The realization of Kitaev's honeycomb magnetic model in real materials has become one of the most pursued topics in condensed matter physics and materials science. If found, it is expected to host exotic quantum phases of matter and offers potential realizations of fault$-$tolerant quantum computations. Over the past years, much effort was made on 4d$-$ or 5d$-$ heavy transition metal compounds because of their intrinsic strong spin$-$orbit coupling. But more recently, there have been growing shreds of evidence that the Kitaev model could also be realized in 3d$-$transition metal systems with much weaker spin$-$orbit coupling. This review intends to serve as a guide to this fast$-$developing field focusing on systems with d$^7$ transition metal occupation. It overviews the current theoretical and experimental progress on realizing the Kitaev model in those systems. We examine the recent experimental observations of candidate materials with Co$^{2+}$ ions: e.g., CoPS$_3$, Na$_3$Co$_2$SbO$_6$, and Na$_2$Co$_2$TeO$_6$, followed by a brief review of theoretical backgrounds. We conclude this article by comparing experimental observations with density functional theory (DFT) calculations. We stress the importance of inter$-t_{2g}$ hopping channels and Hund's coupling in the realization of Kitaev interactions in Co$-$based compounds, which has been overlooked in previous studies. This review suggests future directions in the search for Kitaev physics in 3d cobalt compounds and beyond.
Kitaev physics has recently attracted attention in condensed matter for its anticipated novel quantum spin liquid state. The thermal transport measurement is crucial for probing the novel features of charge-neutral quasiparticles. In this letter, we report a significant thermal Hall effect in $Na_2Co_2TeO_6$ (NCTO), a Kitaev quantum spin liquid candidate, when the magnetic field is applied along the out-of-plane direction of the honeycomb plane. The thermal conductivity ($κ_{xx}$) and thermal Hall conductivity ($κ_{xy}$) in NCTO reveal distinct magnetic field dependences below and above the Ne…
In the quest for materials hosting Kitaev spin liquids, much of the efforts have been focused on the fourth- and fifth-row transition metal compounds, which are spin-orbit coupling assisted Mott insulators. Here, we study the structural and magnetic properties of 3$d$ transition metal oxides, Na$_2$Co$_2$TeO$_6$ and Na$_3$Co$_2$SbO$_6$. The partial occupancy of sodium in former compound is addressed using a cluster expansion, and a honeycomb lattice of sodiums is found to be energetically favored. Starting from the \textit{ab initio} band structures, a many-body second order perturbation theor…
The current efforts to find the materials hosting Kitaev model physics have been focused on Mott insulators of d^5 pseudospin-1/2 ions Ir^{4+} and Ru^{3+} with t_{2g}^5(S=1/2, L=1) electronic configuration. Here we propose that the Kitaev model can be realized in materials based on d^7 ions with t_{2g}^5e_g^2(S=3/2, L=1) configuration such as Co^{2+}, which also host the pseudospin-1/2 magnetism. Considering possible exchange processes, we have derived the d^7 pseudospin-1/2 interactions in 90^{\circ} bonding geometry. The obtained Hamiltonian comprises the bond-directional Kitaev K and isotro…
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 …
This paper reviews the current progress on searching the Kitaev spin liquid state in 3d electron systems. Honeycomb cobaltates were recently proposed as promising candidates to realize the Kitaev spin liquid state, due to the more localized wave functions of 3d ions compared with that of 4d and 5d ions, and also the easy tunability of the exchange Hamiltonian in favor of Kitaev interaction. Several key parameters that have large impacts on the exchange constants, such as the charge-transfer gap and the trigonal crystal field, are identified and discussed. Specifically, tuning crystal field eff…
Dominant Kitaev Interaction and Field-induced Quantum Disordered Phase in the Cobaltate Na$_2$Co$_2$TeO$_6$
Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；School of Physics, Southeast University（东南大学物理学院）；Anhui Key Laboratory of Low-Energy Quantum Materials and Devices, High Magnetic Field Laboratory, Hefei Institutes of Physical Science, Chinese Academy of Sciences（中国科学院合肥物质科学研究院强磁场科学中心）；School of Physical Science and Engineering, Beijing Jiaotong University（北京交通大学物理科学与工程学院）；Center for Quantum Materials, Seoul National University（首尔大学量子材料中心）；School of Physics, Georgia Institute of Technology（佐治亚理工学院物理学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
