Models and materials for generalized Kitaev magnetism

Vacancy Spectroscopy of Non-Abelian Kitaev Spin Liquids

Spin vacancies in the non-Abelian Kitaev spin liquid are known to harbor Majorana zero modes, potentially enabling topological quantum computing at elevated temperatures. Here, we study the spectroscopic signatures of such Majorana zero modes in a scanning tunneling setup where a non-Abelian Kitaev spin liquid with a finite density of spin vacancies forms a tunneling barrier between a tip and a substrate. Our key result is a well-defined peak close to zero bias voltage in the derivative of the tunneling conductance whose voltage and intensity both increase with the density of vacancies. This "quasi-zero-voltage peak" is identified as the closest analog of the zero-voltage peak observed in topological superconductors that additionally reflects the fractionalized nature of spin-liquid-based Majorana zero modes. We further highlight a single-fermion Van Hove singularity at a higher voltage that reveals the energy scale of the emergent Majorana fermions in the Kitaev spin liquid. Our proposed signatures are within reach of current experiments on the candidate material α-RuCl_{3}.

Chirality-selective topological magnon phase transition induced by interplay of anisotropic exchange interactions in honeycomb ferromagnet

A variety of distinct anisotropic exchange interactions commonly exist in one magnetic material due to complex crystal, magnetic and orbital symmetries. Here we investigate the effects of multiple anisotropic exchange interactions on topological magnon in a honeycomb ferromagnet, and find a chirality-selective topological magnon phase transition induced by a complicated interplay of Dzyaloshinsky-Moriya interaction and pseudo-dipolar interaction, accompanied by the bulk gap close and reopen with chiral inversion. Moreover, this novel topological phase transition involves band inversion at high symmetry pointsKandK', which can be regarded as a pseudo-orbital reversal, i.e. magnon valley degree of freedom, implying a new manipulation corresponding to a sign change of the magnon thermal Hall conductivity. Indeed, it can be realized in 4dor 5dcorrelated materials with both spin-orbit coupling and orbital localized states, such as iridates and ruthenates,etc.This novel regulation may have potential applications on magnon devices and topological magnonics.

Beyond Kitaev physics in strong spin-orbit coupled magnets

We review the recent advances and current challenges in the field of strong spin-orbit coupled Kitaev materials, with a particular emphasis on the physics beyond the exactly-solvable Kitaev spin liquid point. To this end, we present a comprehensive overview of the key exchange interactions in candidate materials with a specific focus on systems featuring effectiveJeff=1/2magnetic moments. This includes, but not limited to,5d5iridates,4d5ruthenates and3d7cobaltates. Our exploration covers the microscopic origins of these interactions, along with a systematic attempt to map out the most intriguing correlated regimes of the multi-dimensional parameter space. Our approach is guided by robust symmetry and duality transformations as well as insights from a wide spectrum of analytical and numerical studies. We also survey higher spin Kitaev models and recent exciting results on quasi-one-dimensional models and discuss their relevance to higher-dimensional models. Finally, we highlight some of the key questions in the field as well as future directions.

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.

An exact chiral amorphous spin liquid

Topological insulator phases of non-interacting particles have been generalized from periodic crystals to amorphous lattices, which raises the question whether topologically ordered quantum many-body phases may similarly exist in amorphous systems? Here we construct a soluble chiral amorphous quantum spin liquid by extending the Kitaev honeycomb model to random lattices with fixed coordination number three. The model retains its exact solubility but the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian quantum spin liquid phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.

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.

Confinement of Fractional Excitations in a Triangular Lattice Antiferromagnet

High-resolution neutron and THz spectroscopies are used to study the magnetic excitation spectrum of Cs_{2}CoBr_{4}, a distorted-triangular-lattice antiferromagnet with nearly XY-type anisotropy. What was previously thought of as a broad excitation continuum [L. Facheris et al., Phys. Rev. Lett. 129, 087201 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.087201] is shown to be a series of dispersive bound states reminiscent of "Zeeman ladders" in quasi-one-dimensional Ising systems. At wave vectors where interchain interactions cancel at the mean field level, they can indeed be interpreted as bound finite-width kinks in individual chains. Elsewhere in the Brillouin zone their true two-dimensional structure and propagation are revealed.

