Recent proposals for spin-1 Kitaev materials, such as honeycomb Ni oxides with heavy elements of Bi and Sb, have shown that these compounds naturally give rise to antiferromagnetic (AFM) Kitaev couplings. Conceptual interest in such AFM Kitaev systems has been sparked by the observation of a transition to a gapless $U(1)$ spin liquid at intermediate field strengths in the AFM spin-1/2 Kitaev model. However, all hitherto known spin-1/2 Kitaev materials exhibit ferromagnetic bond-directional exchanges. Here we discuss the physics of the spin-1 Kitaev model in a magnetic field and show, by extensive numerical analysis, that for AFM couplings it exhibits an extended gapless quantum spin liquid at intermediate field strengths. The close analogy to its spin-1/2 counterpart suggests that this gapless spin liquid is a $U(1)$ spin liquid with a neutral Fermi surface, that gives rise to enhanced thermal transport signatures.
In this work, we investigate whether the Kitaev honeycomb model can serve as a starting point to realize the intriguing physics of the Sachdev-Ye-Kitaev (SYK) model. The starting point is to strain the system, which leads to flat bands reminiscent of Landau levels, thereby quenching the kinetic energy. The presence of weak residual perturbations, such as Heisenberg interactions and the $γ$-term, creates effective interactions between the Majorana modes when projected into the flux-free sector. We assume the resulting interactions to be effectively random. This leads to a bipartite Sachdev-Ye-K…
We study the reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the reduced fidelity susceptibility of two nearest site manifest itself a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, and thus suggest that the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen, des…
A range of honeycomb-lattice compounds has been proposed and investigated in the search for a topological Kitaev spin liquid. However, sizable Heisenberg interactions and additional symmetry-allowed exchange anisotropies in the magnetic Hamiltonian of these potential Kitaev materials push them away from the pure Kitaev spin-liquid state. Particularly the Kitaev-to-Heisenberg coupling ratio is essential in this respect. With the help of advanced quantum-chemistry methods, we explore how the magnetic coupling ratios depend on pressure in several honeycomb compounds (Na$_2$IrO$_3$, $β$-Li$_2$IrO$…
The realization of Kitaev spin liquid, where spins on a honeycomb lattice are coupled ferromagnetically by bond-dependent anisotropic interactions, has been a sought-after dream. 5d iridium oxides $α$-Li2IrO3 and $α$-Na2IrO3 with a honeycomb lattice of Jeff = 1/2 moments recently emerged as a possible materialization. Strong signature of Kitaev physics, however, was not captured. Here we report the discovery of a complex iridium oxide $β$-Li2IrO3 with Jeff = 1/2 moments on "hyper-honeycomb" lattice, a three-dimensional analogue of honeycomb lattice. A positive Curie-Weiss temperature $θ_{CW}$ …
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 linea…
We calculate the resonant inelastic X-ray scattering (RIXS) response of the Kitaev honeycomb model, an exactly solvable quantum-spin-liquid model with fractionalized Majorana and flux excitations. We find that the fundamental RIXS channels, the spin-conserving (SC) and the non-spin-conserving (NSC) ones, do not interfere and give completely different responses. SC-RIXS picks up exclusively the Majorana sector with a pronounced momentum dispersion, whereas NSC-RIXS also creates immobile fluxes, thereby rendering the response only weakly momentum dependent, as in the spin structure factor measur…
Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of Kitaev's honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities reflect not the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.