The quest for Kitaev quantum spin liquids has led to great interest in honeycomb quantum magnets with strong spin-orbit coupling. It has been recently proposed that even Mott insulators with $3d$ transition metal ions, having nominally weak spin-orbit coupling, can realize such exotic physics. Motivated by this, we study the rhombohedral honeycomb cobaltates CoTiO$_3$, BaCo$_2$(PO$_4$)$_2$, and BaCo$_2$(AsO$_4$)$_2$, using $\textit{ab initio}$ density functional theory, which takes into account realistic crystal field distortions and chemical information, in conjunction with exact diagonalizat…
We study the exchange interactions and resulting magnetic phases in the honeycomb cobaltates. For a broad range of trigonal crystal fields acting on Co2+ ions, the low-energy pseudospin-1/2 Hamiltonian is dominated by bond-dependent Ising couplings that constitute the Kitaev model. The non-Kitaev terms nearly vanish at small values of trigonal field Δ, resulting in spin liquid ground state. Considering Na3Co2SbO6 as an example, we find that this compound is proximate to a Kitaev spin liquid phase, and can be driven into it by slightly reducing Δby \sim 20 meV, e.g., via strain or pressure cont…
$α$-RuCl$_3$ has been extensively studied recently because of potential bond-dependent Kitaev magnetic exchange interactions and the resulting quantum spin liquid phase that can be realized therein. It has been known that the covalency between Ru 4$d$- and Cl $p$-orbitals is crucial to induce large Kitaev interactions in this compound, therefore replacing Cl into heavier halogen elements such as Br or I can be a promising way to promote the Kitaev interaction even further. In a timely manner, there have been reports on synthesis of $α$-RuBr$_3$ and $α$-RuI$_3$, which are expected to host the s…
Candidate materials for Kitaev spin liquid phase have been intensively studied recently because of their potential applications in fault-tolerant quantum computing. Although most of the studies on Kitaev spin liquid have been done in 4$d$ and 5$d$ based transition metal compounds, recently there has been a growing research interest in Co-based quasi-two-dimensional honeycomb magnets, such as BaCo$_2$(AsO$_4$)$_2$ because of formation of spin-orbit-entangled $J_{\rm eff}$ = 1/2 pseudospin moments at Co$^{2+}$ sites and potential realizations of Kitaev-like magnetism therein. Here, we obtain hig…
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…
Kitaev-Ising-$J_1$-$J_2$ model: a density matrix renormalization group study
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences（俄罗斯科学院理论与应用电动力学研究所）
There has been a great interest in magnetic field induced quantum spin liquids in Kitaev magnets after the discovery of neutron scattering continuum and half quantized thermal Hall conductivity in the material $α$-RuCl$_3$. In this work, we provide a semiclassical analysis of the relevant theoretical models on large system sizes, and compare the results to previous studies on quantum models with small system sizes. We find a series of competing magnetic orders with fairly large unit cells at intermediate magnetic fields, which are most likely missed by previous approaches. We show that quantum…
Quantum spin liquids (QSLs) are topologically ordered exotic states of matter that host fractionalized excitations. Kitaev proposed a particular route towards a QSL via strongly bond-dependent interactions on the hexagonal lattice. A number of candidate Kitaev QSL materials have been pursued, but all have appreciable non-Kitaev interactions, which put these systems far from the QSL regime. Using time-domain terahertz spectroscopy (TDTS) we observed a broad magnetic continuum over a wide range of temperature and field in the honeycomb cobalt-based magnet, BaCo$_2$(AsO$_4$)$_2$, which has been p…
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…
Very recently, a 3$d$ based honeycomb cobaltate Na$_2$Co$_2$TeO$_6$ has garnered tremendous attention due to the proposed proximity to the Kitaev spin-liquid state as its 4$d$/5$d$ counterparts. Here, we use Zn to substitute Co in a broad range and perform systematic studies on Na$_2$Co$_{2-x}$Zn$_x$TeO$_6$ by structural, magnetic, and thermodynamic measurements, and track the doping evolution of its magnetic ground states. Due to the extremely close radii of Zn$^{2+}$ and high-spin Co$^{2+}$ ions, the substitution can be easily achieved. X-ray diffractions reveal no structural transition but …
