Kitaev exchange and field-induced quantum spin-liquid states in honeycomb α-RuCl3

Emergence of the isotropic Kitaev honeycomb latticeα-RuCl3and its magnetic properties

We present a comprehensive investigation of the crystal and magnetic structures of the van der Waals antiferromagnetα-RuCl3using single crystal x-ray and neutron diffraction. The crystal structure at room temperature is a monoclinic (C2/m). However, with decreasing temperature, a remarkable first-order structural phase transition is observed, leading to the emergence of a rhombohedral (R3-) structure characterized by three-fold rotational symmetry forming an isotropic honeycomb lattice. On further cooling, a zigzag-type antiferromagnetic order develops belowTN=6∼6.6K. The critical exponent of the magnetic order parameter was determined to beβ=0.11(1), which is close to the two-dimensional Ising model. Additionally, the angular dependence of the magnetic critical field of the zigzag antiferromagnetic order for the polarized ferromagnetic phase reveals a six-fold rotational symmetry within theab-plane. These findingsreflect the symmetry associated with the Ising-like bond-dependent Kitaev spin interactions and underscore the universality of the Kitaev interaction-dominated antiferromagnetic system.

Signatures of Fractional Statistics in Nonlinear Pump-Probe Spectroscopy

We show that the presence of anyons in the excitation spectrum of a two-dimensional system can be inferred from nonlinear spectroscopic quantities. In particular, we consider pump-probe spectroscopy, where a sample is irradiated by two light pulses with an adjustable time delay between them. The relevant response coefficient exhibits a universal form that originates from the statistical phase acquired when anyons created by the first pulse braid around those created by the second. This behavior is shown to be qualitatively unchanged by nonuniversal physics including nonstatistical interactions and small nonzero temperatures. In magnetic systems, the signal of interest can be measured using currently available terahertz-domain probes, highlighting the potential usefulness of nonlinear spectroscopic techniques in the search for quantum spin liquids.

Dressed jeff-1/2 objects in mixed-valence lacunar spinel molybdates

The lacunar-spinel chalcogenides exhibit magnetic centers in the form of transition-metal tetrahedra. On the basis of density-functional computations, the electronic ground state of an Mo₄¹³⁺ tetrahedron has been postulated as single-configuration a₁² e⁴ t₂⁵, where a₁, e, and t₂ are symmetry-adapted linear combinations of single-site Mo t_2g atomic orbitals. Here we unveil the many-body tetramer wave-function: we show that sizable correlations yield a weight of only 62% for the a₁² e⁴ t₂⁵ configuration. While spin-orbit coupling within the peculiar valence orbital manifold is still effective, the expectation value of the spin-orbit operator and the g factors deviate from figures describing nominal t⁵ j_eff = 1/2 moments. As such, our data documents the dressing of a spin-orbit j_eff = 1/2 object with intra-tetramer excitations. Our results on the internal degrees of freedom of these magnetic moments provide a solid theoretical starting point in addressing the intriguing phase transitions observed at low temperatures in these materials.

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.

Magnetic anisotropy reversal driven by structural symmetry-breaking in monolayer α-RuCl3

Layered α-RuCl₃ is a promising material to potentially realize the long-sought Kitaev quantum spin liquid with fractionalized excitations. While evidence of this state has been reported under a modest in-plane magnetic field, such behaviour is largely inconsistent with theoretical expectations of spin liquid phases emerging only in out-of-plane fields. These predicted field-induced states have been largely out of reach due to the strong easy-plane anisotropy of bulk crystals, however. We use a combination of tunnelling spectroscopy, magnetotransport, electron diffraction and ab initio calculations to study the layer-dependent magnons, magnetic anisotropy, structure and exchange coupling in atomically thin samples. Due to picoscale distortions, the sign of the average off-diagonal exchange changes in monolayer α-RuCl₃, leading to a reversal of spin anisotropy to easy-axis anisotropy, while the Kitaev interaction is concomitantly enhanced. Our work opens the door to the possible exploration of Kitaev physics in the true two-dimensional limit.

A flexible neutron spectrometer concept with a new ultra-high field steady-state vertical-bore magnet

The proposed facility explores materials under ultra-high magnetic fields. By combining the power of high fields to tune materials and of neutron scattering to probe the resulting changes down to the atomic scale, this facility will enable transformative progress in the study of quantum materials and is named for the "TITAN" subset of Greek gods to reflect this transformation. TITAN will offer DC magnetic fields up to at least 20 T. Exploiting the record brightness and bandwidth of the Second Target Station at the Spallation Neutron Source, TITAN will probe atomic-scale responses through high efficiency neutron spectroscopy up to 80 meV energy transfer, high resolution diffraction, and small angle neutron scattering. Focusing neutron optics will maximize flux on accurately positioned samples, while radial collimation and optimized shielding and detection strategies will minimize backgrounds.