The Kitaev spin liquid, a ground state of the bond-dependent Kitaev model in a honeycomb lattice has been a centre of attraction, since a microscopic theory to realize such an interaction in solid-state materials was discovered. A challenge in real materials though is the presence of the Heisenberg and another bond-dependent Gamma interactions detrimental to the Kitaev spin liquid, and there have been many debates on their relative strengths. Here we offer a strategy to extract the Kitaev interaction out of a full microscopic model by utilizing the symmetries of the Hamiltonian. Two tilted mag…
The Kitaev-Heisenberg model defined on both honeycomb and triangular lattices has been studied intensively in recent years as a possible model to describe spin-orbital physics in iridium oxides. In the model, there are many phases characteristic for each lattice. However, there is no study how the phases in the two lattices merge each other when geometry changes from honeycomb lattice to triangular lattice. We investigate the ground state of the Kitaev-Heisenberg model defined on the system connecting the honeycomb and triangular lattices, named a honeycomb-triangular lattice. We obtain a grou…
The properties of Kitaev materials are attracting ever increasing attention owing to their exotic properties. In realistic two-dimensional materials, Kitaev interaction is often accompanied by the Dzyloshinskii-Moriya interaction, which poses a challenge of distinguishing their magnitude separately. In this work, we demonstrate that it can be done by accessing magnonic transport properties. By studying honeycomb ferromagnets exhibiting Dzyaloshinskii-Moriya and Kitaev interactions simultaneously, we reveal non-trivial magnonic topological properties accompanied by intricate magnonic transport …
We show that line defects in a non-Abelian Kitaev spin liquid harbor gapless one-dimensional Majorana modes if the interaction across the defect falls below a critical value. Treating the weak interaction at the line defect within a mean-field approximation, we determine the critical interaction strength as a function of the external magnetic field. In the gapless regime, we use the low-energy effective field theory to calculate the spin-lattice relaxation rate for a nuclear spin near the defect and find a cubic temperature dependence that agrees with experiments in the Kitaev material $α\text…
In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($Γ$) couplings. We investigate the quantum ($S = 1/2$) $K$-$J$-$Γ$ model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic f…
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 dominate in an applied magnetic fiel…
Motivated by recent reports of a field-induced intermediate phase (IP) in the antiferromagnetic honeycomb Kitaev model that may be a spin liquid whose nature is distinct from the Kitaev Z2 phase, we present a detailed numerical study on the nature and dynamical response (such as dynamical spin-structure factors and resonant inelastic x-ray scattering intensities) of this field-induced IP and neighboring phases in a family of Kitaev-based models related by hidden symmetries and duality transformations. We further show that the same field-induced IP can appear in models relevant for $α$-RuCl${_3…
University of Toronto（多伦多大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
Recently, honeycomb cobaltates with 3$d^7$ were proposed to display Kitaev physics despite weak spin-orbit coupling. However, other theoretical and experimental works found leading XXZ Heisenberg and negligible Kitaev interactions in BaCo$_2$(AsO$_4$)$_2$ (BCAO), which calls for a further study to clarify the origin of the discrepancies. Here we derive the analytical expressions of the spin model using strong-coupling perturbation theory. With tight binding parameters obtained by {\it ab-initio} calculations for idealized honeycomb BCAO, we find that the largest intraorbital $t_{2g}-t_{2g}$ ex…
We apply the SU(2) slave fermion formalism to the Kitaev honeycomb lattice model. We show that both the Toric Code phase (the A phase) and the gapless phase of this model (the B phase) can be identified with p-wave superconducting phases of the slave fermions, with nodal lines which, respectively, do not or do intersect the Fermi surface. The non-Abelian Ising anyon phase is a $p+ip$ superconducting phase which occurs when the B phase is subjected to a gap-opening magnetic field. We also discuss the transitions between these phases in this language.
The layered honeycomb magnet $α$-Li$_2$IrO$_3$ has been theoretically proposed as a candidate to display novel magnetic behaviour associated with Kitaev interactions between spin-orbit entangled $j_{\rm eff}=1/2$ magnetic moments on a honeycomb lattice. Here we report single crystal magnetic resonant x-ray diffraction combined with powder magnetic neutron diffraction to reveal an incommensurate magnetic order in the honeycomb layers with Ir magnetic moments counter-rotating on nearest-neighbour sites. This type of magnetic structure has not been reported experimentally before in honeycomb magn…
