Kitaev Spin Liquid Research Articles

Topological magnons for thermal Hall transport in frustrated magnets with bond-dependent interactions

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 the edge heat transport of the 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 $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ 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 depending on the magnetic field orientations, have been proposed to be strong evidence for the non-Abelian Kitaev spin liquid. On the other hand, for in-plane magnetic fields, it has been theoretically shown that such a sign structure can also arise from topological magnons in the field-polarized state. In this paper, we investigate the full implications of topological magnons as heat carriers on thermal transport measurements. We first prove analytically that for any commensurate order with a finite magnetic unit cell, reversing the field direction leads to a sign change in the magnon thermal Hall conductivity in two-dimensional systems. We corroborate this proof numerically with nontrivial magnetic orders as well as the field-polarized state in Kitaev magnets subjected to an in-plane field. In the case of the tilted magnetic field, in which there exists both finite in-plane and out-of-plane field components, we find that the plateau-like behavior of the thermal Hall conductivity and the sign change upon the reversal of the in-plane component of the magnetic field arises in the partially polarized state, as long as the in-plane field contribution to the Zeeman energy is significant. While these results are consistent with the experimental observations, we comment on other aspects that require further investigation in future studies.

Magnetoelectric generation of a Majorana-Fermi surface in Kitaev's honeycomb model

We study the effects of static magnetic and electric fields on Kitaev's honeycomb model. Using the electric polarization operator appropriate for Kitaev materials, we derive the effective Hamiltonian for the emergent Majorana fermions to second-order in both the electric and magnetic fields. We find that while individually each perturbation does not qualitatively alter Kitaev spin liquid, the cross-term induces a finite chemical potential at each Dirac node, giving rise to a Majorana-Fermi surface. We argue this gapless phase is stable and exhibits typical metallic phenomenology, such as linear in temperature heat capacity and finite, but non-quantized, thermal Hall response. Finally, we speculate on the potential for realization of this physics in Kitaev materials.

Hidden Fermi liquidity and topological criticality in the finite temperature Kitaev model

The fate of exotic spin liquid states with fractionalized excitations at finite temperature (T) is of great interest, since signatures of fractionalization manifest in finite-temperature (T) dynamics in real systems, above the tiny magnetic ordering scales. Here, we study a Jordan–Wigner (JW) fermionized Kitaev spin liquid at finite T employing combined exact diagonalization and Monte Carlo simulation methods. We uncover (i) checkerboard or stripy-ordered flux crystals depending on density of flux, and (ii) establish, surprisingly, that: (a) the finite-T version of the T=0 transition from a gapless to gapped phases in the Kitaev model is a Mott transition of the fermions, belonging to the two-dimensional Ising universality class. These transitions correspond to a topological transition between a string condensate and a dilute closed string state (b) the Mott “insulator” phase is a precise realization of Laughlin’s gossamer (here, p-wave) superconductor (g-SC), and (c) the Kitaev Toric Code phase (TC) is adiabatically connected to the g-SC, and is a fully Gutzwiller-projected fermi sea of JW fermions. These findings establish the finite-T quantum spin liquid phases in the d=2 to be hidden Fermi liquid(s) of neutral fermions.

Revealing the phase diagram of Kitaev materials by machine learning: Cooperation and competition between spin liquids

Kitaev materials are promising materials for hosting quantum spin liquids and investigating the interplay of topological and symmetry-breaking phases. We use an unsupervised and interpretable machine-learning method, the tensorial-kernel support vector machine, to study the honeycomb Kitaev-$\Gamma$ model in a magnetic field. Our machine learns the global classical phase diagram and the associated analytical order parameters, including several distinct spin liquids, two exotic $S_3$ magnets, and two modulated $S_3 \times Z_3$ magnets. We find that the extension of Kitaev spin liquids and a field-induced suppression of magnetic order already occur in the large-$S$ limit, implying that critical parts of the physics of Kitaev materials can be understood at the classical level. Moreover, the two $S_3 \times Z_3$ orders are induced by competition between Kitaev and $\Gamma$ spin liquids and feature a different type of spin-lattice entangled modulation, which requires a matrix description instead of scalar phase factors. Our work provides a direct instance of a machine detecting new phases and paves the way towards the development of automated tools to explore unsolved problems in many-body physics.

