics of the S=1 system. Using an exact diagonalization technique, we show a finite regime of S=1 spin liquid in the presence of the Heisenberg interaction. Candidate materials are proposed, and generalization to higher spins is discussed.

Hidden Plaquette Order in a Classical Spin Liquid Stabilized by Strong Off-Diagonal Exchange

We report a new classical spin liquid in which the collective flux degrees of freedom break the translation symmetry of the honeycomb lattice. This exotic phase exists in the frustrated spin-orbit magnets where a dominant off-diagonal exchange, the so-called Γ term, results in a macroscopic ground-state degeneracy at the classical level. We demonstrate that the system undergoes a phase transition driven by thermal order by disorder at a critical temperature T_{c}≈0.04|Γ|. This transition reduces the emergent spherical spin symmetry to a cubic one: spins point predominantly toward the cubic axes, yet seem to remain disordered at T<T_{c}. Importantly, we show that the phase transition corresponds to a hidden plaquette ordering of hexagonal fluxes, which explicitly breaks the cubic symmetry, a scenario that is confirmed by our extensive Monte Carlo simulations. We further compute the dynamical structure factors of the spin-liquid phase and reveal unusual dynamical properties of the hexagonal flux parameters.

Theory of the field-revealed Kitaev spin liquid

Elementary excitations in entangled states such as quantum spin liquids may exhibit exotic statistics different from those obeyed by fundamental bosons and fermions. Non-Abelian anyons exist in a Kitaev spin liquid-the ground state of an exactly solvable model. A smoking-gun signature of these excitations, namely a half-integer quantized thermal Hall conductivity, was recently reported in α-RuCl₃. While fascinating, a microscopic theory for this phenomenon remains elusive because the pure Kitaev model cannot display this effect in an intermediate magnetic field. Here we present a microscopic theory of the Kitaev spin liquid emerging between the low- and high-field states. Essential to this result is an antiferromagnetic off-diagonal symmetric interaction which allows the Kitaev spin liquid to protrude from the ferromagnetic Kitaev limit under a magnetic field. This generic model displays a strong field anisotropy, and we predict a wide spin liquid regime when the field is perpendicular to the honeycomb plane.

Sawtooth Torque in Anisotropic j_{eff}=1/2 Magnets: Application to α-RuCl_{3}

The so-called "Kitaev candidate" materials based on 4d^{5} and 5d^{5} metals have recently emerged as magnetic systems displaying strongly anisotropic exchange interactions reminiscent of the Kitaev's honeycomb model. Recently, these materials have been shown to commonly display a distinct sawtooth angular dependence of the magnetic torque over a wide range of magnetic fields. While higher order chiral spin interactions have been considered as a source of this observation, we show here that bilinear anisotropic interactions and/or g anisotropy are each sufficient to explain the observed torque response, which may be distinguished on the basis of high-field measurements. These findings unify the understanding of magnetic torque experiments in a variety of Kitaev candidate materials.

Particle–hole fluctuations and possible superconductivity in doped α-RuCl3*

We study various particle–hole excitations and possible superconducting pairings mediated by these fluctuations in doped α-RuCl3 by using multi-band Hubbard model with all t2g orbitals. By performing a random-phase-approximation (RPA) analysis, we find that among all particle–hole excitations, the j eff = 1/2 pseudospin fluctuations are dominant, suggesting the robustness of j eff = 1/2 picture even in the doped systems. We also find that the most favorable superconducting state has a d-wave pairing symmetry.

Quasi-degenerate magnetic states in α-RuCl3

Exploring quantum spin liquid (QSL) state has both fundamental scientific value and realistic application potential. Recently, α-RuCl₃ was experimentally observed to hold in-plane zigzag antiferromagnetic (AFM) order at low temperature, which was further proposed to be proximate to a Kitaev QSL ground state. We have studied the magnetic properties of α-RuCl₃ in the framework of electronic structure calculation based on density functional theory (DFT) with Hubbard U correction (DFT+U) and spin-orbit coupling. When the intra-orbital Hubbard interaction U and the inter-orbital Hund's coupling J adopt the commonly accepted values of U = 2.0 eV and J = 0.4 eV, the zigzag AFM order indeed owns the minimum energy, consistent with the experimental observation. More importantly, we find that compared with the ferromagnetic order in the previous theoretical studies, there exist a series of magnetic configurations energetically even closer to the zigzag AFM ground state. The further calculations and analysis indicate that these low-energy magnetic states are closely related to the electronic correlation effect of Ru 4d orbitals. By decreasing U and increasing J with just about 0.2 eV, they become energetically degenerate with the zigzag AFM order, inducing strong magnetic frustration and then yielding a state without long-range magnetic order but with nonzero local moments. Considering the facts that theoretically the pressure usually reduces the intra-orbital Hubbard interaction and meanwhile enhances the inter-orbital Hund's coupling, while experimentally the pressure drives α-RuCl₃ into a quantum disordered phase, our results provide a perspective to understand the exotic magnetic behaviors of α-RuCl₃.

Uniaxial and hydrostatic pressure effects in α-RuCl3 single crystals via thermal-expansion measurements

We present high-resolution thermal-expansion and specific-heat measurements of single crystalline α-RuCl₃. An extremely hysteretic structural transition expanding over 100 K is observed by thermal-expansion along both crystallographic axes, which we attribute to a change of stacking sequence of the RuCl₃ layers. Three magnetic transitions are observed, which we link to the different stacking sequences. Using our data and thermodynamic relations, we derive the uniaxial and hydrostatic pressure derivatives of all three magnetic transitions. Our results demonstrate that magnetic order of the major transition at  ∼7 K should be totally suppressed by very moderate pressures of 0.3 GPa. Finally, we discuss why our results differ from recent hydrostatic pressure measurements and suggest a possible route to reaching the spin-liquid state in α-RuCl₃.

Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic Field

Motivated by recent experimental observations in α-RuCl_{3}, we study the K-Γ model on the honeycomb lattice in an external magnetic field. By a slave-particle representation and variational Monte Carlo calculations, we reproduce the phase transition from zigzag magnetic order to a field-induced disordered phase. The nature of this state depends crucially on the field orientation. For particular field directions in the honeycomb plane, we find a gapless Dirac spin liquid, in agreement with recent experiments on α-RuCl_{3}. For a range of out-of-plane fields, we predict the existence of a Kalmeyer-Laughlin-type chiral spin liquid, which would show an integer-quantized thermal Hall effect.

Ultralow-Temperature Thermal Conductivity of the Kitaev Honeycomb Magnet α-RuCl_{3} across the Field-Induced Phase Transition

Recently, there have been increasingly hot debates on whether there exists a quantum spin liquid in the Kitaev honeycomb magnet α-RuCl_{3} in a high magnetic field. To investigate this issue, we perform ultralow-temperature thermal conductivity measurements on single crystals of α-RuCl_{3} down to 80 mK and up to 9 T. Our experiments clearly show a field-induced phase transition occurring at μ_{0}H_{c}≈7.5  T, above which the magnetic order is completely suppressed. The minimum of thermal conductivity at 7.5 T is attributed to the strong scattering of phonons by magnetic fluctuations. Most importantly, above 7.5 T, we do not observe any significant contribution of thermal conductivity from gapless magnetic excitations, which puts a strong constraint on the nature of the high-field phase of α-RuCl_{3}.

Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb

We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points [Formula: see text] and [Formula: see text] are singular. The ϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.