The main question we address is how to probe the fractionalized excitations of a quantum spin liquid (QSL), for example, in the Kitaev honeycomb model. By analyzing the energy spectrum and entanglement entropy, for antiferromagnetic couplings and a field along either [111] or [001], we find a gapless QSL phase sandwiched between the non-Abelian Kitaev QSL and polarized phases. Increasing the field strength towards the polarized limit destroys this intermediate QSL phase, resulting in a considerable reduction in the number of frequency modes and the emergence of a beating pattern in the local d…
Due to the combination of a substantial spin-orbit coupling and correlation effects, iridium oxides hold a prominent place in the search for novel quantum states of matter, including, e.g., Kitaev spin liquids and topological Weyl states. We establish the promise of the very recently synthesized hyper-honeycomb iridate $β$-Li2IrO3 in this regard. A detailed theoretical analysis reveals the presence of large ferromagnetic first-neighbor Kitaev interactions, while a second-neighbor antiferromagnetic Heisenberg exchange drives the ground state from ferro to zigzag order via a three-dimensional Ki…
We investigate the ground state properties of the spin S=1 Kitaev honeycomb model under a magnetic field based on the density matrix renormalization group (DMRG) calculation. With the time reversal symmetry breaking due to the magnetic field, a gapped Kitaev spin liquid is identified for both ferromagnetic (FM) and antiferromagnetic (AFM) Kitaev couplings. The topological nature of such Kitaev spin liquid is manifested by the nearly quantized Wilson loop, degeneracy in the entanglement spectra and existence of edge modes. While the FM Kitaev spin liquid is destroyed by a weaker magnetic field …
We investigate the generic features of the low energy dynamical spin structure factor of the Kitaev honeycomb quantum spin liquid perturbed away from its exact soluble limit by generic symmetry-allowed exchange couplings. We find that the spin gap persists in the Kitaev-Heisenberg model, but generally vanishes provided more generic symmetry-allowed interactions exist. We formulate the generic expansion of the spin operator in terms of fractionalized Majorana fermion operators according to the symmetry enriched topological order of the Kitaev spin liquid, described by its projective symmetry gr…
We derive and study a spin one-half Hamiltonian on a honeycomb lattice describing the exchange interactions between Ir$^{4+}$ ions in a family of layered iridates $A_2$IrO$_3$ ($A$=Li,Na). Depending on the microscopic parameters, the Hamiltonian interpolates between the Heisenberg and exactly solvable Kitaev models. Exact diagonalization and a complementary spin-wave analysis reveal the presence of an extended spin-liquid phase near the Kitaev limit and a conventional Néel state close to the Heisenberg limit. The two phases are separated by an unusual stripy antiferromagnetic state, which is t…
Kitaev quantum spin liquid, massively quantum entangled states, is so scarce in nature that searching for new candidate systems remains a great challenge. Honeycomb heterostructure could be a promising route to realize and utilize such an exotic quantum phase by providing additional controllability of Hamiltonian and device compatibility, respectively. Here, we provide epitaxial honeycomb oxide thin film Na3Co2SbO6, a candidate of Kitaev quantum spin liquid proposed recently. We found a spin glass and antiferromagnetic ground states depending on Na stoichiometry, signifying not only the import…
We study the gapped phase of Kitaev's honeycomb model (a Z_2 spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z_2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z_2 vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.
Intensive studies of the interplay between spin-orbit coupling (SOC) and electronic correlations in transition metal compounds have recently been undertaken. In particular, $j_{\rm eff}$ = 1/2 bands on a honeycomb lattice provide a pathway to realize Kitaev's exactly solvable spin model. However, since current wisdom requires strong atomic SOC to make $j_{\rm eff}=1/2$ bands, studies have been limited to iridium oxides. Contrary to this expectation, we demonstrate how Kitaev interactions arise in 4$d$-orbital honeycomb $α$-RuCl$_3$, despite having significantly weaker SOC than the iridium oxid…
Recently, realizations of Kitaev physics have been sought in the A2IrO3 family of honeycomb iridates, originating from oxygen-mediated exchange through edge-shared octahedra. However, for the j = 1/2 Mott insulator in these materials exchange from direct d-orbital overlap is relevant, and it was proposed that a Heisenberg term should be added to the Kitaev model. Here we provide the generic nearest-neighbour spin Hamiltonian when both oxygen-mediated and direct overlap are present, containing a bond dependent off-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyze this com…
We consider the quasi-two-dimensional pseudo-spin-1/2 Kitaev - Heisenberg model proposed for A2IrO3 (A=Li, Na) compounds. The spin-wave excitation spectrum, the sublattice magnetization, and the transition temperatures are calculated in the random phase approximation (RPA) for four different ordered phases, observed in the parameter space of the model: antiferomagnetic, stripe, ferromagnetic, and zigzag phases. The Néel temperature and temperature dependence of the sublattice magnetization are compared with the experimental data on Na2IrO3.
