Erratum: Thermal Transport in the Kitaev Model [Phys. Rev. Lett. 119, 127204 (2017)].

This corrects the article DOI: 10.1103/PhysRevLett.119.127204.

Antiferromagnet-Semiconductor Van Der Waals Heterostructures: Interlayer Interplay of Exciton with Magnetic Ordering.

Van der Waals (vdW) heterostructures have attracted great interest because of their rich material combinations. The discovery of two-dimensional magnets has provided a new platform for magnetic vdW heterointerfaces; however, research on magnetic vdW heterointerfaces has been limited to those with ferromagnetic surfaces. Here, we report a magnetic vdW heterointerface using layered intralayer-antiferromagnetic MPSe3 (M = Mn, Fe) and monolayer transition-metal dichalcogenides (TMDs). We found an anomalous upshift of the excitonic peak in monolayer TMDs below the antiferromagnetic transition temperature in the MPSe3, capturing a signature of the interlayer exciton-magnon coupling. This is a concept extended from single materials to heterointerfaces. Moreover, this coupling strongly depends on the in-plane magnetic structure and stacking direction, showing its sensitivity to their magnetic interfaces. Our finding offers an opportunity to investigate interactions between elementary excitations in different materials across interfaces and to search for new functions of magnetic vdW heterointerfaces.

Electric Toroidal Quadrupoles in the Spin-Orbit-Coupled Metal Cd_{2}Re_{2}O_{7}.

We report our theoretical results on the order parameters for the pyrochlore metal Cd_{2}Re_{2}O_{7}, which undergoes enigmatic phase transitions with inversion symmetry breaking. By carefully examining active electronic degrees of freedom based on the lattice symmetry, we propose that two parity-breaking phases at ambient pressure are described by unconventional multipoles, electric toroidal quadrupoles (ETQs) with different components, x^{2}-y^{2} and 3z^{2}-r^{2}, in the pyrochlore tetrahedral unit. We elucidate that the ETQs are activated by bond or spin-current order on Re─Re bonds. Our ETQ scenario provides a key to reconciling the experimental contradictions, by measuring ETQ specific phenomena, such as peculiar spin splittings in the electronic band structure, magnetocurrent effect, and nonreciprocal transport under a magnetic field.

Néel- and Bloch-Type Magnetic Vortices in Rashba Metals.

We theoretically study noncoplanar spin textures in polar magnetic conductors. Starting from the Kondo lattice model with the Rashba spin-orbit coupling, we derive an effective spin model with generalized Ruderman-Kittel-Kasuya-Yosida interactions including the anisotropic and antisymmetric exchange interactions. By performing simulated annealing for the effective model, we find that a vortex crystal of Néel type is stabilized even in the absence of a magnetic field. Moreover, we demonstrate that a Bloch-type vortex crystal, which is usually associated with the Dresselhaus spin-orbit coupling, can also be realized in our Rashba-based model. A magnetic field turns the vortex crystals into Néel- and Bloch-type Skyrmion-like crystals. Our results underscore that the interplay between the spin-orbit coupling and itinerant magnetism brings fertile possibilities of noncoplanar magnetic orderings.

Zero-Field Skyrmions with a High Topological Number in Itinerant Magnets.

Magnetic Skyrmions are swirling spin textures with topologically protected noncoplanarity. Recently, Skyrmions with the topological number of unity have been extensively studied in both experiment and theory. We here show that a Skyrmion crystal with an unusually high topological number of two is stabilized in itinerant magnets at a zero magnetic field. The results are obtained for a minimal Kondo lattice model on a triangular lattice by an unrestricted large-scale numerical simulation and variational calculations. We find that the topological number can be switched by a magnetic field as 2↔1↔0. The Skyrmion crystals are formed by the superpositions of three spin density waves induced by the Fermi surface effect, and hence, the size of Skyrmions can be controlled by the band structure and electron filling. We also discuss the charge and spin textures of itinerant electrons in the Skyrmion crystals which are directly obtained in our numerical simulations.

Thermal Transport in the Kitaev Model.

