It is now well established that the Kitaev honeycomb model in a magnetic field along the $[111]$-direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-abelian QSL at low fields $H< H_{c1}$ and a partially polarized phase at high fields $H> H_{c2}$. Here, we analyze the low field and high field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization (ED) and density matrix renormalization group (DMRG) methods. We find that the energy to create a bound state of two-magnons $Δ_p$ becomes lower than the energy to create a single spin flip $Δ_s$ near $H_{c2}$. In the entire Kitaev spin liquid $Δ_p<Δ_s$ and both gaps vanish at $H_{c2}$. We make testable predictions for magnon pairing that could be observable in Raman scattering measurements on Kitaev QSL candidate materials.
We study periodically driven pure Kitaev model and ferromagnetic phase of the Kitaev-Heisenberg model on the honeycomb lattice by off-resonant linearly and circularly-polarized lights at zero magnetic field. Using a combination of linear spin wave and Floquet theories, we show that the effective time-independent Hamiltonians in the off-resonant regime map onto the corresponding anisotropic static spin model, plus a tunable photoinduced magnetic field along the $[111]$ direction, which precipitates Floquet topological magnons and chiral magnon edge modes. They are tunable by the light amplitude…
Topological phases in magnetic materials offer novel tunability of topological properties via varying the underlying magnetism. We show that three dimensional Kitaev materials 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 zig-zag ordered state in the hyperhoneycomb $β\text{-}\mathrm{Li_2IrO_3}$, w…
A number of topological phases are found to emerge in the ferromagnetic Kitaev-Heisenberg model on CaVO lattice in the presence of Dzyaloshinskii-Moriya interaction. Heisenberg and Kitaev terms have been considered on nearest and next-nearest neighbor bonds in a variety of ways. Both isotropic and anisotropic couplings are taken into account. Topological phases are characterized by Chern numbers for the distinct magnon bands as well as the number of modes for topologically protected gapless magnon edge states. Band structure, dispersion relation along the high-symmetric points of first Brillou…
We study the Kitaev-Heisenberg-$Γ$-$Γ'$ model that describes the magnetism in strong spin-orbit coupled honeycomb lattice Mott insulators. In strong $[111]$ magnetic fields that bring the system into the fully polarized paramagnetic phase, we find that the spin wave bands carry nontrivial Chern numbers over large regions of the phase diagram implying the presence of chiral magnon edge states. In contrast to other topological magnon systems, the topological nontriviality of these systems results from the presence of magnon number non-conserving terms in the Hamiltonian. Since the effects of int…
Kitaev quantum spin liquids host Majorana fermions via the fractionalization of spins. In a magnetic field, the Majorana fermions were predicted to comprise a topological state, which has attracted great attention by the discovery of the half-quantized thermal Hall conductivity. Nevertheless, a reliable theory remains elusive for the field effect, especially at finite temperature. Here we present unbiased large-scale numerical results for the Kitaev model in a wide range of magnetic field and temperature. We find that the unconventional paramagnetic region showing fractional spin dynamics exte…
We calculate magnon dispersions and damping in the Kitaev-Heisenberg model with an off-diagonal exchange $Γ$ and isotropic third-nearest-neighbor interaction $J_3$ on a honeycomb lattice. This model is relevant to a description of the magnetic properties of iridium oxides $α$-Li$_2$IrO$_3$ and Na$_2$IrO$_3$, and Ru-based materials such as $α$-RuCl$_3$. We use an unconventional parametrization of the spin-wave expansion, in which each Holstein-Primakoff boson is represented by two conjugate hermitian operators. This approach gives us an advantage over the conventional one in identifying paramet…
The thermal Hall effect, defined as a heat current response transversal to an applied temperature gradient, is a central experimental probe of exotic electrically insulating phases of matter. A key question is how the interplay between magnetic and structural degrees of freedom gives rise to a nonzero thermal Hall conductivity (THC). Here, we present evidence for an intrinsic thermal Hall effect in the Heisenberg-Kitaev antiferromagnet and spin-liquid candidate Na$_2$Co$_2$TeO$_6$ brought about by the quantum-geometric Berry curvature of so-called magnon polarons, resulting from magnon-phonon …