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

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

Rare-Earth Chalcohalides: A Family of van der Waals Layered Kitaev Spin Liquid CandidatesSupported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0302904 and 2016YFA0300504), the National Natural Science Founation of China (Grant Nos. U1932215 and 11774419), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33010100).

The Kitaev spin liquid (KSL) system has attracted tremendous attention in recent years because of its fundamental significance in condensed matter physics and promising applications in fault-tolerant topological quantum computation. Material realization of such a system remains a major challenge in the field due to the unusual configuration of anisotropic spin interactions, though great effort has been made before. Here we reveal that rare-earth chalcohalides REChX (RE = rare earth; Ch = O, S, Se, Te; X = F, Cl, Br, I) can serve as a family of KSL candidates. Most family members have the typical SmSI-type structure with a high symmetry of , and rare-earth magnetic ions form an undistorted honeycomb lattice. The strong spin-orbit coupling of 4f electrons intrinsically offers anisotropic spin interactions as required by the Kitaev model. We have grown the crystals of YbOCl and synthesized the polycrystals of SmSI, ErOF, HoOF and DyOF, and made careful structural characterizations. We carry out magnetic and heat capacity measurements down to 1.8 K and find no obvious magnetic transition in all the samples but DyOF. The van der Waals interlayer coupling highlights the true two-dimensionality of the family which is vital for the exact realization of Abelian/non-Abelian anyons, and the graphene-like feature will be a prominent advantage for developing miniaturized devices. The family is expected to act as an inspiring material platform for the exploration of KSL physics.

Effect of structural disorder on the Kitaev magnet Ag3LiIr2O6

The search for an ideal Kitaev spin liquid candidate with anyonic excitations and long-range entanglement has motivated the synthesis of a new family of intercalated Kitaev magnets such as ${\mathrm{H}}_{3}\mathrm{Li}{\mathrm{Ir}}_{2}{\mathrm{O}}_{6}$, ${\mathrm{Cu}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$, and ${\mathrm{Ag}}_{3}\mathrm{Li}{\mathrm{Ir}}_{2}{\mathrm{O}}_{6}$. The absence of a susceptibility peak and a two-step release of the magnetic entropy in these materials have been proposed as evidence of proximity to the Kitaev spin liquid. Here we present a comparative study of the magnetic susceptibility, heat capacity, and muon spin relaxation ($\ensuremath{\mu}\mathrm{SR}$) between two samples of ${\mathrm{Ag}}_{3}\mathrm{Li}{\mathrm{Ir}}_{2}{\mathrm{O}}_{6}$ in the clean and disordered limits. In the disordered limit, the absence of a peak in either susceptibility or heat capacity and the lack of zero-field muon precession in the $\ensuremath{\mu}\mathrm{SR}$ signal give the impression of a proximate spin liquid state. However, in the clean limit, peaks are resolved in both susceptibility and heat capacity, and spontaneous oscillations appear in the $\ensuremath{\mu}\mathrm{SR}$ signal, confirming long-range antiferromagnetic order in the ground state. The $\ensuremath{\mu}\mathrm{SR}$ oscillations fit to a Bessel function, characteristic of incommensurate order, as reported in the parent compound $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Li}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$. Our results clarify the role of structural disorder in the intercalated Kitaev magnets.