In conventional insulating magnets, heat is carried by magnons and phonons. In contrast, when the magnets harbor a quantum spin liquid state, emergent quasiparticles from the fractionalization of quantum spins can carry heat. Here, we investigate unconventional thermal transport yielded by such exotic carriers, in both longitudinal and transverse components, for the Kitaev model, whose ground state is exactly shown to be a quantum spin liquid with fractional excitations described as itinerant Majorana fermions and localized Z_{2} fluxes. We find that the longitudinal thermal conductivity exhibits a single peak at a high temperature, while the nonzero frequency component has a peak at a low temperature, reflecting the spin fractionalization. On the other hand, we show that the transverse thermal conductivity is induced by the magnetic field in a wide temperature range up to the energy scale of the bare exchange coupling; while increasing temperature, the transverse response divided by temperature decreases from the quantized value expected for the topologically nontrivial ground state and shows nonmonotonic temperature dependence. These characteristic behaviors provide experimentally accessible evidence of fractional excitations in the proximity to the Kitaev quantum spin liquid.

Spin-Liquid-to-Spin-Liquid Transition in Kitaev Magnets Driven by Fractionalization.

While phase transitions between magnetic analogs of the three states of matter-a long-range ordered state, paramagnet, and spin liquid-are extensively studied, the possibility of "liquid-liquid" transitions, namely, between different spin liquids, remains elusive. By introducing the additional Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid turns into a spin-nematic quantum paramagnet before a magnetic order is established by the Ising coupling. The quantum phase transition between the two liquid states accompanies a topological change driven by fractionalized excitations, the Z_{2} gauge fluxes, and is of first order. At finite temperatures, this yields a persisting first-order transition line that terminates at a critical point located deep inside the regime where quantum spins are fractionalized. It is suggested that similar transitions may occur in other perturbed Kitaev magnets with bond asymmetry.

Magnetoelectric Behavior from S=1/2 Asymmetric Square Cupolas.

Magnetoelectric properties are studied by a combined experimental and theoretical study of a quasi-two-dimensional material composed of square cupolas, Ba(TiO)Cu_{4}(PO_{4})_{4}. The magnetization is measured up to the field above the saturation, and several anomalies are observed depending on the field directions. We propose a S=1/2 spin model with Dzyaloshinskii-Moriya interactions, which reproduces the full magnetization curves well. Elaborating the phase diagram of the model, we show that the anomalies are explained by magnetoelectric phase transitions. Our theory also accounts for the scaling of the dielectric anomaly observed in the experiments. The results elucidate the crucial role of the in-plane component of Dzyaloshinskii-Moriya interactions, which is induced by the noncoplanar buckling of a square cupola. We also predict a "hidden" phase and another magnetoelectric response, both of which appear in a nonzero magnetic field.

Fractional Spin Fluctuations as a Precursor of Quantum Spin Liquids: Majorana Dynamical Mean-Field Study for the Kitaev Model.

Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte Carlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the quantum spin liquid. We find that dynamical spin correlations show peculiar temperature and frequency dependence even below the temperature where static correlations saturate. The results provide the experimentally accessible symptoms of the fluctuating fractionalized spins evincing the quantum spin liquids.

Theory of Valence Transition in BiNiO_{3}.

Motivated by the colossal negative thermal expansion recently found in BiNiO_{3}, the valence transition accompanied by the charge transfer between the Bi and Ni sites is theoretically studied. We introduce an effective model for Bi-6s and Ni-3d orbitals taking into account the valence skipping of Bi cations, and investigate the ground-state and finite-temperature phase diagrams within the mean-field approximation. We find that the valence transition is caused by commensurate locking of the electron filling in each orbital associated with charge and magnetic orderings, and the critical temperature and the nature of the transitions are strongly affected by the relative energy between the Bi and Ni levels and the effective electron-electron interaction in the Bi sites. The obtained phase diagram well explains the temperature- and pressure-driven valence transitions in BiNiO_{3} and the systematic variation of valence states for a series of Bi and Pb perovskite oxides.