Amorphous and Polycrystalline Routes toward a Chiral Spin Liquid

We show that a chiral spin liquid spontaneously emerges in partially amorphous, polycrystalline, or ion-irradiated Kitaev materials. In these systems, time-reversal symmetry is broken spontaneously due to a nonzero density of plaquettes with an odd number of edges n_{odd}. This mechanism opens a sizable gap, at small n_{odd} compatible with that of typical amorphous materials and polycrystals, and which can alternatively be induced by ion irradiation. We find that the gap is proportional to n_{odd}, saturating at n_{odd}∼40%. Using exact diagonalization, we find that the chiral spin liquid is approximately as stable to Heisenberg interactions as Kitaev's honeycomb spin-liquid model. Our results open up a significant number of noncrystalline systems where chiral spin liquids can emerge without external magnetic fields.

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.

Excitations in the Ordered and Paramagnetic States of Honeycomb Magnet Na_{2}Co_{2}TeO_{6}

Na_{2}Co_{2}TeO_{6} is a proposed approximate Kitaev magnet, yet its actual magnetic interactions are elusive due to a lack of knowledge on the full excitation spectrum. Here, using inelastic neutron scattering and single crystals, we determine the system's temperature-dependent magnetic excitations over the entire Brillouin zone. Without committing to specific models, we unveil a distinct signature of the third-nearest-neighbor coupling in the spin waves, which signifies the associated distance as an emerging effective link in the ordered state. The presence of at least six nonoverlapping spin-wave branches is at odds with all models proposed to date. Above the ordering temperature, persisting dynamic correlations can be described by equal-time magnetic structure factors of a hexagonal cluster, which reveal the leading instabilities. Our result sets definitive constraints on theoretical models for Na_{2}Co_{2}TeO_{6} and provides new insight for the materialization of the Kitaev model.

Disorder, Low-Energy Excitations, and Topology in the Kitaev Spin Liquid

The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. However, deviations from the original Kitaev model are expected to appear in real materials. In this Letter, we investigate the fate of Kitaev's spin liquid in the presence of disorder-bond defects or vacancies-for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states with a nonuniversal exponent. We link this power-law distribution of energy scales to weakly coupled droplets inside the bulk, in an uncanny similarity to the Griffiths phase often present in the vicinity of disordered quantum phase transitions. If time-reversal symmetry is broken, we find that power-law singularities are tied to the destruction of the topological phase of the Kitaev model in the presence of bond disorder alone. However, there is a transition from this topologically trivial phase with power-law singularities to a topologically nontrivial one for weak to moderate site dilution. Therefore, diluted Kitaev materials are potential candidates to host Kitaev's chiral spin-liquid phase.

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.

One-to-one correspondence between thermal structure factors and coupling constants of general bilinear Hamiltonians

A theorem that establishes a one-to-one relation between zero-temperature static spin-spin correlators and coupling constants for a general class of quantum spin Hamiltonians bilinear in the spin operators has been recently established by Quintanilla, using an argument in the spirit of the Hohenberg-Kohn theorem in density functional theory. Quintanilla's theorem gives a firm theoretical foundation to quantum spin Hamiltonian learning using spin structure factors as input data. Here we extend the validity of the theorem in two directions. First, following the same approach as Mermin, the proof is extended to the case of finite-temperature spin structure factors, thus ensuring that the application of this theorem to experimental data is sound. Second, we note that this theorem applies to all types of Hamiltonians expressed as sums of bilinear operators, so that it can also relate the density-density correlators to the Coulomb matrix elements for interacting electrons in the lowest Landau level.

Low-field magnetic anisotropy of Sr2IrO4

Magnetic anisotropy in strontium iridate (Sr₂IrO₄) is essential because of its strong spin-orbit coupling and crystal field effect. In this paper, we present a detailed mapping of the out-of-plane (OOP) magnetic anisotropy in Sr₂IrO₄for different sample orientations using torque magnetometry measurements in the low-magnetic-field region before the isospins are completely ordered. Dominant in-plane anisotropy was identified at low fields, confirming thebaxis as an easy magnetization axis. Based on the fitting analysis of the strong uniaxial magnetic anisotropy, we observed that the main anisotropic effect arises from a spin-orbit-coupled magnetic exchange interaction affecting the OOP interaction. The effect of interlayer exchange interaction results in additional anisotropic terms owing to the tilting of the isospins. The results are relevant for understanding OOP magnetic anisotropy and provide a new way to analyze the effects of spin-orbit-coupling and interlayer magnetic exchange interactions. This study provides insight into the understanding of bulk magnetic, magnetotransport, and spintronic behavior on Sr₂IrO₄for future studies.