Angle-dependent thermodynamics of α−RuCl3

Thermodynamics of the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ is studied for different directions of in-plane magnetic field using measurements of the magnetic Gr\"uneisen parameter $\Gamma_B$ and specific heat $C$. We identify two critical fields $B_c^{\rm AF1}$ and $B_c^{\rm AF2}$ corresponding, respectively, to a transition between two magnetically ordered states and the loss of magnetic order toward a quantum paramagnetic state. The $B_c^{AF2}$ phase boundary reveals a narrow region of magnetic fields where inverse melting of the ordered phase may occur. No additional transitions are detected above $B_c^{\rm AF2}$ for any direction of the in-plane field, although a shoulder anomaly in $\Gamma_B$ is observed systematically at $8-10$ T. Large field-induced entropy effects imply additional low-energy excitations at low fields and/or strongly field-dependent phonon entropies. Our results establish universal features of $\alpha$-RuCl$_3$ in high magnetic fields and challenge the presence of a field-induced Kitaev spin liquid in this material.

Testing topological phase transitions in Kitaev materials under in-plane magnetic fields: Application to α−RuCl3

Material realization of the non-Abelian Kitaev spin liquid phase - an example of Ising topological order (ITO) - has been the subject of intense research in recent years. The $4d$ honeycomb Mott insulator $\alpha$-RuCl$_3$ has emerged as a leading candidate, as it enters a field-induced magnetically disordered state where a half-integer quantized thermal Hall conductivity $\kappa_{xy}$ was reported. Further, a recent report of a sign change in the quantized $\kappa_{xy}$ across a certain crystallographic direction is strong evidence for a topological phase transition between two ITOs with opposite Chern numbers. Although this is a fascinating result, independent verification remains elusive, and one may ask if there is a thermodynamic quantity sensitive to the phase transition. Here we propose that the magnetotropic coefficient $k$ under in-plane magnetic fields would serve such a purpose. We report a singular feature in $k$ that indicates a topological phase transition across the $\hat{b}$-axis where ITO is prohibited by a $C_2$ symmetry. If the transition in $\alpha$-RuCl$_3$ is indeed a direct transition between ITOs, then this feature in $k$ should be observable.

Vacancy-Induced Low-Energy Density of States in the Kitaev Spin Liquid

The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the 5d-electron compound H$_{3}$LiIr$_{2}$O$_{6}$ has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of H$_{3}$LiIr$_{2}$O$_{6}$. Physically, the vacancies can originate from various sources such as missing magnetic moments or the presence of non-magnetic impurities (true vacancies), or from local weak couplings of magnetic moments due to strong but rare bond randomness (quasivacancies). We show numerically that the vacancy effect is readily detectable even at low vacancy concentrations and that it is not very sensitive neither to nature of vacancies nor to different flux backgrounds. We also study the response of the site-diluted Kitaev spin liquid to the three-spin interaction term, which breaks time-reversal symmetry and imitates an external magnetic field. We propose a field-induced flux-sector transition where the ground state becomes flux free for larger fields, resulting in a clear suppression of the low temperature specific heat. Finally, we discuss the effect of dangling Majorana fermions in the case of true vacancies and show that their coupling to an applied magnetic field via the Zeeman interaction can also account for the scaling behavior in the high-field limit observed in H$_{3}$LiIr$_{2}$O$_{6}$.

Generic field-driven phenomena in Kitaev spin liquids: Canted magnetism and proximate spin liquid physics

Topological spin liquids in two spatial dimensions are stable phases in the presence of a small magnetic field, but may give way to field-induced phenomena at intermediate field strengths. Sandwiched between the low-field spin liquid physics and the high-field spin-polarized phase, the exploration of magnetic phenomena in this intermediate regime however often remains elusive to controlled analytical approaches. Here we numerically study such intermediate-field magnetic phenomena for two representative Kitaev models (on the square-octagon and decorated honeycomb lattice) that exhibit either Abelian or non-Abelian topological order in the low-field limit. Using a combination of exact diagonalization and density matrix renormalization group techniques, as well as linear spin-wave theory, we establish the generic features of Kitaev spin liquids in an external magnetic field. While ferromagnetic models typically exhibit a direct transition to the polarized state at a relatively low field strength, antiferromagnetic couplings not only substantially stabilizes the topological spin liquid phase, but generically lead to the emergence of a distinct field-induced intermediate regime, separated by a crossover from the high-field polarized regime. Our results suggest that, for most lattice geometries, this regime generically exhibits significant spin canting, antiferromagnetic spin-spin correlations, and an extended proximate spin liquid regime at finite temperatures. Notably, we identify a symmetry obstruction in the original honeycomb Kitaev model that prevents, at least for certain field directions, the formation of such canted magnetism without breaking symmetries -- consistent with the recent numerical observation of an extended gapless spin liquid in this case.