Thermodynamics of Chiral Spin Liquids with Abelian and Non-Abelian Anyons.

Thermodynamic properties of chiral spin liquids are investigated for a variant of the Kitaev model defined on a decorated honeycomb lattice. Using the quantum Monte Carlo simulation, we find that the model exhibits a finite-temperature phase transition associated with the time reversal symmetry breaking, in both topologically trivial and nontrivial regions. Numerical results for the Chern number and the thermal Hall conductivity indicate that the phase transition changes from a continuous to a discontinuous transition as we vary the coupling constants to reach the non-Abelian phase coming from the Abelian phase of the model. In addition, we find as a diagnostic of the chiral spin liquids, successive crossovers with multistage entropy release above the critical temperature, which indicates that the hierarchical fractionalization of a quantum spin occurs differently between the two regions.

Vaporization of Kitaev spin liquids.

The quantum spin liquid is an exotic quantum state of matter in magnets. This state is a spin analog of liquid helium that does not solidify down to the lowest temperature due to strong quantum fluctuations. In conventional fluids, the liquid and gas possess the same symmetry and adiabatically connect to each other by bypassing the critical end point. We find that the situation is qualitatively different in quantum spin liquids realized in a three-dimensional Kitaev model; both gapless and gapped quantum spin liquid phases at low temperatures are always distinguished from the high-temperature paramagnet (spin gas) by a phase transition. The results challenge the common belief that the absence of thermodynamic singularity down to the lowest temperature is a symptom of a quantum spin liquid.

Charge order in a two-dimensional Kondo lattice model.

The possibility of charge order is theoretically examined for the Kondo lattice model in two dimensions, which does not include bare repulsive interactions. Using two complementary numerical methods, we find that charge order appears at quarter filling in an intermediate Kondo coupling region. The charge ordered ground state is an insulator exhibiting an antiferromagnetic order at charge-poor sites, while the paramagnetic charge-ordered state at finite temperatures is metallic with pseudogap behavior. We confirm that the stability of charge order is closely related with the local Kondo-singlet formation at charge-rich sites. Our results settle the controversy on charge order in the Kondo lattice model in realistic spatial dimensions.

Thermally-robust spatiotemporal parallel reservoir computing by frequency filtering in frustrated magnets.

Physical reservoir computing is a framework for brain-inspired information processing that utilizes nonlinear and high-dimensional dynamics in non-von-Neumann systems. In recent years, spintronic devices have been proposed for use as physical reservoirs, but their practical application remains a major challenge, mainly because thermal noise prevents them from retaining short-term memory, the essence of neuromorphic computing. Here, we propose a framework for spintronic physical reservoirs that exploits frequency domain dynamics in interacting spins. Through the effective use of frequency filters, we demonstrate, for a model of frustrated magnets, both robustness to thermal fluctuations and feasibility of frequency division multiplexing. This scheme can be coupled with parallelization in spatial domain even down to the level of a single spin, yielding a vast number of spatiotemporal computational units. Furthermore, the nonlinearity via the exchange interaction allows information processing among different frequency threads. Our findings establish a design principle for high-performance spintronic reservoirs with the potential for highly integrated devices.

Berry curvature contributions of kagome-lattice fragments in amorphous Fe-Sn thin films.

Amorphous semiconductors are widely applied to electronic and energy-conversion devices owing to their high performance and simple fabrication processes. The topological concept of the Berry curvature is generally ill-defined in amorphous solids, due to the absence of long-range crystalline order. Here, we demonstrate that the Berry curvature in the short-range crystalline order of kagome-lattice fragments effectively contributes to the anomalous electrical and magneto-thermoelectric properties in Fe-Sn amorphous films. The Fe-Sn films on glass substrates exhibit large anomalous Hall and Nernst effects comparable to those of the single crystals of topological semimetals Fe3Sn2 and Fe3Sn. With modelling, we reveal that the Berry curvature contribution in the amorphous state likely originates from randomly distributed kagome-lattice fragments. This microscopic interpretation sheds light on the topology of amorphous materials, which may lead to the realization of functional topological amorphous electronic devices.