Direct observation of the Mottness and p-d orbital hybridization in the epitaxial monolayer α-RuCl3

α-RuCl₃, a promising material to accomplish the Kitaev honeycomb model, has attracted enormous interest recently. Mottness and p-d bonds play vital roles in generating Kitaev interactions and underpinning the potential exotic states of quantum magnets, and the van der Waals monolayer is considered to be a better platform to approach a two-dimensional Kitaev model than the bulk. Here, we worked out the growth art of an α-RuCl₃ monolayer on a graphite substrate and studied its electronic structure, particularly the delicate orbital occupations, through scanning tunneling microscopy and spectroscopy. An in-plane lattice expansion of 2.67 ± 0.83% is observed and the pronounced t_2g-p_π and e_g-p_σ hybridization are visualized. The Mott nature is unveiled by an ∼0.6 eV full gap at the Fermi level located inside the t_2g-p_π manifold which is further verified by the density functional theory calculations. The monolayer phase of α-RuCl₃ fulfills the a priori criteria of recent theoretical predictions of tuning the relevant properties in this material and provides a novel platform to explore the Kitaev physics.

The non-centrosymmetric layered compounds IrTe2I and RhTe2I

The previously unreported layered compounds IrTe₂I and RhTe₂I were prepared by a high-pressure synthesis method. Single crystal X-ray and powder X-ray diffraction studies find that the compounds are isostructural, crystallizing in a layered orthorhombic structure in the non-centrosymmetric, non-symmorphic space group Pca2₁ (#29). Characterization reveals diamagnetic, high resistivity, semiconducting behavior for both compounds, consistent with the +3 chemical valence and d⁶ electronic configurations for both iridium and rhodium and the Te-Te dimers seen in the structural study. Electronic band structures are calculated for both compounds, showing good agreement with the experimental results.

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.

Identification of a Kitaev quantum spin liquid by magnetic field angle dependence

Quantum spin liquids realize massive entanglement and fractional quasiparticles from localized spins, proposed as an avenue for quantum science and technology. In particular, topological quantum computations are suggested in the non-abelian phase of Kitaev quantum spin liquid with Majorana fermions, and detection of Majorana fermions is one of the most outstanding problems in modern condensed matter physics. Here, we propose a concrete way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence. Topologically protected critical lines exist on a plane of magnetic field angles, and their shapes are determined by microscopic spin interactions. A chirality operator plays a key role in demonstrating microscopic dependences of the critical lines. We also show that the chirality operator can be used to evaluate topological properties of the non-abelian Kitaev quantum spin liquid without relying on Majorana fermion descriptions. Experimental criteria for the non-abelian spin liquid state are provided for future experiments.

Non-Abelian phase of Kitaev quantum spin liquid is promising for topological quantum computation. Here, the authors propose a way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence, providing criteria for such a state for future experiments.

Spin-orbital entangled state and realization of Kitaev physics in 3dcobalt compounds: a progress report

The realization of Kitaev's honeycomb magnetic model in real materials has become one of the most pursued topics in condensed matter physics and materials science. If found, it is expected to host exotic quantum phases of matter and offers potential realizations of fault-tolerant quantum computations. Over the past years, much effort has been made on 4d- or 5d-heavy transition metal compounds because of their intrinsic strong spin-orbit coupling. But more recently, there have been growing shreds of evidence that the Kitaev model could also be realized in 3d-transition metal systems with much weaker spin-orbit coupling. This review intends to serve as a guide to this fast-developing field focusing on systems withd⁷transition metal occupation. It overviews the current theoretical and experimental progress on realizing the Kitaev model in those systems. We examine the recent experimental observations of candidate materials with Co²⁺ions: e.g., CoPS₃, Na₃Co₂SbO₆, and Na₂Co₂TeO₆, followed by a brief review of theoretical backgrounds. We conclude this article by comparing experimental observations with density functional theory calculations. We stress the importance of inter-t_2ghopping channels and Hund's coupling in the realization of Kitaev interactions in Co-based compounds, which has been overlooked in previous studies. This review suggests future directions in the search for Kitaev physics in 3dcobalt compounds and beyond.