Topological Z2 invariant in Kitaev spin liquids: Classification of gapped spin liquids beyond projective symmetry group

A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids. However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete. We demonstrate the classification beyond PSG by utilizing the Kitaev model on the squareoctagon lattice, where two gapped spin liquids are distinguished by a topological $Z_2$ invariant. This $Z_2$ invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively. Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time. This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.

Competing antiferromagnetic-ferromagnetic states in a d7 Kitaev honeycomb magnet.

The Kitaev model is a rare example of an analytically solvable and physically instantiable Hamiltonian yielding a topological quantum spin liquid ground state. Here we report signatures of Kitaev spin liquid physics in the honeycomb magnet Li3Co2SbO6, built of high-spin (Co2+) ions, in contrast to the more typical low-spin electron configurations in the presence of large spin-orbit coupling. Neutron powder diffraction measurements, heat capacity, and magnetization studies support the development of a long-range antiferromagnetic order space group of , below at . The magnetic entropy recovered between and 50 K is estimated to be , in good agreement with the value expected for systems close to a Kitaev quantum spin liquid state. The temperature-dependent magnetic order parameter demonstrates a value of 0.19(3), consistent with XY anisotropy and in-plane ordering, with Ising-like interactions between layers. Further, we observe a spin-flop-driven crossover to ferromagnetic order with space group of under an applied magnetic field of at . Magnetic structure analysis demonstrates these magnetic states are competing at finite applied magnetic fields even below the spin-flop transition. Both the compass model, a quantitative comparison of the specific heat of Li3Co2SbO6, and related honeycomb cobaltates to the anisotropic Kitaev model further support proximity to a Kitaev spin liquid state. This material demonstrates the rich playground of high-spin systems for spin liquid candidates and complements known Ir- and Ru-based materials.

Further insights into the thermodynamics of the Kitaev honeycomb model

Here we revisit the thermodynamics of the Kitaev quantum spin liquid realized on the honeycomb lattice. We address two main questions: First, we investigate whether there are observable thermodynamic signatures of the topological Majorana boundary modes of the Kitaev honeycomb model. We argue that for the time-reversal invariant case the residual low-temperature entropy is the primary thermodynamic signature of these Majorana edge modes, and verify using large-scale Monte Carlo simulations that this residual entropy is present in the full Kitaev model. When time-reversal symmetry is broken, the Majorana edge modes are potentially observable in more direct thermodynamic measurements such as the specific heat, though only at temperatures well below the bulk gap. % Second, we study the energetics, and the corresponding thermodynamic signatures, of the flux excitations in the Kitaev model. Specifically, we study the flux interactions on both cylinder and torus geometries numerically, and quantify their impact on the thermodynamics of the Kitaev spin liquid by using a polynomial fit for the average flux energy as a function of flux density and extrapolating it to the thermodynamic limit. By comparing this model to Monte Carlo simulations, we find that flux interactions have a significant quantitative impact on the shape and the position of the low-temperature peak in the specific heat.

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.

Kitaev Spin Liquid Candidate OsxCl3 Comprised of Honeycomb Nano-Domains

An osmium chloride with the chemical formula of OsxCl3 (x = 0.81) was synthesized and its crystal structure and thermodynamic properties were investigated. OsxCl3 crystallizes in a layered CdCl2-type structure with the triangular lattice partially occupied by Os ions on average. However, on microscopic length scales, the triangular lattice is composed of nano-domains with a honeycomb arrangement of Os ions, as observed by electron microscopy and Raman scattering experiments. Magnetization and heat capacity measurements revealed an absence of magnetic long-range order down to 0.08 K, while a broad peak in heat capacity at 0.15 K may indicate a short-range order in the local honeycomb lattice. OsxCl3 may exhibit certain aspects of the Kitaev spin liquid that are expected for a perfect honeycomb lattice of osmium trichloride.