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.

Nonmagnetic J=0 State and Spin-Orbit Excitations in K_{2}RuCl_{6}

Spin-orbit Mott insulators composed of t_{2g}^{4} transition metal ions may host excitonic magnetism due to the condensation of spin-orbital J=1 triplons. Prior experiments suggest that the 4d antiferromagnet Ca_{2}RuO_{4} embodies this notion, but a J=0 nonmagnetic state as a basis of the excitonic picture remains to be confirmed. We use Ru L_{3}-edge resonant inelastic x-ray scattering to reveal archetypal J multiplets with a J=0 ground state in the cubic compound K_{2}RuCl_{6}, which are well described within the LS-coupling scheme. This result highlights the critical role of unquenched orbital moments in 4d-electron compounds and calls for investigations of quantum criticality and excitonic magnetism on various crystal lattices.

Spin-orbit effects in pentavalent iridates: models and materials

Spin-orbit effects in heavy 5dtransition metal oxides, in particular, iridates, have received enormous current interest due to the prediction as well as the realization of a plethora of exotic and unconventional magnetic properties. While a bulk of these works are based on tetravalent iridates (d⁵), where the counter-intuitive insulating state of the rather extended 5dorbitals are explained by invoking strong spin-orbit coupling, the recent quest in iridate research has shifted to the other valencies of Ir, of which pentavalent iridates constitute a notable representative. In contrast to the tetravalent iridates, spin-orbit entangled electrons ind⁴systems are expected to be confined to theJ= 0 singlet state without any resultant moment or magnetic response. However, it has been recently predicted that, magnetism ind⁴systems may occur via magnetic condensation of excitations across spin-orbit-coupled states. In reality, the magnetism in Ir⁵⁺systems are often quite debatable both from theoretical as well as experimental point of view. Here we provide a comprehensive overview of the spin-orbit coupledd⁴model systems and its implications in the studied pentavalent iridates. In particular, we review here the current experimental and theoretical understanding of the double perovskite (A₂BYIrO₆,A= Sr, Ba,B= Y, Sc, Gd), 6H-perovskite (Ba₃MIr₂O₉,M= Zn, Mg, Sr, Ca), post-perovskite (NaIrO₃), and hexagonal (Sr₃MIrO₆) iridates, along with a number of open questions that require future investigation.

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.

Inferring hidden symmetries of exotic magnets from detecting explicit order parameters

An unconventional magnet may be mapped onto a simple ferromagnet by the existence of a high-symmetry point. Knowledge of conventional ferromagnetic systems may then be carried over to provide insight into more complex orders. Here we demonstrate how an unsupervised and interpretable machine-learning approach can be used to search for potential high-symmetry points in unconventional magnets without any prior knowledge of the system. The method is applied to the classical Heisenberg-Kitaev model on a honeycomb lattice, where our machine learns the transformations that manifest its hidden O(3) symmetry, without using data of these high-symmetry points. Moreover, we clarify that, in contrast to the stripy and zigzag orders, a set of D_{2} and D_{2h} ordering matrices provides a more complete description of the magnetization in the Heisenberg-Kitaev model. In addition, our machine also learns the local constraints at the phase boundaries, which manifest a subdimensional symmetry. This paper highlights the importance of explicit order parameters to many-body spin systems and the property of interpretability for the physical application of machine-learning techniques.

Half-integer quantized anomalous thermal Hall effect in the Kitaev material candidate α-RuCl3

Half-integer thermal quantum Hall conductance has recently been reported for the two-dimensional honeycomb material α-RuCl₃ We found that the half-integer thermal Hall plateau appears even for a magnetic field with no out-of-plane components. The measured field-angular variation of the quantized thermal Hall conductance has the same sign structure as the topological Chern number of the pure Kitaev spin liquid. This observation suggests that the non-Abelian topological order associated with fractionalization of the local magnetic moments persists even in the presence of non-Kitaev interactions in α-RuCl₃.

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.