On the proximate Kitaev quantum-spin liquid α-RuCl3: thermodynamics, excitations and continua

This topical review provides an overview over recent thermodynamic, infrared, and THz results on the proximate Kitaev spin-liquid. Quantum-spin liquids are exotic phases characterized by the absence of magnetic ordering even at the lowest temperatures and by the occurrence of fractionalized spin excitations. Among those, Kitaev spin liquids are most fascinating as they belong to the rare class of model systems, that can be solved analytically by decomposing localized spinsS= 1/2 into Majorana fermions. The main aim of this review is to summarize experimental evidence obtained by THz spectroscopy and utilizing heat-capacity experiments, which point to the existence of fractionalized excitations in the spin-liquid state, which in α-RuCl₃exists at temperatures just above the onset of magnetic order or at in-plane magnetic fields just beyond the quantum-critical point where antiferromagnetic order becomes suppressed. Thermodynamic and spectroscopic results are compared to theoretical predictions and model calculations. In addition, we document recent progress in elucidating the sub-gap (<1 eV) electronic structure of the 4d⁵ruthenium electrons to characterize their local electronic configuration. The on-site excitation spectra of thedelectrons below the optical gap can be consistently explained using a spin-orbit coupling constant of ∼170 meV and the concept of multiple spin-orbital excitations. Furthermore, we discuss the phonon spectra of the title compound including rigid-plane shear and compression modes of the single molecular layers. In recent theoretical concepts it has been shown that phonons can couple to Majorana fermions and may play a substantial role in establishing the half-integer thermal quantum Hall effect observed in this material.

Learning spin liquids on a honeycomb lattice with artificial neural networks

Machine learning methods provide a new perspective on the study of many-body system in condensed matter physics and there is only limited understanding of their representational properties and limitations in quantum spin liquid systems. In this work, we investigate the ability of the machine learning method based on the restricted Boltzmann machine in capturing physical quantities including the ground-state energy, spin-structure factor, magnetization, quantum coherence, and multipartite entanglement in the two-dimensional ferromagnetic spin liquids on a honeycomb lattice. It is found that the restricted Boltzmann machine can encode the many-body wavefunction quite well by reproducing accurate ground-state energy and structure factor. Further investigation on the behavior of multipartite entanglement indicates that the residual entanglement is richer in the gapless phase than the gapped spin-liquid phase, which suggests that the residual entanglement can characterize the spin-liquid phases. Additionally, we confirm the existence of a gapped non-Abelian topological phase in the spin liquids on a honeycomb lattice with a small magnetic field and determine the corresponding phase boundary by recognizing the rapid change of the local magnetization and residual entanglement.

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.

Scattering Signatures of Bond-Dependent Magnetic Interactions

Bond-dependent magnetic interactions can generate exotic phases such as Kitaev spin-liquid states. Experimentally determining the values of bond-dependent interactions is a challenging but crucial problem. Here, I show that each symmetry-allowed nearest-neighbor interaction on triangular and honeycomb lattices has a distinct signature in paramagnetic neutron-diffraction data, and that such data contain sufficient information to determine the spin Hamiltonian unambiguously via unconstrained fits. Moreover, I show that bond-dependent interactions can often be extracted from powder-averaged data. These results facilitate experimental determination of spin Hamiltonians for materials that do not show conventional magnetic ordering.

Majorana-Mediated Spin Transport in Kitaev Quantum Spin Liquids

We study the spin transport through the quantum spin liquid (QSL) by investigating the real-time and real-space dynamics of the Kitaev spin system with zigzag edges using the time-dependent Majorana mean-field theory. After the magnetic-field pulse is introduced to one of the edges, spin moments are excited in the opposite edge region although spin moments are never induced in the Kitaev QSL region. This unusual spin transport originates from the fact that the S=1/2 spins are fractionalized into the itinerant and localized Majorana fermions in the Kitaev system. Although both Majorana fermions are excited by the magnetic pulse, only the itinerant ones flow through the bulk regime without spin excitations, resulting in the spin transport in the Kitaev system despite the presence of a nonzero spin gap. We also demonstrate that this phenomenon can be observed in the system with small Heisenberg interactions using the exact diagonalization.

