Breakdown of magnons in a strongly spin-orbital coupled magnet

Periodic DFT calculations to compute the attributes of a quantum material: honeycomb ruthenium trichloride

Quantum spin liquids (QSLs) have become prominent materials of interest in the pursuit of fault-tolerant materials for quantum computing applications. This is due to the fact that these materials are theorized to host an interesting variety of quantum phenomena such as quasi-particles that may behave as anyons as a result of the high entangled nature of the spin states within the systems. Computing the electronic and magnetic properties of these materials is necessary in order to understand the underlying interactions of the materials. In this paper, the structural, electronic, and magnetic properties including lattice parameters, bandgap, Heisenberg coupling constants, and Curie temperatures for α-RuCl3, a promising candidate for the Kitaev QSL model, are computed using periodic density functional theory. Furthermore, various parameters of the calculations (i.e. functional choice, basis set, k-point density, and Hubbard correction) are varied in order to determine what effect, if any, the computational setup has on the computed properties. The results of this study indicate that PBE functional with Hubbard corrections of 1.5-2.5 eV with a k-point density of 3.0 points per Å-1 appear to be the best parameters to compute Heisenberg coupling constants for α-RuCl3. These parameters with the addition of spin orbit coupling works well for computing Curie temperatures for α-RuCl3. Distinct differences are noted in the computations of the bulk structure vs. monolayer structures, indicating that interactions between the layers play a role in the material properties and changes to the inter-layer spacing may result in interesting and unique magnetic properties that require further investigation.

Theory of Moiré Magnetism in Twisted Bilayer α-RuCl3

Motivated by the recent developments in moiré superlattices of van der Waals magnets and the desire to control the magnetic interactions of α-RuCl3, here we present a comprehensive theory of the long-range ordered magnetic phases of twisted bilayer α-RuCl3. Using a combination of first-principles calculations and atomistic simulations, we show that the stacking-dependent interlayer exchange gives rise to an array of magnetic phases that can be realized by controlling the twist angle. In particular, we discover a complex hexagonal domain structure in which multiple zigzag orders coexist. This multidomain order minimizes the interlayer energy while enduring the energy cost due to domain wall formation. Further, we show that quantum fluctuations can be enhanced across the phase transitions. Our results indicate that magnetic frustration due to stacking-dependent interlayer exchange in moiré superlattices can be exploited to tune quantum fluctuations and the magnetic ground state of α-RuCl3.

Triple-q Order in Na_{2}Co_{2}TeO_{6} from Proximity to Hidden-SU(2)-Symmetric Point

In extended Heisenberg-Kitaev-Gamma-type spin models, hidden-SU(2)-symmetric points are isolated points in parameter space that can be mapped to pure Heisenberg models via nontrivial duality transformations. Such points generically feature quantum degeneracy between conventional single-q and exotic multi-q states. We argue that recent single-crystal inelastic neutron scattering data place the honeycomb magnet Na_{2}Co_{2}TeO_{6} in proximity to such a hidden-SU(2)-symmetric point. The low-temperature order is identified as a triple-q state arising from the Néel antiferromagnet with staggered magnetization in the out-of-plane direction via a 4-sublattice duality transformation. This state naturally explains various distinctive features of the magnetic excitation spectrum, including its surprisingly high symmetry and the dispersive low-energy and flat high-energy bands. Our result demonstrates the importance of bond-dependent exchange interactions in cobaltates, and illustrates the intriguing magnetic behavior resulting from them.

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.

Disorder-Enriched Magnetic Excitations in a Heisenberg-Kitaev Quantum Magnet Na_{2}Co_{2}TeO_{6}

Using optical magnetospectroscopy, we investigate the magnetic excitations of Na_{2}Co_{2}TeO_{6} in a broad magnetic field range (0  T≤B≤17.5  T) at low temperature. Our measurements reveal rich spectra of in-plane magnetic excitations with a surprisingly large number of modes, even in the high-field spin-polarized state. Theoretical calculations find that the Na-occupation disorder in Na_{2}Co_{2}TeO_{6} plays a crucial role in generating these modes. Our Letter demonstrates the necessity to consider disorder in the spin environment in the search for Kitaev quantum spin liquid states in practicable materials.