Multiple pseudo-plateaux states and antiferromagnetic pair selection in the XY model in the highly frustrated honeycomb lattice

We present a study of the low temperature magnetic phases of the classicalXYmodel with third nearest neighbor interactions on the honeycomb lattice at the maximally frustrated point under an external magnetic field by extensive Monte Carlo simulations. We focus on the characterization of the emergent low temperature phases, which are a direct consequence of the unusually high numbers of spins per plaquette in the model. Specifically, we show that, since thermal fluctuations partially lift the ground-state degeneracy and select the most collinear states, the selected states are those with the highest number of 'antiferromagnetic pairs' (AFp) compatible with the external magnetic field. These AFp are formed in such a way that they maximize the degeneracy of the selected submanifold of ground states. Moreover, two collinear pseudoplateaux emerge atM= 1/3 andM= 2/3. To characterize the magnetization process, we employ Monte Carlo simulations and calculate relevant order parameters to construct the complete temperature vs magnetic field phase diagram.

Anomalous Quantum Oscillations in a Heterostructure of Graphene on a Proximate Quantum Spin Liquid

The quasi-two-dimensional Mott insulator α-RuCl_{3} is proximate to the sought-after Kitaev quantum spin liquid (QSL). In a layer of α-RuCl_{3} on graphene, the dominant Kitaev exchange is further enhanced by strain. Recently, quantum oscillation (QO) measurements of such α-RuCl_{3} and graphene heterostructures showed an anomalous temperature dependence beyond the standard Lifshitz-Kosevich (LK) description. Here, we develop a theory of anomalous QO in an effective Kitaev-Kondo lattice model in which the itinerant electrons of the graphene layer interact with the correlated magnetic layer via spin interactions. At low temperatures, a heavy Fermi liquid emerges such that the neutral Majorana fermion excitations of the Kitaev QSL acquire charge by hybridizing with the graphene Dirac band. Using ab initio calculations to determine the parameters of our low-energy model, we provide a microscopic theory of anomalous QOs with a non-LK temperature dependence consistent with our measurements. We show how remnants of fractionalized spin excitations can give rise to characteristic signatures in QO experiments.

Honeycomb layered oxides: structure, energy storage, transport, topology and relevant insights

The advent of nanotechnology has hurtled the discovery and development of nanostructured materials with stellar chemical and physical functionalities in a bid to address issues in energy, environment, telecommunications and healthcare. In this quest, a class of two-dimensional layered materials consisting of alkali or coinage metal atoms sandwiched between slabs exclusively made of transition metal and chalcogen (or pnictogen) atoms arranged in a honeycomb fashion have emerged as materials exhibiting fascinatingly rich crystal chemistry, high-voltage electrochemistry, fast cation diffusion besides playing host to varied exotic electromagnetic and topological phenomena. Currently, with a niche application in energy storage as high-voltage materials, this class of honeycomb layered oxides serves as ideal pedagogical exemplars of the innumerable capabilities of nanomaterials drawing immense interest in multiple fields ranging from materials science, solid-state chemistry, electrochemistry and condensed matter physics. In this review, we delineate the relevant chemistry and physics of honeycomb layered oxides, and discuss their functionalities for tunable electrochemistry, superfast ionic conduction, electromagnetism and topology. Moreover, we elucidate the unexplored albeit vastly promising crystal chemistry space whilst outlining effective ways to identify regions within this compositional space, particularly where interesting electromagnetic and topological properties could be lurking within the aforementioned alkali and coinage-metal honeycomb layered oxide structures. We conclude by pointing towards possible future research directions, particularly the prospective realisation of Kitaev-Heisenberg-Dzyaloshinskii-Moriya interactions with single crystals and Floquet theory in closely-related honeycomb layered oxide materials.

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.

Electrical Access to Ising Anyons in Kitaev Spin Liquids

We show that spin-spin correlations in a non-Abelian Kitaev spin liquid are associated with a characteristic inhomogeneous charge density distribution in the vicinity of Z_{2} vortices. This density profile and the corresponding local electric fields are observable, e.g., by means of surface probe techniques. Conversely, by applying bias voltages to several probe tips, one can stabilize Ising anyons (Z_{2} vortices harboring a Majorana zero mode) at designated positions, where we predict a clear Majorana signature in energy absorption spectroscopy.