Materials design of Kitaev spin liquids beyond the Jackeli-Khaliullin mechanism

The Kitaev spin liquid provides a rare example of well-established quantum spin liquids in more than one dimension. It is obtained as the exact ground state of the Kitaev spin model with bond-dependent anisotropic interactions. The peculiar interactions can be yielded by the synergy of spin-orbit coupling and electron correlations for specific electron configuration and lattice geometry, which is known as the Jackeli-Khaliullin mechanism. Based on this mechanism, there has been a fierce race for the materialization of the Kitaev spin liquid over the last decade, but the candidates have been still limited mostly to 4d- and 5d-electron compounds including cations with the low-spind5electron configuration, such as Ir4+and Ru3+. Here we discuss recent efforts to extend the material perspective beyond the Jackeli-Khaliullin mechanism, by carefully reexamining the two requisites, formation of thejeff= 1/2 doublet and quantum interference between the exchange processes, for not onlyd- but alsof-electron systems. We present three examples: the systems including Co2+and Ni3+with the high-spind7electron configuration, Pr4+with thef1-electron configuration, and polar asymmetry in the lattice structure. In particular, the latter two are intriguing since they may realize the antiferromagnetic Kitaev interactions, in contrast to the ferromagnetic ones in the existing candidates. This partial overview would stimulate further material exploration of the Kitaev spin liquids and its topological properties due to fractional excitations.

Resonant inelastic x-ray scattering study of [Formula: see text]-RuCl3: a progress report

Ru M₃-edge resonant inelastic x-ray scattering (RIXS) measurements of [Formula: see text] with 27 meV resolution reveals a spin-orbit exciton without noticeable splitting. We extract values for the spin-orbit coupling constant ([Formula: see text] meV) and trigonal distortion field energy ([Formula: see text] meV) which support the [Formula: see text] nature of [Formula: see text]. We demonstrate the feasibility of M-edge RIXS for 4d systems, which allows ultra high-resolution RIXS of 4d systems until instrumentation for L-edge RIXS improves.

Magnetic field induced quantum phases in a tensor network study of Kitaev magnets

Recent discovery of the half quantized thermal Hall conductivity in [Formula: see text]-RuCl[Formula: see text], a candidate material for the Kitaev spin liquid, suggests the presence of a highly entangled quantum state in external magnetic fields. This field induced phase appears between the low field zig-zag magnetic order and the high field polarized state. Motivated by this experiment, we study possible field induced quantum phases in theoretical models of the Kitaev magnets, using the two dimensional tensor network approach or infinite tensor product states. We find various quantum ground states in addition to the chiral Kitaev spin liquid occupying a small area in the phase diagram. They form a band of emergent quantum phases in an intermediate window of external magnetic fields, somewhat reminiscent of the experiment. We discuss the implications of these results in view of the experiment and previous theoretical studies.

Electronic Properties of α-RuCl_{3} in Proximity to Graphene

In the pursuit of developing routes to enhance magnetic Kitaev interactions in α-RuCl_{3}, as well as probing doping effects, we investigate the electronic properties of α-RuCl_{3} in proximity to graphene. We study α-RuCl_{3}/graphene heterostructures via ab initio density functional theory calculations, Wannier projection, and nonperturbative exact diagonalization methods. We show that α-RuCl_{3} becomes strained when placed on graphene and charge transfer occurs between the two layers, making α-RuCl_{3} (graphene) lightly electron doped (hole doped). This gives rise to an insulator-to-metal transition in α-RuCl_{3} with the Fermi energy located close to the bottom of the upper Hubbard band of the t_{2g} manifold. These results suggest the possibility of realizing metallic and even exotic superconducting states. Moreover, we show that in the strained α-RuCl_{3} monolayer the Kitaev interactions are enhanced by more than 50% compared to the unstrained bulk structure. Finally, we discuss scenarios related to transport experiments in α-RuCl_{3}/graphene heterostructures.

Nonsymmorphic-Symmetry-Protected Topological Magnons in Three-Dimensional Kitaev Materials

Topological phases in magnetic materials offer novel tunability of topological properties via varying the underlying magnetism. We show that three-dimensional Kitaev materials with nonsymmorphic symmetries can provide a great opportunity for controlling symmetry-protected topological nodal magnons. These materials are originally considered as strong candidates for the Kitaev quantum spin liquid due to the bond-dependent frustrating spin-exchange interactions. As a concrete example, we consider the symmetry and topology of the magnons in the canted zigzag ordered state in the hyperhoneycomb β-Li_{2}IrO_{3}, which can be obtained by applying a magnetic field in the counter-rotating spiral state at zero field. It is shown that the magnetic glide symmetries and the non-Hermitian nature of the bosonic magnons lead to unique topological protection that is different from the case of their fermionic counterparts. We investigate how such topological magnons can be controlled by changing the symmetry of the underlying spin-exchange interactions.

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.