Instabilities of heavy magnons in an anisotropic magnet

The search for new elementary particles is one of the most basic pursuits in physics, spanning from subatomic physics to quantum materials. Magnons are the ubiquitous elementary quasiparticle to describe the excitations of fully-ordered magnetic systems. But other possibilities exist, including fractional and multipolar excitations. Here, we demonstrate that strong quantum interactions exist between three flavors of elementary quasiparticles in the uniaxial spin-one magnet FeI₂. Using neutron scattering in an applied magnetic field, we observe spontaneous decay between conventional and heavy magnons and the recombination of these quasiparticles into a super-heavy bound-state. Akin to other contemporary problems in quantum materials, the microscopic origin for unusual physics in FeI₂ is the quasi-flat nature of excitation bands and the presence of Kitaev anisotropic magnetic exchange interactions.

Field-free spin-orbit torque switching assisted by in-plane unconventional spin torque in ultrathin [Pt/Co]N

Electrical manipulation of magnetization without an external magnetic field is critical for the development of advanced non-volatile magnetic-memory technology that can achieve high memory density and low energy consumption. Several recent studies have revealed efficient out-of-plane spin-orbit torques (SOTs) in a variety of materials for field-free type-z SOT switching. Here, we report on the corresponding type-x configuration, showing significant in-plane unconventional spin polarizations from sputtered ultrathin [Pt/Co]_N, which are either highly textured on single crystalline MgO substrates or randomly textured on SiO₂ coated Si substrates. The unconventional spin currents generated in the low-dimensional Co films result from the strong orbital magnetic moment, which has been observed by X-ray magnetic circular dichroism (XMCD) measurement. The x-polarized spin torque efficiency reaches up to -0.083 and favors complete field-free switching of CoFeB magnetized along the in-plane charge current direction. Micromagnetic simulations additionally demonstrate its lower switching current than type-y switching, especially in narrow current pulses. Our work provides additional pathways for electrical manipulation of spintronic devices in the pursuit of high-speed, high-density, and low-energy non-volatile memory.

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.

Dynamical Anyon Generation in Kitaev Honeycomb Non-Abelian Spin Liquids

Relativistic Mott insulators known as "Kitaev materials" potentially realize spin liquids hosting non-Abelian anyons. Motivated by fault-tolerant quantum-computing applications in this setting, we introduce a dynamical anyon-generation protocol that exploits universal edge physics. The setup features holes in the spin liquid, which define energetically cheap locations for non-Abelian anyons, connected by a narrow bridge that can be tuned between spin liquid and topologically trivial phases. We show that modulating the bridge from trivial to spin liquid over intermediate time scales-quantified by analytics and extensive simulations-deposits non-Abelian anyons into the holes with O(1) probability. The required bridge manipulations can be implemented by integrating the Kitaev material into magnetic tunnel junction arrays that engender locally tunable exchange fields. Combined with existing readout strategies, our protocol reveals a path to topological qubit experiments in Kitaev materials at zero applied magnetic field.

The Magnetic Genome of Two-Dimensional van der Waals Materials

Magnetism in two-dimensional (2D) van der Waals (vdW) materials has recently emerged as one of the most promising areas in condensed matter research, with many exciting emerging properties and significant potential for applications ranging from topological magnonics to low-power spintronics, quantum computing, and optical communications. In the brief time after their discovery, 2D magnets have blossomed into a rich area for investigation, where fundamental concepts in magnetism are challenged by the behavior of spins that can develop at the single layer limit. However, much effort is still needed in multiple fronts before 2D magnets can be routinely used for practical implementations. In this comprehensive review, prominent authors with expertise in complementary fields of 2D magnetism (i.e., synthesis, device engineering, magneto-optics, imaging, transport, mechanics, spin excitations, and theory and simulations) have joined together to provide a genome of current knowledge and a guideline for future developments in 2D magnetic materials research.