We use time-domain terahertz spectroscopy to measure the low energy conductivity and magnons in RuCl$_3$ under external magnetic field. At zero field, an oscillation with a frequency of 0.62 THz is clearly observed in time-domain spectrum below T$_N$, which is identified as a magnon excitation in the magnetic order state. The magnon excitation is not affected by the external magnetic field $\textbf{H}_{DC}$ when it is applied along the c-axis, but is clearly suppressed when $\textbf{H}_{DC}$ is applied within ab plane. More interestingly, when the magnetic component of THz wave $\textbf{h}(t)$…
We introduce an exactly solvable SU(2) symmetric Kitaev model with exotic spin excitations. With time reversal symmetry (TRS), the ground state is a spin liquid with gapless or gapped spin-1 but fermionic excitations. When TRS is broken, the resulting spin liquid exhibits deconfined vortex excitations which carry spin-1/2 quantum number and obey non-Abelian statistics. We show that this SU(2) invariant non-Abelian spin liquid exhibits spin quantum Hall effect with quantized spin Hall conductivity σ^s_{xy}=\hbar/2π, and that the spin response is effectively described the SO(3) level-1 Chern-Sim…
The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, while applying finite magnetic fields induces a topological spin liquid with non-Abelian anyonic excitations. This latter phase has been much sought after in Kitaev candidate materials, such as $α$-RuCl$_3$. Currently, two competing scenarios exist for the intermediate field phase of this compound ($B=7-10$ T), based on experimental as well as theoretical results: (i) conventional multiparticle magnetic excitations of integer quantum number vs. (ii) Majorana fermionic excitations of possibly non-Abelian natu…
There have been intensive studies on Kitaev materials for the sake of the realization of exotic states such as quantum spin liquid and topological orders. In realistic materials, the Kitaev interaction may coexist with the Dzyaloshinskii-Moriya interaction (DMI), and it is a challenge to distinguish their magnitudes separately. Here, we study the topological magnon excitations and related thermal Hall conductivity of kagome magnet exhibiting Heisenberg, Kitaev and DM interactions exposed to a magnetic field. In a strong magnetic field perpendicular to the plane of the lattice ([111] direction)…
We study the nature of the debated thermal Hall effect in the candidate Kitaev material $α$-RuCl$_3$. Without assuming the existence of a gapped spin liquid, we show that a realistic minimal spin model in the canted zigzag phase suffices, at the level of linear spin-wave theory, to qualitatively explain the observed temperature and magnetic field dependence of the non-quantized thermal Hall conductivity $κ_{xy}$, with its origin lying in the Berry curvature of the magnon bands. The magnitude of the effect is however too small compared to the measurement by Czajka et al. [Nat. Mater. 22, 36-41 …
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-…
We have investigated the field-angle behaviors of magnetic excitations under an in-plane magnetic field for proximate Kitaev systems. By employing the exact diagonalization method in conjunction with the linear spin wave theory, we have demonstrated that the magnetic excitation gap in the polarized phase is determined by the magnon excitation at $M$ points and has a strong anisotropy with respect to the field direction in the vicinity of the critical field limit. The specific heat from this magnon excitation bears qualitatively the same anisotropic behaviors as expected one for the non-Abelian…
Proposed effective Hamiltonians from the literature for the material $α$-RuCl${}_3$ are used to compute the magnon thermal Hall conductivity, $κ_{xy}$, using linear spin wave theory for the magnetically ordered state. No model previously proposed that was tested explains published experimental data. Models with Kitaev interaction $K>0$ are seen to predict $κ_{xy}\gtrsim 0$, which is inconsistent with the data. Fluctuations toward a Kitaev-type spin liquid would have the wrong sign to explain the data. However, a slight variant of a previously proposed model predicts a large $κ_{xy}$, demonstra…
The family of edge-sharing tri-coordinated iridates and ruthenates has emerged in recent years as a major platform for Kitaev spin liquid physics, where spins fractionalize into emergent magnetic fluxes and Majorana fermions with Dirac-like dispersions. While such exotic states are usually pre-empted by long-range magnetic order at low temperatures, signatures of Majorana fermions with long coherent times have been predicted to manifest at intermediate and higher energy scales, similar to the observation of spinons in quasi-1D spin chains. Here we present a Resonant Inelastic X-ray Scattering …
We theoretically show that the Kitaev interaction generates a novel class of spin texture in the excitation spectrum of the antiferromagnetic insulator found in the Kitaev-Heisenberg-$Γ$ model. In conducting electronic systems, there is a series of vortex type of spin texture along the Fermi surface induced by Rashba and Dresselhaus spin-orbit coupling. Such spin textures are rarely found in magnetic insulators, since there had been no systematic ways to control the kinetics of its quasi-particle called magnon using a magnetic field or spacially asymmetric exchange couplings. Here, we propose …