Vison Crystals in an Extended Kitaev Model on the Honeycomb Lattice

We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and, hence, the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric π-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature Monte Carlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.

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.

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.

Detuning the Honeycomb of the α-RuCl3 Kitaev Lattice: A Case of Cr3+ Dopant

Fine-tuning chemistry by doping with transition metals enables new perspectives for exploring Kitaev physics on a two-dimensional (2D) honeycomb lattice of α-RuCl₃, which is promising in the field of quantum information protection and quantum computation. The key parameters to vary by doping are both Heisenberg and Kitaev components of the nearest-neighbor exchange interaction between the J_eff = ¹/₂ Ru³⁺ spins, depending strongly on the peculiarities of the crystal structure. Here, we present crystal growth by chemical vapor transport and structure elucidation of a solid solution series Ru_{1- x}Cr _xCl₃ (0 ≤ x ≤ 1), with Cr³⁺ ions coupled to the Ru³⁺ Kitaev host. The Cr³⁺ substitution preserves the honeycomb type lattice of α-RuCl₃ and creates mixed occupancy of Ru/Cr sites without cationic order within the layers as confirmed by single-crystal X-ray diffraction and transmission electron microscopy investigations. In contrast to high-quality single crystals of α-RuCl₃ with ABAB-stacked layers, the ternary compounds demonstrate a significant stacking disorder along the c-axis direction as evidenced by X-ray diffraction and high resolution scanning transmission electron microscopy (HR-STEM). Raman spectra of substituted samples are in line with the symmetry conservation of the parent lattice upon chromium doping. At the same time, our magnetic susceptibility data indicate that the Kitaev physics of α-RuCl₃ is increasingly suppressed by the dominant spin-only driven magnetism of Cr³⁺ ( S = ³/₂) in Ru_{1- x}Cr _xCl₃.

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.

Nontrivial Triplon Topology and Triplon Liquid in Kitaev-Heisenberg-type Excitonic Magnets

The combination of strong spin-orbit coupling and correlations, e.g., in ruthenates and iridates, has been proposed as a means to realize quantum materials with nontrivial topological properties. We discuss here Mott insulators where on-site spin-orbit coupling favors a local J_{tot}=0 singlet ground state. We investigate excitations into a low-lying triplet, triplons, and find them to acquire nontrivial band topology in a magnetic field. We also comment on magnetic states resulting from triplon condensation, where we find, in addition to the same ordered phases known from the J_{tot}=1/2 Kitaev-Heisenberg model, a triplon liquid taking the parameter space of Kitaev's spin liquid.

Engineering Kitaev exchange in stacked iridate layers: impact of inter-layer species on in-plane magnetism

Electrostatic effects involving the inter-layer species are important: the largest Kitaev interactions comewith a more isotropic distribution of inter-layer cations around a given ligand.

Novel functionalities may be achieved in oxide electronics by appropriate stacking of planar oxide layers of different metallic species, MO_p and M′O_q. The simplest mechanism allowing the tailoring of the electronic states and physical properties of such heterostructures is of electrostatic nature—charge imbalance between the M and M′ cations. Here we clarify the effect of interlayer electrostatics on the anisotropic Kitaev exchange in H₃LiIr₂O₆, a recently proposed realization of the Kitaev spin liquid. By quantum chemical calculations, we show that the precise position of H⁺ cations between magnetically active [LiIr₂O₆]^3– honeycomb-like layers has a strong impact on the magnitude of Kitaev interactions. In particular, it is found that stacking with straight interlayer O–H–O links is detrimental to in-plane Kitaev exchange since coordination by a single H-ion of the O ligand implies an axial Coulomb potential at the O site and unfavorable polarization of the O 2p orbitals mediating the Ir–Ir interactions. Our results therefore provide valuable guidelines for the rational design of Kitaev quantum magnets, indicating unprecedented Kitaev interactions of ≈40 meV if the linear interlayer linkage is removed.