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.

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.

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.

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.

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.

Proximate ferromagnetic state in the Kitaev model material α-RuCl3

α-RuCl₃ is a major candidate for the realization of the Kitaev quantum spin liquid, but its zigzag antiferromagnetic order at low temperatures indicates deviations from the Kitaev model. We have quantified the spin Hamiltonian of α-RuCl₃ by a resonant inelastic x-ray scattering study at the Ru L₃ absorption edge. In the paramagnetic state, the quasi-elastic intensity of magnetic excitations has a broad maximum around the zone center without any local maxima at the zigzag magnetic Bragg wavevectors. This finding implies that the zigzag order is fragile and readily destabilized by competing ferromagnetic correlations. The classical ground state of the experimentally determined Hamiltonian is actually ferromagnetic. The zigzag state is stabilized by quantum fluctuations, leaving ferromagnetism – along with the Kitaev spin liquid – as energetically proximate metastable states. The three closely competing states and their collective excitations hold the key to the theoretical understanding of the unusual properties of α-RuCl₃ in magnetic fields.

RuCl₃ has stood out as a prime candidate in the search for quantum spin liquids; however, its antiferromagnetic ordering at low temperature suggests deviations from typical QSL models. Here, using resonant inelastic x-ray scattering, the authors provide a comprehensive determination of the low energy effective Hamiltonian.

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.

Giant phonon anomalies in the proximate Kitaev quantum spin liquid α-RuCl3

The Kitaev quantum spin liquid epitomizes an entangled topological state, for which two flavors of fractionalized low-energy excitations are predicted: the itinerant Majorana fermion and the Z2 gauge flux. It was proposed recently that fingerprints of fractional excitations are encoded in the phonon spectra of Kitaev quantum spin liquids through a novel fractional-excitation-phonon coupling. Here, we detect anomalous phonon effects in α-RuCl3 using inelastic X-ray scattering with meV resolution. At high temperature, we discover interlaced optical phonons intercepting a transverse acoustic phonon between 3 and 7 meV. Upon decreasing temperature, the optical phonons display a large intensity enhancement near the Kitaev energy, JK~8 meV, that coincides with a giant acoustic phonon softening near the Z2 gauge flux energy scale. These phonon anomalies signify the coupling of phonon and Kitaev magnetic excitations in α-RuCl3 and demonstrates a proof-of-principle method to detect anomalous excitations in topological quantum materials.

Sign Structure of Thermal Hall Conductivity and Topological Magnons for In-Plane Field Polarized Kitaev Magnets

The appearance of half-quantized thermal Hall conductivity in α-RuCl_{3} in the presence of in-plane magnetic fields has been taken as a strong evidence for the Kitaev spin liquid. Apart from the quantization, the observed sign structure of the thermal Hall conductivity is also consistent with predictions from the exact solution of the Kitaev honeycomb model. Namely, the thermal Hall conductivity changes sign when the field direction is reversed with respect to the heat current, which is perpendicular to one of the three nearest neighbor bonds on the honeycomb lattice. On the other hand, the thermal Hall conductivity is almost zero when the field is applied along the bond direction. Here, we theoretically demonstrate that such a peculiar sign structure of the thermal Hall conductivity is a generic property of the polarized state in the presence of in-plane magnetic fields. In this case, the thermal Hall effect arises from topological magnons with finite Chern numbers, and the sign structure follows from the symmetries of the momentum space Berry curvature. Using a realistic spin model with bond-dependent interactions, we show that the thermal Hall conductivity can have a magnitude comparable to that observed in the experiments. Hence, the sign structure alone cannot make a strong case for the Kitaev spin liquid. The quantization at very low temperatures, however, will be a decisive test as the magnon contribution vanishes in the zero temperature limit.