Anderson\u2013Kitaev spin liquid

The bond-disordered Kitaev model attracts much attention due to the experimental relevance in α-RuCl3 and A3LiIr2O6 (A = H, D, Ag, etc.). Applying a magnetic field to break the time-reversal symmetry leads to a strong modulation in mass terms for Dirac cones. Because of the smallness of the flux gap of the Kitaev model, a small bond disorder can have large influence on itinerant Majorana fermions. The quantization of the thermal Hall conductivity κxy/T disappears by a quantum Hall transition induced by a small disorder, and κxy/T shows a rapid crossover into a state with a negligible Hall current. We call this immobile liquid state Anderson–Kitaev spin liquid (AKSL). Especially, the critical disorder strength δJc1 ~ 0.05 in the unit of the Kitaev interaction would have many implications for the stability of Kitaev spin liquids.

Multinode quantum spin liquids on the honeycomb lattice

Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $\Gamma'$ terms. To fully understand the effect of the $\Gamma'$ interactions, we investigate the quantum $K$-$\Gamma$-$\Gamma'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $\Gamma'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that a $\pi$-flux state with a distinct PSG appears at larger $\Gamma'$. The $\pi$-flux state is characterized by an enhanced periodic structure in the spinon dispersion in the original Brillouin zone (BZ), which is experimentally observable. Interestingly, two PKSL8 states are competing with the $\pi$-flux state and one of them can be stabilized by six-spin ring-exchange interactions. The physical properties of these nodal QSLs are studied by applying magnetic fields and the results depend on the number of cones. Our study infers that there exist a family of zero-flux QSLs that contain $6n+2, n\in\mathbb Z$ Majorana cones and a family of $\pi$-flux QSLs containing $4(6n+2)$ cones in the original BZ. It provides guidelines for experimental realization of non-Kitaev QSLs in relevant materials.

Kitaev quasiparticles in a proximate spin liquid: A many-body localization perspective

We study the stability of Kitaev quasiparticles in the presence of a perturbing Heisenberg interaction as a Fock space localization phenomenon. We identify parameter regimes where Kitaev states are localized, fractal or delocalized in the Fock space of exact eigenstates, with the first two implying quasiparticle stability. Finite temperature calculations show that a vison gap, and a nonzero plaquette Wilson loop at low temperatures, both characteristic of the deconfined Kitaev spin liquid phase, persist far into the neighboring phase that has a concomitant stripy spin-density wave (SDW) order. The key experimental implication for Kitaev materials is that below a characteristic energy scale, unrelated to the SDW ordering, Kitaev quasiparticles are stable.

Thermodynamic classification of three-dimensional Kitaev spin liquids

In the field of frustrated magnetism, Kitaev models provide a unique framework to study the phenomena of spin fractionalization and emergent lattice gauge theories in two and three spatial dimensions. Their ground states are quantum spin liquids, which can typically be described in terms of a Majorana band structure and an ordering of the underlying $\mathbb{Z}_2$ gauge structure. Here we provide a comprehensive classification of the "gauge physics" of a family of elementary three-dimensional Kitaev models, discussing how their thermodynamics and ground state order depends on the underlying lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations we show that the ground-state gauge order can generally be understood in terms of the length of elementary plaquettes -- a result which extends the applicability of Lieb's theorem to lattice geometries beyond its original scope. At finite temperatures, the proliferation of (gapped) vison excitations destroys the gauge order at a critical temperature scale, which we show to correlate with the size of vison gap for the family of three-dimensional Kitaev models. We also discuss two notable exceptions where the lattice structure gives rise to "gauge frustration" or intertwines the gauge ordering with time-reversal symmetry breaking. In a more general context, the thermodynamic gauge transitions in such 3D Kitaev models are one of the most natural settings for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.