Sub-gap optical response in the Kitaev spin-liquid candidate α-RuCl3

We report detailed optical experiments on the layered compound α-RuCl₃ focusing on the THz and sub-gap optical response across the structural phase transition from the monoclinic high-temperature to the rhombohedral low-temperature structure, where the stacking sequence of the molecular layers is changed. This type of phase transition is characteristic for a variety of tri-halides crystallizing in a layered honeycomb-type structure and so far is unique, as the low-temperature phase exhibits the higher symmetry. One motivation is to unravel the microscopic nature of THz and spin-orbital excitations via a study of temperature and symmetry-induced changes. The optical studies are complemented by thermal expansion experiments. We document a number of highly unusual findings: A characteristic two-step hysteresis of the structural phase transition, accompanied by a dramatic change of the reflectivity. A complex dielectric loss spectrum in the THz regime, which could indicate remnants of Kitaev physics. Orbital excitations, which cannot be explained based on recent models, and an electronic excitation, which appears in a narrow temperature range just across the structural phase transition. Despite significant symmetry changes across the monoclinic to rhombohedral phase transition and a change of the stacking sequence, phonon eigenfrequencies and the majority of spin-orbital excitations are not strongly influenced. Obviously, the symmetry of a single molecular layer determines the eigenfrequencies of most of these excitations. Only one mode at THz frequencies, which becomes suppressed in the high-temperature monoclinic phase and one phonon mode experience changes in symmetry and stacking. Finally, from this combined terahertz, far- and mid-infrared study we try to shed some light on the so far unsolved low energy (<1 eV) electronic structure of the ruthenium 4d ⁵ electrons in α-RuCl₃.

Strong Effect of Hydrogen Order on Magnetic Kitaev Interactions in H_{3}LiIr_{2}O_{6}

Very recently a quantum liquid was reported to form in H_{3}LiIr_{2}O_{6}, an iridate proposed to be a close realization of the Kitaev honeycomb model. To test this assertion we perform detailed quantum chemistry calculations to determine the magnetic interactions between Ir moments. We find that weakly bond dependent ferromagnetic Kitaev exchange dominates over other couplings, but still is substantially lower than in Na_{2}IrO_{3}. This reduction is caused by the peculiar position of the interlayer species: removing hydrogen cations next to a Ir_{2}O_{2} plaquette increases the Kitaev exchange by more than a factor of 3 on the corresponding Ir─Ir link. Consequently, any lack of hydrogen order will have a drastic effect on the magnetic interactions and strongly promote spin disordering.

Antiferromagnetic spin-frustrated layers of corner-sharing Cu4 tetrahedra on the kagome lattice in volcanic minerals Cu5O2(VO4)2(CuCl), NaCu5O2(SeO3)2Cl3, and K2Cu5Cl8(OH)4·2H2O

The objective of the present work was to analyze the possibility of realization of quantum spin liquids in three volcanic minerals-averievite (Cu₅O₂(VO₄)₂(CuCl)), ilinskite (NaCu₅O₂(SeO₃)₂Cl₃), and avdononite (K₂Cu₅Cl₈(OH)₄·2H₂O)-from the crystal chemistry point of view. Based on the structural data, the sign and strength of magnetic interactions have been calculated and the geometric frustrations serving as the main reason of the existence of spin liquids have been investigated. According to our calculations, the magnetic structures of averievite and ilinskite are composed of antiferromagnetic (AFM) spin-frustrated layers of corner-sharing Cu₄ tetrahedra on the kagome lattice. However, the direction of nonshared corners of tetrahedra is different in them. The oxygen ions centering the OCu₄ tetrahedra in averievite and ilinskite provide the main contribution to the formation of AFM interactions along the tetrahedra edges. The local electric polarization in averievite and the possibility of spin configuration fluctuations due to vibrations of tetrahedra-centering oxygen ions have been discussed. The existence of structural phase transitions accompanied with magnetic transitions was assumed in ilinskite because of the effect of a lone electron pair by Se⁴⁺ ions. As was demonstrated through comparison of averievite and avdoninite, at the removal of centering oxygen ions from tetrahedra, the magnetic structure of the pyrochlore layer present in averievite transformed into an openwork curled net with large cells woven from corner-sharing open AFM spin-frustrated tetrahedra ('butterflies') in avdoninite.

Breakdown of Magnetic Order in the Pressurized Kitaev Iridate β-Li_{2}IrO_{3}

Temperature-pressure phase diagram of the Kitaev hyperhoneycomb iridate β-Li_{2}IrO_{3} is explored using magnetization, thermal expansion, magnetostriction, and muon spin rotation measurements, as well as single-crystal x-ray diffraction under pressure and ab initio calculations. The Néel temperature of β-Li_{2}IrO_{3} increases with the slope of 0.9  K/GPa upon initial compression, but the reduction in the polarization field H_{c} reflects a growing instability of the incommensurate order. At 1.4 GPa, the ordered state breaks down upon a first-order transition, giving way to a new ground state marked by the coexistence of dynamically correlated and frozen spins. This partial freezing in the absence of any conspicuous structural defects may indicate the classical nature of the resulting pressure-induced spin liquid, an observation paralleled to the increase in the nearest-neighbor off-diagonal exchange Γ under pressure.