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.

Thermodynamic Perspective on Field-Induced Behavior of α-RuCl_{3}

Measurements of the magnetic Grüneisen parameter (Γ_{B}) and specific heat on the Kitaev material candidate α-RuCl_{3} are used to access in-plane field and temperature dependence of the entropy up to 12 T and down to 1 K. No signatures corresponding to phase transitions are detected beyond the boundary of the magnetically ordered region, but only a shoulderlike anomaly in Γ_{B}, involving an entropy increment as small as 10^{-5}Rlog2. These observations put into question the presence of a phase transition between the purported quantum spin liquid and the field-polarized state of α-RuCl_{3}. We show theoretically that at low temperatures Γ_{B} is sensitive to crossings in the lowest excitations within gapped phases, and identify the measured shoulderlike anomaly as being of such origin. Exact diagonalization calculations demonstrate that the shoulderlike anomaly can be reproduced in extended Kitaev models that gain proximity to an additional phase at finite field without entering it. We discuss manifestations of this proximity in other measurements.

Kitaev Spin Liquid in 3d Transition Metal Compounds

We study the exchange interactions and resulting magnetic phases in the honeycomb cobaltates. For a broad range of trigonal crystal fields acting on Co^{2+} ions, the low-energy pseudospin-1/2 Hamiltonian is dominated by bond-dependent Ising couplings that constitute the Kitaev model. The non-Kitaev terms nearly vanish at small values of trigonal field Δ, resulting in spin liquid ground state. Considering Na_{3}Co_{2}SbO_{6} as an example, we find that this compound is proximate to a Kitaev spin liquid phase, and can be driven into it by slightly reducing Δ by ∼20  meV, e.g., via strain or pressure control. We argue that, due to the more localized nature of the magnetic electrons in 3d compounds, cobaltates offer the most promising search area for Kitaev model physics.

Nature of Magnetic Excitations in the High-Field Phase of α-RuCl_{3}

We present comprehensive electron spin resonance (ESR) studies of in-plane oriented single crystals of α-RuCl_{3}, a quasi-two-dimensional material with honeycomb structure, focusing on its high-field spin dynamics. The measurements were performed in magnetic fields up to 16 T, applied along the [110] and [100] directions. Several ESR modes were detected. Combining our findings with recent inelastic neutron- and Raman-scattering data, we identified most of the observed excitations. Most importantly, we show that the low-temperature ESR response beyond the boundary of the magnetically ordered region is dominated by single- and two-particle processes with magnons as elementary excitations. The peculiarities of the excitation spectrum in the vicinity of the critical field are discussed.

Temperature dependence of the(π,0)anomaly in the excitation spectrum of the 2D quantum Heisenberg antiferromagnet

It is well established that in the low-temperature limit, the two-dimensional quantum Heisenberg antiferromagnet on a square lattice (2DQHAFSL) exhibits an anomaly in its spectrum at short-wavelengths on the zone-boundary. In the vicinity of thepoint the pole in the one-magnon response exhibits a downward dispersion, is heavily damped and attenuated, giving way to an isotropic continuum of excitations extending to high energies. The origin of the anomaly and the presence of the continuum are of current theoretical interest, with suggestions focused around the idea that the latter evidences the existence of spinons in a two-dimensional system. Here we present the results of neutron inelastic scattering experiments and Quantum Monte Carlo calculations on the metallo-organic compound Cu(DCOO)D2O (CFTD), an excellent physical realisation of the 2DQHAFSL, designed to investigate how the anomaly atevolves up to finite temperatures. Our data reveal that on warming the anomaly survives the loss of long-range, three-dimensional order, and that it is thus a robust feature of the two-dimensional system. With further increase of temperature the zone-boundary response gradually softens and broadens, washing out theanomaly. This is confirmed by a comparison of our data with the results of finite-temperature Quantum Monte Carlo simulations where the two are found to be in good accord. In the vicinity of the antiferromagnetic zone centre, there was no significant softening of the magnetic excitations over the range of temperatures investigated.