Motivated by the experiments on the rare-earth double perovskites, we propose a generalized Kitaev-Heisenberg model to describe the generic interaction between the spin-orbit-entangled Kramers' doublets of the rare-earth moments. We carry out a systematic analysis of the mean-field phase diagram of this new model. In the phase diagram, there exist large regions with a continuous U(1) or O(3) degeneracy. Since no symmetry of the model protects such a continuous degeneracy, we predict that the quantum fluctuation lifts the continuous degeneracy and favors various magnetic orders in the phase dia…
The description of quantized collective excitations stands as a landmark in the quantum theory of condensed matter. A prominent example occurs in conventional magnets, which support bosonic magnons - quantized harmonic fluctuations of the ordered spins. In striking contrast is the recent discovery that strongly spin-orbital coupled magnets, such as $α$-RuCl$_3$, may display a broad excitation continuum inconsistent with conventional magnons. Due to incomplete knowledge of the underlying interactions unraveling the nature of this continuum remains challenging. The most discussed explanation ref…
We study the spin-$S$ Kitaev model in the classical ($S \to \infty$) limit using Monte Carlo simulations combined with semi-classical spin dynamics. We discuss differences and similarities in the dynamical structure factors of the spin-$1/2$ and the classical Kitaev liquids. Interestingly, the low-temperature and low-energy spectrum of the classical model exhibits a finite energy peak, which is the precursor of the one produced by the Majorana modes of the $S=1/2$ model. The classical peak is spectrally narrowed compared to the quantum result and can be explained by magnon excitations within f…
The topological properties of massive and massless fermionic quasiparticles have been intensively investigated over the past decade in topological materials without magnetism. Recently, the bosonic analogs of such quasiparticles arising from spin waves have been reported in the two-dimensional (2D) honeycomb lattice ferromagnet/antiferromagnet and the 3D antiferromagnet. Here we use time-of-flight inelastic neutron scattering to study spin waves of the S = 1 honeycomb lattice antiferromagnet BaNi2(AsO4)2, which has a zig-zag antiferromagnetic (AF) ground state identical to that of the Kitaev q…
Searching for Kitaev quantum spin liquid (QSL) is a fascinating and challenging problem. Much effort has been devoted to honeycomb lattice candidates with strong spin-orbit coupling in 5d-electron iridates and 4delectron RuCl3. Recently, theoretical studies suggested that the 3d7 Co-based honeycomb materials with high spin state S=3/2 and effective orbital angular momentum L=1 could also be promising candidates of Kitaev QSL. One of the candidates, BaCo2(AsO4)2, was revisited recently. The long range magnetic order in BaCo2(AsO4)2 can be suppressed by very weak in-plane magnetic field, suggest…
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 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 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, it is almost zero…
In spite of flourishing studies on the topology of spin waves, a generic framework to classify and compute magnon band topology in non-collinear magnets is still missing. In this work we provide such a theory framework, by mapping an arbitrary linear spin wave into a local free-fermion Hamiltonian with exactly the same spectrum, symmetry implementation and band topology, which allows for a full classification and calculation on any topological properties of magnon bands. We apply this fermionization approach to honeycomb Kitaev magnet $α$-RuCl$_3$, and show the existence of topologically prote…
Proximate quantum spin liquids and Majorana continua in magnetically ordered Kitaev magnets
Technical University of Munich（慕尼黑工业大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
We identify fingerprints of a proximate quantum spin-liquid (QSL), observable by finite-temperature dynamical thermal transport within a minimal version of the idealized Kitaev model on a two-leg ladder, if subjected to inevitably present Heisenberg couplings. Using exact diagonalization and quantum typicality, we uncover (i) an insulator-conductor crossover induced by fracton recombination at infinitesimal Heisenberg coupling, (ii) low- and high-energy signatures of fractons, which survive far off the pure QSL point, and (iii) a non-monotonous current life-time versus Heisenberg couplings. Gu…