Magnetism and ion diffusion in honeycomb layered oxide [Formula: see text]

In the quest for developing novel and efficient batteries, a great interest has been raised for sustainable K-based honeycomb layer oxide materials, both for their application in energy devices as well as for their fundamental material properties. A key issue in the realization of efficient batteries based on such compounds, is to understand the K-ion diffusion mechanism. However, investigation of potassium-ion (K[Formula: see text]) dynamics in materials using e.g. NMR and related techniques has so far been very challenging, due to its inherently weak nuclear magnetic moment, in contrast to other alkali ions such as lithium and sodium. Spin-polarised muons, having a high gyromagnetic ratio, make the muon spin rotation and relaxation ([Formula: see text]SR) technique ideal for probing ions dynamics in these types of energy materials. Here we present a study of the low-temperature magnetic properties as well as K[Formula: see text] dynamics in honeycomb layered oxide material [Formula: see text] using mainly the [Formula: see text]SR technique. Our low-temperature [Formula: see text]SR results together with complementary magnetic susceptibility measurements find an antiferromagnetic transition at [Formula: see text] K. Further [Formula: see text]SR studies performed at higher temperatures reveal that potassium ions (K[Formula: see text]) become mobile above 200 K and the activation energy for the diffusion process is obtained as [Formula: see text] meV. This is the first time that K[Formula: see text] dynamics in potassium-based battery materials has been measured using [Formula: see text]SR. Assisted by high-resolution neutron diffraction, the temperature dependence of the K-ion self diffusion constant is also extracted. Finally our results also reveal that K-ion diffusion occurs predominantly at the surface of the powder particles. This opens future possibilities for potentially improving ion diffusion as well as K-ion battery device performance using nano-structuring and surface coatings of the particles.

Structural stability of CuAl2O4under pressure

Structural properties of CuAl2O4, which was recently argued to show unusual suppression of the Jahn-Teller distortions by the spin-orbit coupling, are investigated under pressures up to 6 GPa. Analysis of x-ray powder diffraction experiments shows that CuAl2O4gets unstable and decomposes onto CuO and Al2O3at pressures ∼6 GPa and temperature ∼1000 K. This finding is complemented by the density-functional theory +U+ spin-orbit coupling calculations, which demonstrate that this instability is partially driven by a (relatively) large compressibility of strongly Jahn-Teller distorted CuO.

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.

Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons

We provide an exact study of dynamical correlations for the quantum spin-orbital liquid phases of an SU(2)-symmetric Kitaev honeycomb lattice model. We show that the spin dynamics in this Kugel-Khomskii type model is exactly the density-density correlation function of S=1 fermionic magnons, which could be probed in resonant inelastic x-ray scattering experiments. We predict the characteristic signatures of spin-orbital fractionalization in inelastic scattering experiments and compare them to the ones of the spin-anisotropic Kitaev honeycomb spin liquid. In particular, the resonant inelastic x-ray scattering response shows a characteristic momentum dependence directly related to the dispersion of fermionic excitations. The neutron scattering cross section displays a mixed response of fermionic magnons as well as spin-orbital excitations. The latter has a bandwidth of broad excitations and a vison gap that is three times larger than that of the spin-1=2 Kitaev model.

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.

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.

Pulsed spallation neutron spectroscopy of low dimensional magnets: past, present, and future

The early 1990s saw the first useful application of pulsed neutron spectroscopy to the study of excitations in low dimensional magnetic systems, with Roger Cowley as a key participant in important early experiments. Since that time the technique has blossomed as a powerful tool utilizing vastly improved neutron instrumentation coupled with more powerful pulsed sources. Here we review representative experiments illustrating some of the fascinating physics that has been revealed in quasi-one and two dimensional systems.

Novel Chiral Quantum Spin Liquids in Kitaev Magnets

Quantum magnets with pure Kitaev spin exchange interactions can host a gapped quantum spin liquid with a single Majorana edge mode propagating in the counterclockwise direction when a small positive magnetic field is applied. Here, we show how under a sufficiently strong positive magnetic field a topological transition into a gapped quantum spin liquid with two Majorana edge modes propagating in the clockwise direction occurs. The Dzyaloshinskii-Moriya interaction is found to turn the nonchiral Kitaev's gapless quantum spin liquid into a chiral one with equal Berry phases at the two Dirac points. Thermal Hall conductance experiments can provide evidence of the novel topologically gapped quantum spin liquid states predicted.

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.