Unusual Phonon Heat Transport in α-RuCl_{3}: Strong Spin-Phonon Scattering and Field-Induced Spin Gap

The honeycomb Kitaev-Heisenberg model is a source of a quantum spin liquid with Majorana fermions and gauge flux excitations as fractional quasiparticles. Here we unveil the highly unusual low-temperature heat conductivity κ of α-RuCl_{3}, a prime candidate for realizing such physics: beyond a magnetic field of B_{c}≈7.5  T, κ increases by about one order of magnitude, both for in-plane as well as out-of-plane transport. This clarifies the unusual magnetic field dependence unambiguously to be the result of severe scattering of phonons off putative Kitaev-Heisenberg excitations in combination with a drastic field-induced change of the magnetic excitation spectrum. In particular, an unexpected, large energy gap arises, which increases linearly with the magnetic field, reaching remarkable ℏω_{0}/k_{B}≈50  K at 18 T.

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.

Models and materials for generalized Kitaev magnetism

The exactly solvable Kitaev model on the honeycomb lattice has recently received enormous attention linked to the hope of achieving novel spin-liquid states with fractionalized Majorana-like excitations. In this review, we analyze the mechanism proposed by Jackeli and Khaliullin to identify Kitaev materials based on spin-orbital dependent bond interactions and provide a comprehensive overview of its implications in real materials. We set the focus on experimental results and current theoretical understanding of planar honeycomb systems (Na₂IrO₃, α-Li₂IrO₃, and α-RuCl₃), three-dimensional Kitaev materials (β- and γ-Li₂IrO₃), and other potential candidates, completing the review with the list of open questions awaiting new insights.

Destabilization of Magnetic Order in a Dilute Kitaev Spin Liquid Candidate

The insulating honeycomb magnet α-RuCl_{3} exhibits fractionalized excitations that signal its proximity to a Kitaev quantum spin liquid state; however, at T=0, fragile long-range magnetic order arises from non-Kitaev terms in the Hamiltonian. Spin vacancies in the form of Ir^{3+} substituted for Ru are found to destabilize this long-range order. Neutron diffraction and bulk characterization of Ru_{1-x}Ir_{x}Cl_{3} show that the magnetic ordering temperature is suppressed with increasing x, and evidence of zizag magnetic order is absent for x>0.3. Inelastic neutron scattering demonstrates that the signature of fractionalized excitations is maintained over the full range of x investigated. The depleted lattice without magnetic order thus hosts a spin-liquid-like ground state that may indicate the relevance of Kitaev physics in the magnetically dilute limit of RuCl_{3}.

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

α-RuCl_{3} is a leading candidate material for the observation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that α-RuCl_{3} undergoes a quantum phase transition to a QSL in a magnetic field of 7.5 T applied in the ab plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of α-RuCl_{3}.

Antiferromagnetic Resonance and Terahertz Continuum in α-RuCl_{3}

We report measurements of optical absorption in the zigzag antiferromagnet α-RuCl_{3} as a function of temperature T, magnetic field B, and photon energy ℏω in the range ∼0.3-8.3 meV, using time-domain terahertz spectroscopy. Polarized measurements show that threefold rotational symmetry is broken in the honeycomb plane from 2 to 300 K. We find a sharp absorption peak at 2.56 meV upon cooling below the Néel temperature of 7 K at B=0 that we identify as the magnetic-dipole excitation of a zero-wave-vector magnon, or antiferromagnetic resonance (AFMR). With the application of B, the AFMR broadens and shifts to a lower frequency as long-range magnetic order is lost in a manner consistent with transitioning to a spin-disordered phase. From a direct, internally calibrated measurement of the AFMR spectral weight, we place an upper bound on the contribution to the dc susceptibility from a magnetic excitation continuum.

Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in α-RuCl_{3}

We report on terahertz spectroscopy of quantum spin dynamics in α-RuCl_{3}, a system proximate to the Kitaev honeycomb model, as a function of temperature and magnetic field. We follow the evolution of an extended magnetic continuum below the structural phase transition at T_{s2}=62  K. With the onset of a long-range magnetic order at T_{N}=6.5  K, spectral weight is transferred to a well-defined magnetic excitation at ℏω_{1}=2.48  meV, which is accompanied by a higher-energy band at ℏω_{2}=6.48  meV. Both excitations soften in a magnetic field, signaling a quantum phase transition close to B_{c}=7  T, where a broad continuum dominates the dynamical response. Above B_{c}, the long-range order is suppressed, and on top of the continuum, emergent magnetic excitations evolve. These excitations follow clear selection rules and exhibit distinct field dependencies, characterizing the dynamical properties of a possibly field-induced quantum spin liquid.