The theoretical inception of the Kitaev honeycomb model has had defining influence on the experimental hunt for quantum spin liquids, bringing unprecedented focus onto the synthesis of materials with bond-directional interactions. Numerous Kitaev materials, which are believed to harbor ground states parametrically close to the Kitaev spin liquid, have been investigated since. Yet, in these materials the Kitaev interaction often comes hand in hand with off-diagonal $Γ$ interactions -- with the competition of the two potentially giving rise to a magnetically ordered ground state. In an attempt t…
We study the Kitaev-Heisenberg model with spin-1 local degree of freedom on a two-dimensional honeycomb lattice numerically by density matrix renormalization group method. By tuning the relative value of the Kitaev and Heisenberg exchange couplings, we obtain the whole phase diagram with two spin liquid phases and four symmetry broken phases. We identify that the spin liquid phases are gapless by calculating the central charge at the pure Kitaev points without Heisenberg interaction. Comparing to its spin-1/2 counterpart, the position and number of gapless modes of the spin-1 case are quite di…
We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg ($J=\cosϕ)$ and Kitaev ($K=\sinϕ$) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-$z$ , ferromagnetic-$xy$, staggered-$xy$, Néel-$z$, and two liquid states: Tomonaga-Luttinger liquid and spiral-$xy$. The two Kitaev points $ϕ=\fracπ{2}$ and $ϕ=\frac{3π}{2}$ are singular. The $ϕ$-dependent phase dia…
Compounds of transition metal ions with strong spin-orbit coupling recently attracted attention due to the possibility to host frustrated bond-dependent anisotropic magnetic interactions. In general, such interactions lead to complex phase diagrams that may include exotic phases, e.g. the Kitaev spin liquid. Here we report on our comprehensive analysis of the global phase diagram of the extended Kitaev-Heisenberg model relevant to honeycomb lattice compounds Na2IrO3 and alpha-RuCl3. We have utilized recently developed method based on spin coherent states that enabled us to resolve arbitrary sp…
Motivated by recent experiments on the Kitaev honeycomb magnet $α\text{-RuCl}_3$, we introduce time-domain probes of the edge and quasiparticle content of non-Abelian spin liquids. Our scheme exploits ancillary quantum spins that communicate via time-dependent tunneling of energy into and out of the spin liquid's chiral Majorana edge state. We show that the ancillary-spin dynamics reveals the edge-state velocity and, in suitable geometries, detects individual non-Abelian anyons and emergent fermions via a time-domain counterpart of quantum-Hall anyon interferometry. We anticipate applications …
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 br…
We theoretically propose a family of $f$-electron based magnets that realizes Kitaev-type bond-dependent anisotropic interactions. Based on {\it ab initio} calculations, we show that $A_2$PrO$_3$ ($A$: alkali metals) crystalize in a triclinic structure with honeycomb layers of edge-sharing PrO$_6$ octahedra. Each Pr$^{4+}$ cation has a $4f$ electron in the $Γ_7$ doublet, which comprises a spin-orbital entangled Kramers pair with the effective moment $J_{\rm eff}=1/2$. By using the Wannier orbitals from the {\it ab initio} calculations, we find that the effective interactions between the $J_{\r…
Quantum spin liquid is a disordered magnetic state with fractional spin excitations. Its clearest example is found in an exactly solved Kitaev honeycomb model where a spin flip fractionalizes into two types of anyons, quasiparticles that are neither fermions nor bosons: a pair of gauge fluxes and a Majorana fermion. Here we demonstrate this kind of fractionalization in the Kitaev paramagnetic state of the honeycomb magnet $α$-RuCl$_3$. The spin-excitation gap measured by nuclear magnetic resonance consists of the predicted Majorana fermion contribution following the cube of the applied magneti…
We study the extended Kitaev-Heisenberg (EKH) quantum spin model by adding bond-dependent off-diagonal Heisenberg term into the original KH model, which was recently proposed to describe the honeycomb Iridates. A rigorous mathematical mapping of spin operators reveals the intrinsic symmetry of the model Hamiltonian. By employing an unbiased numerical entanglement renormalization method based on tensor network ansatz, we obtain the global phase diagram containing eight distinct quantum phases. By using the dual mapping of spin operators, each of the individual magnetic phase in the global phase…
We compute the quantum Fisher information (QFI) associated with two different local operators in the Kitaev honeycomb model, and find divergent behaviour in the second derivatives of these quantities with respect to the driving parameter at the quantum phase transition between the gapped and gapless phases for both fully anti-ferromagnetic and fully ferromagnetic exchange couplings. The QFI associated with a local magnetization operator behaves differently from that associated with a local bond operator depending on whether the critical point is approached from the gapped or gapless side. We s…
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, th…
Based on the Dirac spinor representation of the SO(4) group, we discuss the relationship between three types of representation of spin in terms of Majorana fermions, namely the Kitaev representation, the SO(3) representation and the SO(4) chiral representation. Comparing the three types, we show that the Hilbert space of the SO(3) representation is different from the other two by requiring pairing of sites, but it has the advantage over the other two in that no unphysical states are involved. As an example of its application, we present a new alternative solution of the Kitaev honeycomb model.…