Spin liquids in geometrically perfect triangular antiferromagnets

The cradle of quantum spin liquids, triangular antiferromagnets show strong proclivity to magnetic order and require deliberate tuning to stabilize a spin-liquid state. In this brief review, we juxtapose recent theoretical developments that trace the parameter regime of the spin-liquid phase, with experimental results for Co-based and Yb-based triangular antiferromagnets. Unconventional spin dynamics arising from both ordered and disordered ground states are discussed, and the notion of a geometrically perfect triangular system is scrutinized to demonstrate non-trivial imperfections that may assist magnetic frustration in stabilizing dynamic spin states with peculiar excitations.

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.

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.

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.

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.

Role of Hydrogen in the Spin-Orbital-Entangled Quantum Liquid Candidate H_{3}LiIr_{2}O_{6}

Motivated by recent reports of H_{3}LiIr_{2}O_{6} as a spin-orbital-entangled quantum liquid, we investigate via a combination of density functional theory and nonperturbative exact diagonalization the microscopic nature of its magnetic interactions. We find that while the interlayer O─H─O bond geometry strongly affects the local magnetic couplings, these bonds are likely to remain symmetrical due to large zero-point fluctuations of the H positions. In this case, the estimated magnetic model lies close to the classical tricritical point between ferromagnetic, zigzag, and incommensurate spiral orders, what may contribute to the lack of magnetic ordering. However, we also find that substitution of H by D (deuterium) as well as disorder-induced inhomogeneities destabilizes the O─H or D─O bonds, modifying strongly the local magnetic couplings. These results suggest that the magnetic response in H_{3}LiIr_{2}O_{6} is likely sensitive to both the stoichiometry and the microstructure of the samples and emphasize the importance of a careful treatment of hydrogen for similar systems.

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.

Diagnosing Fractionalization from the Spin Dynamics of Z_{2} Spin Liquids on the Kagome Lattice by Quantum Monte Carlo Simulations

Based on large-scale quantum Monte Carlo simulations, we examine the dynamical spin structure factor of the Balents-Fisher-Girvin kagome lattice spin-1/2 model, which is known to harbor an extended Z_{2} quantum spin liquid phase. We use a correlation-matrix sampling scheme combined with a stochastic analytic continuation method to resolve the spectral functions of this anisotropic quantum spin model with a three-site unit cell. Based on this approach, we monitor the spin dynamics throughout the phase diagram of this model, from the XY-ferromagnetic region to the Z_{2} quantum spin liquid regime. In the latter phase, we identify a gapped two-spinon continuum in the transverse scattering channel, which is faithfully modeled by an effective spinon tight-binding model. Within the longitudinal channel, we identify gapped vison excitations and exhibit indications for the translational symmetry fractionalization of the visons via an enhanced spectral periodicity.

Unusual Thermal Hall Effect in a Kitaev Spin Liquid Candidate α-RuCl_{3}

The Kitaev quantum spin liquid displays the fractionalization of quantum spins into Majorana fermions. The emergent Majorana edge current is predicted to manifest itself in the form of a finite thermal Hall effect, a feature commonly discussed in topological superconductors. Here we report on thermal Hall conductivity κ_{xy} measurements in α-RuCl_{3}, a candidate Kitaev magnet with the two-dimensional honeycomb lattice. In a spin-liquid (Kitaev paramagnetic) state below the temperature characterized by the Kitaev interaction J_{K}/k_{B}∼80  K, positive κ_{xy} develops gradually upon cooling, demonstrating the presence of highly unusual itinerant excitations. Although the zero-temperature property is masked by the magnetic ordering at T_{N}=7  K, the sign, magnitude, and T dependence of κ_{xy}/T at intermediate temperatures follows the predicted trend of the itinerant Majorana excitations.

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}.

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.

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.