Revisiting the Role of Hydrogen Bonding in the Strong Dimer Superexchange of a 2D Copper(II) Halide Honeycomb-Like Lattice: Structural and Magnetic Study

The title compound H₂L(CuCl₃H₂O)Cl (H₂L = 1-(4'-pyridinium)pyridin-4-ol-ium), 1) was synthesized and investigated structurally and magnetically as well as via a first-principles, bottom-up theoretical analysis of the potential magnetic superexchange pathways. Compound 1 can be described structurally as a well-isolated, distorted 2D-honeycomb lattice with two potential exchange pathways: a dimeric interaction via hydrogen-bonded pairs of (CuCl₃H₂O) ions and a chain structure via bridging chloride ions. Surprisingly, the experimental magnetic data are best fitted using both a simple dimer model with a Curie-Weiss correction for interdimer exchange (J_dimer = -107.4(1) K, θ = -1.22(4) K) and a strong-rung ladder model (J_rung = -105.8(7) K, J_rail = 2(7) K). Theoretical analysis at the UB3LYP/6-31+G(d) level supports the strong exchange observed through the [CuCl₄(H₂O)]^2- dimer moiety superexchange pathway (-102 K = -71 cm^-1). However, the apparent vanishingly small exchange through the single halide bridge is merely a brute average of competing ferromagnetic (FM) (+24.8 K = +17.0 cm^-1) and antiferromagnetic (AFM) (-21.0 K = -14.6 cm^-1) exchange interactions. Our computational study shows that these fitting parameters carry no physical meaning since a honeycomb plaquette must be taken as magnetic building block for 1. The competition between FM and AFM pair interactions leads to geometrical frustration in 1 and could induce interesting magnetic response at low temperatures, if the magnetic exchange is adequately tuned by modifying substituents in ligands and, in turn, interactions within the crystal packing.

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

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

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.

Theory of Two-Dimensional Nonlinear Spectroscopy for the Kitaev Spin Liquid

Unambiguous identification of fractionalized excitations in quantum spin liquids has been a long-standing issue in correlated topological phases. Conventional spectroscopic probes, such as the dynamical spin structure factor, can only detect composites of fractionalized excitations, leading to a broad continuum in energy. Lacking a clear signature in conventional probes has been the biggest obstacle in the field. In this work, we theoretically investigate what kinds of distinctive signatures of fractionalized excitations can be probed in two-dimensional nonlinear spectroscopy by considering the exactly solvable Kitaev spin liquids. We demonstrate the existence of a number of salient features of the Majorana fermions and fluxes in two-dimensional nonlinear spectroscopy, which provide crucial information about such excitations.

Octupolar versus Néel Order in Cubic 5d^{2} Double Perovskites

We report time-of-flight neutron spectroscopy and neutron and x-ray diffraction studies of the 5d^{2} double perovskite magnets, Ba_{2}MOsO_{6} (M=Zn,Mg,Ca). These materials host antiferromagnetically coupled 5d^{2} Os^{6+} ions decorating a face-centered cubic (fcc) lattice and are found to remain cubic down to the lowest temperatures. They all exhibit thermodynamic anomalies consistent with a single phase transition at a temperature T^{*}, and a gapped magnetic excitation spectrum with spectral weight concentrated at wave vectors typical of type-I antiferromagnetic orders. However, while muon spin resonance experiments show clear evidence for time-reversal symmetry breaking below T^{*}, we observe no corresponding magnetic Bragg scattering signal. These results are shown to be consistent with ferro-octupolar symmetry breaking below T^{*}, and are discussed in the context of other 5d double perovskite magnets and theories of exotic orders driven by multipolar interactions.

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

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

Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI_{3}

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality CrI_{3} single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with K∼-5.2  meV, 25 times larger than the Heisenberg exchange J∼-0.2  meV, and responsible for opening the ∼5  meV gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy Γ∼-67.5  μeV, though small, is crucial for opening a ∼0.3  meV gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a μeV-scale quadrupolar contribution to the S=3/2 magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher T_{C} as well as magnetically frustrated quantum spin liquids based on Kitaev physics.

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

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

Pressure-Tuned Interactions in Frustrated Magnets: Pathway to Quantum Spin Liquids?

Quantum spin liquids are prime examples of strongly entangled phases of matter with unconventional exotic excitations. Here, strong quantum fluctuations prohibit the freezing of the spin system. On the other hand, frustrated magnets, the proper platforms to search for the quantum spin liquid candidates, still show a magnetic ground state in most of the cases. Pressure is an effective tuning parameter of structural properties and electronic correlations. Nevertheless, the ability to influence the magnetic phases should not be forgotten. We review experimental progress in the field of pressure-tuned magnetic interactions in candidate systems. Elaborating on the possibility of tuned quantum phase transitions, we further show that chemical or external pressure is a suitable parameter in these exotic states of matter.

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.