We describe scanning tunneling spectroscopic signatures of Majorana zero modes (MZMs) in Kitaev spin liquids. The tunnel conductance is determined by the dynamical spin correlations of the spin liquid, which we compute exactly, and by spin-anisotropic cotunneling form factors. Near a $\mathbb{Z}_2$ vortex, the tunnel conductance has a staircase voltage dependence, where conductance steps arise from MZMs and (at higher voltages) from additional vortex configurations. By scanning the probe tip position, one can detect the vortex locations. Our analysis suggests that topological magnon bound stat…
We investigate both the longitudinal thermal conductivity ($κ_{xx}$) and the planar thermal Hall conductivity ($κ_{xy}$) in the Kitaev spin liquid candidate of Co-based honeycomb antiferromagnet Na$_2$Co$_2$TeO$_6$ in a magnetic field ($B$) applied along the $a$ and $a^*$ axes. A finite $κ_{xy}$ is resolved for both field directions in the antiferromagnetic (AFM) phase below the Néel temperature of 27 K. The temperature dependence of $κ_{xy}/T$ shows the emergence of topological bosonic excitations. In addition, the field dependence of $κ_{xy}$ shows sign reversals at the critical fields in th…
We calculate the two-magnon Raman scattering spectra in antiferromagnetic phases of several frustrated spin models defined on the honeycomb lattice. These include the Néel antiferromagnetic phase of a $J_1$-$J_2$-$J_3$ model and the stripe phase of the Heisenberg-Kitaev model. We show that both the magnetic frustration and the anisotropy of interactions may significantly affect the Raman spectra. We further discuss the implications of our results to the magnetic excitations of the iron-based compound BaFe$_2$Se$_2$O and show how the magnetic interactions can be extracted from fit to the Raman …
We show that the mesoscopic incommensurate $\mathbb{Z}_2$ vortex crystals proposed for layered triangular anisotropic magnets can be most saliently identified by two distinctive signatures in dynamical spin response experiments: The presence of pseudo-Goldstone `phonon' modes at low frequencies $ω$, associated with the collective vibrations of the vortex cores, and a characteristic multi-scattered intensity profile at higher $ω$, arising from a large number of Bragg reflections and magnon bandgaps. These are direct fingerprints of the large vortex sizes and magnetic unit cells and the solitoni…
Incommensurate spiral order is a common occurrence in frustrated magnetic insulators. Typically, all magnetic moments rotate uniformly, through the same wavevector. However the honeycomb iridates family Li2IrO3 shows an incommensurate order where spirals on neighboring sublattices are counter-rotating, giving each moment a different local environment. Theoretically describing its spin dynamics has remained a challenge: the Kitaev interactions proposed to stabilize this state, which arise from strong spin-orbit effects, induce magnon umklapp scattering processes in spin-wave theory. Here we pro…
We revisit the theory of magnetic Raman scattering in Mott insulators with strong spin-orbit coupling, with a major focus on Kitaev materials. We show that Kitaev materials with bond-anisotropic interactions are generally expected to show both one- and two-magnon responses. It is further shown that, in order to obtain the correct leading contributions to the Raman vertex operator $\mc{R}$, one must take into account the precise, photon-assisted microscopic hopping processes of the electrons and that, in systems with multiple hopping paths, $\mc{R}$ contains terms beyond those appearing in the …
Thermal Hall conductivity from semiclassical spin dynamics simulations: implementation and applications to chiral ferromagnets and Kitaev magnets
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；University of Münster（明斯特大学）；Imperial College London（帝国理工学院）
Thermal transport in topologically-ordered phases of matter provides valuable insights as it can detect the charge-neutral quasiparticles that would not directly couple to electromagnetic probes. An important example is edge heat transport of Majorana fermions in a chiral spin liquid, which leads to a half-quantized thermal Hall conductivity. This signature is precisely what has recently been measured in $α$-RuCl$_3$ under external magnetic fields. The plateau-like behavior of the half-quantized thermal Hall conductivity as a function of external magnetic field, and the peculiar sign change de…
Recently, Co-based honeycomb magnets have been proposed as promising candidate materials to host the Kitaev spin liquid state. One of the front-runners is BaCo$_2$(AsO$_4$)$_2$ (BCAO), where it was suggested that the exchange processes between Co$^{2+}$ ions via the surrounding edge-sharing oxygen octahedra could give rise to bond-dependent Kitaev interactions. In this work, we present and analyze comprehensive inelastic neutron scattering studies of BCAO with fields in the honeycomb plane. Combining the constraints from the magnon excitations in the high-field polarized state and the inelasti…