Dirac half-metal in a triangular ferrimagnet.

An idea is proposed for realizing a fully spin-polarized Dirac semimetal in frustrated itinerant magnets. We show that itinerant electrons on a triangular lattice exhibit the Dirac cone dispersion with half-metallic behavior in the presence of a three-sublattice ferrimagnetic order. The Dirac nodes have the same structure as those of graphene. By variational calculation and Monte Carlo simulation, we demonstrate that the ferrimagnetic order with the Dirac node spontaneously emerges in a simple Kondo lattice model with Ising anisotropy. The realization will be beneficial for spintronics as a candidate for a spin-current generator.

Partial disorder in an Ising-spin Kondo lattice model on a triangular lattice.

A phase diagram of an Ising-spin Kondo lattice model on a triangular lattice near 1/3 filling is investigated by Monte Carlo simulation. We identify a partially disordered phase with the coexistence of magnetic order and paramagnetic moments, which was unstable in two-dimensional Ising models with localized spins only. The partial disorder emerges in the competing regime between a two-sublattice stripe phase and three-sublattice ferrimagnetic phase, at finite temperatures above an electronic phase separation. The peculiar magnetic structure accompanies a charge order and develops a gap in the electronic structure. The results manifest a crucial role of the nonperturbative interplay between spin and charge degrees of freedom in stabilizing the partial disorder.

Non-Kondo mechanism for resistivity minimum in spin ice conduction systems.

We present a mechanism of resistivity minimum in conduction electron systems coupled with localized moments, which is distinguished from the Kondo effect. Instead of the spin-flip process in the Kondo effect, electrons are elastically scattered by local spin correlations which evolve in a particular way under geometrical frustration as decreasing temperature. This is demonstrated by the cellular dynamical mean-field theory for a spin-ice-type Kondo lattice model on a pyrochlore lattice. Peculiar temperature dependences of the resistivity, specific heat, and magnetic susceptibility in the non-Kondo mechanism are compared with the experimental data in metallic Ir pyrochlore oxides.

Hidden multiple-spin interactions as an origin of spin scalar chiral order in frustrated Kondo lattice models.

We reveal the significance of kinetic-driven multiple-spin interactions hidden in geometrically frustrated Kondo lattice models. Carefully examining the perturbation in terms of the spin-charge coupling up to the fourth order, we find that a positive biquadratic interaction is critically enhanced and plays a crucial role on stabilizing a spin scalar chiral order near 1/4 filling in a triangular lattice case. This is a generalized Kohn anomaly, appearing only when the second-order perturbation is inefficient because of the degeneracy under frustration. The mechanism is potentially common to frustrated spin-charge coupled systems, leading to emergence of unusual magnetic orders.

Zoology of Multiple-Q Spin Textures in a Centrosymmetric Tetragonal Magnet with Itinerant Electrons.

Magnetic skyrmion is a topologically stable particle-like swirling spin texture potentially suitable for high-density information bit, which was first observed in noncentrosymmetric magnets with Dzyaloshinskii-Moriya interaction. Recently, nanometric skyrmion has also been discovered in centrosymmetric rare-earth compounds, and the identification of their skyrmion formation mechanism and further search of nontrivial spin textures are highly demanded. Here, magnetic structures in a prototypical skyrmion-hosting centrosymmetric tetragonal magnet GdRu2 Si2 is exhaustively studied by performing the resonant X-ray scattering experiments. A rich variety of double-Q magnetic structures, including the antiferroic order of meron(half-skyrmion)/anti-meron-like textures with fractional local topological charges are identified. The observed intricate magnetic phase diagram is successfully reproduced by the theoretical framework considering the four-spin interaction mediated by itinerant electrons and magnetic anisotropy. The present results will contribute to the better understanding of the novel skyrmion formation mechanism in this centrosymmetric rare-earth compound, and suggest that itinerant electrons can ubiquitously host a variety of unique multiple-Q spin orders in a simple crystal lattice system.