The Kitaev model, characterized by bond-dependent Ising spin interactions among spin-orbit entangled dipole moments in Mott insulators, offered a new approach to quantum spin liquids. Motivated by another type of bond-dependent interaction among quadrupole moments in $5d^2$ Mott insulators, we provide a microscopic route to uncover the Kitaev multipolar liquid, featuring fractionalized excitations out of non-Kramers doublets carrying multipole moments. The key ingredient is the magnetic field that allows for bond-anisotropic quadrupole-octupole interactions via mixing with the excited triplet …
The exactly solvable Kitaev honeycomb lattice model is realized as the low energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low energy effective Hamiltonian is exact, without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments, and contains short-range interactions up to six-spin(or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments, but as pseudo-spin of the two-dimensional spin singlet sector of the f…
We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second-neighbor Kitaev coupling $K_2$, which has recently been shown to be the dominant perturbation away from the nearest-neighbor model in iridate Na$_2$IrO$_3$, and may also play a role in $α$-RuCl$_3$ and Li$_2$IrO$_3$. This coupling naturally explains the zigzag ordering (without introducing unrealistically large longer-range Heisenberg exchange terms) and the special entanglement between real and spin space observed recently in Na$_2$IrO$_3$. Moreover, the minimal $K_1$-$K_2$ model t…
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, $J$ and $\tilde J$, where $J$ is for the conventional spin and and $\tilde J$ for rotated spin. When $\tilde J=0$, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order($\tilde J<0$, ferromagnetic for rotated spin) or a zig-zag antiferromagnetic order ($\tilde J>0$, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or two antiferromagnetic orders induces Kitaev's spin liquid phase characterized by the exactly sol…
In extended Kitaev models on the honeycomb lattice, off-diagonal interactions (e.g. the $Γ, Γ^{'}$ terms) give rise to non-Kitaev quantum spin liquid (QSL) and several magnetically ordered phases. In the present work, we dope holes to the system and study the resultant $t$-$K$-$Γ$-$Γ^{'}$ model by mean field theory. The interplay between the charge and spin degrees of freedom results in a rich phase diagram. Similar to doped cuprates, superconductors, pseudogap phases, fermi liquid, strange metal and paramagnetic phase are generated. What is different is that, more than one superconducting pha…
We demonstrate that $\mathbb{Z}_2$ gauge transformations and lattice deformations in Kitaev's honeycomb lattice model can have the same description in the continuum limit of the model in terms of chiral gauge fields. The chiral gauge fields are coupled to the Majorana fermions that satisfy the Dirac dispersion relation in the non-Abelian sector of the model. For particular values, the effective chiral gauge field becomes equivalent to the $\mathbb{Z}_2$ gauge field, enabling us to associate effective fluxes to lattice deformations. Motivated by this equivalence, we consider Majorana-bounding $…
Motivated by recent synthesis of the hyper-honeycomb material $β$-$\mathrm{Li_2 Ir O_3}$, we study the dynamical structure factor (DSF) of the corresponding 3D Kitaev quantum spin-liquid (QSL), whose fractionalised degrees of freedom are Majorana fermions and emergent flux-loops. Properties of this 3D model are known to differ in important ways from those of its 2D counterpart -- it has finite-temperature phase transition, as well as distinct features in Raman response. We show, however, that the qualitative behaviour of the DSF is broadly dimension-independent. Characteristics of the 3D DSF i…
We develop high temperature series expansions for the thermodynamic properties of the honeycomb-lattice Kitaev-Heisenberg model. Numerical results for uniform susceptibility, heat capacity and entropy as a function of temperature for different values of the Kitaev coupling $K$ and Heisenberg exachange coupling $J$ (with $|J|\le |K|$) are presented. These expansions show good convergence down to a temperature of a fraction of $K$ and in some cases down to $T=K/10$. In the Kitaev exchange dominated regime, the inverse susceptibility has a nearly linear temperature dependence over a wide temperat…
Co$^{2+}$ ions in an octahedral crystal field, stabilise a j$_{eff}$ = 1/2 ground state with an orbital degree of freedom and have been recently put forward for realising Kitaev interactions, a prediction we have tested by investigating spin dynamics in two cobalt honeycomb lattice compounds, Na$_2$Co$_2$TeO$_6$ and Na$_3$Co$_2$SbO$_6$, using inelastic neutron scattering. We used linear spin wave theory to show that the magnetic spectra can be reproduced with a spin Hamiltonian including a dominant Kitaev nearest-neighbour interaction, weaker Heisenberg interactions up to the third neighbour a…
In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of $\mathbb{Z}_2$ quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gaple…
We study the gapped phase of Kitaev's honeycomb model (a $Z_2$ spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a $Z_2$ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a $Z_2$ vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics …
Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of qu…
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…
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$, $K_{X}+K_{Y}+K_{Z}=-3K$, and $K_{Z}<-K$, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=gμ_{B}H_{z1}/K$, to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$, and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$, respectively. These indicate that magnetic fi…