We report detailed measurements of the Onsager-like planar thermal Hall conductivity $κ_{xy}$ in $α$-RuCl$_3$, a spin-liquid candidate of topical interest. With the thermal current ${\bf J}_{\rm Q}$ and magnetic field $\bf B\parallel a$ (zigzag axis), the observed $κ_{xy}/T$ varies strongly with temperature $T$ (1-10 K). The results are well-described by bosonic edge excitations which evolve to topological magnons at large $B$. Fits to $κ_{xy}/T$ yield a Chern number $\sim 1$ and a band energy $ω_1\sim$1 meV, in agreement with sharp modes seen in electron spin-resonance experiments. The bosoni…
The magnetic properties of 4d and 5d transition-metal insulating compounds with the honeycomb structure are believed to be described by the Heisenberg-Kitaev model, which contains both the isotropic Heisenberg interaction J and anisotropic Kitaev interaction K. In this paper, to investigate the charge dynamics in these materials, we study the single-hole propagation of the t-J-K model in various magnetically ordered phases by the self-consistent Born approximation. We find that there are low-energy coherent quasiparticle (QP) excitations in all of these phases which appear firstly around the K…
We present a theoretical study of the static and dynamical properties of the Kitaev magnet $β$-Li$_2$IrO$_3$ based on the minimal $J$-$K$-$Γ$ model. We argue that the incommensurate magnetic order of $β$-Li$_2$IrO$_3$ can be qualitatively understood in terms of a long-wavelength twisting of a nearby commensurate, period-3 state, that shares the same symmetry, propagation vector, counter-rotation of the two spin sublattices, and non-coplanarity. This period-3 state shows very different structure when either the Kitaev interaction $K$ or the off-diagonal exchange anisotropy $Γ$ is dominant, feat…
Finding new materials with antiferromagnetic (AFM) Kitaev interaction is an urgent issue to broaden and enrich the quantum magnetism research significantly. By carrying out inelastic neutron scattering experiments and subsequent analysis, we conclude that Na$_3$Co$_2$SbO$_6$ and Na$_2$Co$_2$TeO$_6$ are new honeycomb cobalt-based AFM Kitaev systems. The spin-orbit excitons at 20-28~meV in both compounds strongly supports the idea that Co$^{2+}$ ions of both compounds have a spin-orbital entangled $J_\rm{eff}=1/2$ state. Furthermore, we found that a generalized Kitaev-Heisenberg Hamiltonian can …
In the search for experimental realizations of bond-anisotropic Kitaev interactions and resulting spin-liquid phases, the layered magnet $α$-RuCl$_3$ is a prime candidate. Its modelling typically involves Heisenberg, Kitaev, and symmetric off-diagonal $Γ$ interactions on the two-dimensional honeycomb lattice. However, recent neutron-scattering experiments point towards a sizeable magnetic interlayer coupling. Here we study three-dimensional exchange models for $α$-RuCl$_3$, for both possible $R\bar{3}$ and $C2/m$ crystal structures. We discuss the symmetry constraints on the interlayer couplin…
Revealing the spin excitations of complex quantum magnets is key to developing a minimal model that explains the underlying magnetic correlations in the ground state. We investigate the low-energy magnons in $α$-RuCl$_3$ by combining time-domain terahertz spectroscopy under an external magnetic field and model Hamiltonian calculations. We observe two absorption peaks around 2.0 and 2.4 meV, which we attribute to zone-center spin waves. Using linear spin-wave theory with only nearest-neighbor terms of the exchange couplings, we calculate the antiferromagnetic resonance frequencies and reveal th…
What is the correct low-energy spin Hamiltonian description of $α$-RuCl$_3$? The material is a promising Kitaev spin liquid candidate, but is also known to order magnetically, the description of which necessitates additional interaction terms. The nature of these interactions, their magnitudes and even signs, remain an open question. In this work we systematically investigate dynamical and thermodynamic magnetic properties of proposed effective Hamiltonians. We calculate zero-temperature inelastic neutron scattering (INS) intensities using exact diagonalization, and magnetic specific heat usin…
In the quest for realizations of quantum spin liquids, the exploration of Kitaev materials - spin-orbit entangled Mott insulators with strong bond-directional exchanges - has taken center stage. However, in these materials the local spin-orbital j=1/2 moments typically show long-range magnetic order at low temperature, thus defying the formation of a spin-liquid ground state. Using resonant inelastic x-ray scattering (RIXS), we here report on a proximate spin liquid regime with clear fingerprints of Kitaev physics in the magnetic excitations of the honeycomb iridates alpha-Li2IrO3 and Na2IrO3.…
Thermal Hall conductivity originating from topological magnons is observed in the Kitaev candidate $α$-RuCl$_3$ in broad intervals of temperature and in-plane magnetic field, raising questions on the role of the Majorana mode in heat conduction.