Breakdown of magnons in a strongly spin-orbital coupled magnet

The description of quantized collective excitations stands as a landmark in the quantum theory of condensed matter. A prominent example occurs in conventional magnets, which support bosonic magnons-quantized harmonic fluctuations of the ordered spins. In striking contrast is the recent discovery that strongly spin-orbital-coupled magnets, such as α-RuCl3, may display a broad excitation continuum inconsistent with conventional magnons. Due to incomplete knowledge of the underlying interactions unraveling the nature of this continuum remains challenging. The most discussed explanation refers to a coherent continuum of fractional excitations analogous to the celebrated Kitaev spin liquid. Here, we present a more general scenario. We propose that the observed continuum represents incoherent excitations originating from strong magnetic anharmonicity that naturally occurs in such materials. This scenario fully explains the observed inelastic magnetic response of α-RuCl3 and reveals the presence of nontrivial excitations in such materials extending well beyond the Kitaev state.

Dynamics of the Kitaev-Heisenberg Model

We introduce a matrix-product state based method to efficiently obtain dynamical response functions for two-dimensional microscopic Hamiltonians. We apply this method to different phases of the Kitaev-Heisenberg model and identify characteristic dynamical features. In the ordered phases proximate to the spin liquid, we find significant broad high-energy features beyond spin-wave theory, which resemble those of the Kitaev model. This establishes the concept of a proximate spin liquid, which was recently invoked in the context of inelastic neutron scattering experiments on α-RuCl_{3}. Our results provide an example of a natural path for proximate spin liquid features to arise at high energies above a conventionally ordered state, as the diffuse remnants of spin-wave bands intersect to yield a broad peak at the Brillouin zone center.

Evidence for a Field-Induced Quantum Spin Liquid in α-RuCl_{3}

We report a ^{35}Cl nuclear magnetic resonance study in the honeycomb lattice α-RuCl_{3}, a material that has been suggested to potentially realize a Kitaev quantum spin liquid (QSL) ground state. Our results provide direct evidence that α-RuCl_{3} exhibits a magnetic-field-induced QSL. For fields larger than ∼10  T, a spin gap opens up while resonance lines remain sharp, evidencing that spins are quantum disordered and locally fluctuating. The spin gap increases linearly with an increasing magnetic field, reaching ∼50  K at 15 T, and is nearly isotropic with respect to the field direction. The unusual rapid increase of the spin gap with increasing field and its isotropic nature are incompatible with conventional magnetic ordering and, in particular, exclude that the ground state is a fully polarized ferromagnet. The presence of such a field-induced gapped QSL phase has indeed been predicted in the Kitaev model.

Anomalous Thermal Conductivity and Magnetic Torque Response in the Honeycomb Magnet α-RuCl_{3}

We report on the unusual behavior of the in-plane thermal conductivity κ and torque τ response in the Kitaev-Heisenberg material α-RuCl_{3}. κ shows a striking enhancement with linear growth beyond H=7  T, where magnetic order disappears, while τ for both of the in-plane symmetry directions shows an anomaly at the same field. The temperature and field dependence of κ are far more complex than conventional phonon and magnon contributions, and require us to invoke the presence of unconventional spin excitations whose properties are characteristic of a field-induced spin-liquid phase related to the enigmatic physics of the Kitaev model in an applied magnetic field.

Atomic-scale observation of structural and electronic orders in the layered compound α-RuCl3

A pseudospin-1/2 Mott phase on a honeycomb lattice is proposed to host the celebrated two-dimensional Kitaev model which has an elusive quantum spin liquid ground state, and fascinating physics relevant to the development of future templates towards topological quantum bits. Here we report a comprehensive, atomically resolved real-space study by scanning transmission electron and scanning tunnelling microscopies on a novel layered material displaying Kitaev physics, α-RuCl₃. Our local crystallography analysis reveals considerable variations in the geometry of the ligand sublattice in thin films of α-RuCl₃ that opens a way to realization of a spatially inhomogeneous magnetic ground state at the nanometre length scale. Using scanning tunnelling techniques, we observe the electronic energy gap of ≈0.25 eV and intra-unit cell symmetry breaking of charge distribution in individual α-RuCl₃ surface layer. The corresponding charge-ordered pattern has a fine structure associated with two different types of charge disproportionation at Cl-terminated surface.