The recently discovered three-dimensional hyperhoneycomb iridate, $β$-Li$_2$IrO$_3$, has raised hopes for the realization of dominant Kitaev interaction between spin-orbit entangled local moments due to its near-ideal lattice structure. If true, this material may lie close to the sought-after quantum spin liquid phase in three dimensions. Utilizing ab-initio electronic structure calculations, we first show that the spin-orbit entangled basis, $j_{\rm eff}$=1/2, correctly captures the low energy electronic structure. The effective spin model derived in the strong coupling limit supplemented by …
By considering two {\it ab initio}-based complementary approaches, we analyze the electronic structure and extract effective spin models of BaCo$_2$(AsO$_4$)$_2$, a honeycomb material which has been proposed as a candidate for Kitaev physics. Both methods show that the dominant direct hopping makes the bond-dependent Kitaev term negligible averting the material away from the sought-after spin-liquid regime. As a result, we present a simple three-parameter exchange model to describe the interactions of the lowest doublet of the honeycomb cobaltate BaCo$_2$(AsO$_4$)$_2$. Remarkably, it is the th…
Relevance of on-site and intersite Coulomb interactions in the Kitaev-Heisenberg magnet Na$_3$Co$_2$SbO$_6$
Institute for Theoretical Solid State Physics, Leibniz IFW Dresden（德累斯顿亥姆霍兹中心莱布尼茨研究所理论固态物理研究所）；Department of Physics, Karpagam Academy of Higher Education（卡尔帕甘高等教育学院物理系）；Centre for Computational Physics, Karpagam Academy of Higher Education（卡尔帕甘高等教育学院计算物理中心）；School of Physics, University of Melbourne（墨尔本大学物理学院）；Department of Physics, Technical University Dresden（德累斯顿工业大学物理系）
Quantum spin liquids (QSLs) are highly entangled, disordered magnetic states that arise in frustrated Mott insulators and host exotic fractional excitations such as spinons and chargons. Despite being charge insulators some QSLs are predicted to exhibit gapless itinerant spinons that yield metallic behavior in the spin channel. We have deposited isolated magnetic atoms onto single-layer (SL) 1T-TaSe$_2$, a gapless QSL candidate, to experimentally probe how itinerant spinons couple to impurity spin centers. Using scanning tunneling spectroscopy we observe the emergence of new, impurity-induced …
We construct a series of one-dimensional non-unitary dynamics consisting of both unitary and imaginary evolutions based on the Sachdev-Ye-Kitaev model. Starting from a short-range entangled state, we analyze the entanglement dynamics using the path integral formalism in the large $N$ limit. Among all the results that we obtain, two of them are particularly interesting: (1) By varying the strength of the imaginary evolution, the interacting model exhibits a first order phase transition from the highly entangled volume law phase to an area law phase; (2) The one-dimensional free fermion model di…
Cobaltates with 3$d$ based layered honeycomb structure were recently proposed as Kitaev magnets and putative candidates to host the long-sought Kitaev spin liquid. Here we present inelastic neutron scattering results down to 50 mK for powder samples of Na$_3$Co$_2$SbO$_6$ and Na$_2$Co$_2$TeO$_6$, with high resolution in regions of low momentum and energy transfers. We compare the experimental data below the antiferromagnetic zigzag ordering temperature with dynamical structure factors obtained within spin wave theory. We search the wide parameter range of a $K$-$J_1$-$Γ$-$Γ^{\prime}$-$J_3$ spi…
Significant efforts have focused on the magnetic excitations of relativistic Mott insulators, predicted to realize the Kitaev quantum spin liquid (QSL). This exactly solvable model involves a highly entangled state resulting from bond-dependent Ising interactions that produce excitations which are non-local in terms of spin flips. A key challenge in real materials is identifying the relative size of the non-Kitaev terms and their role in the emergence or suppression of fractional excitations. Here, we identify the energy and temperature boundaries of non-Kitaev interactions by direct compariso…
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…
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…
We study the orbitally frustrated singlet-triplet models that emerge in the context of spin-orbit coupled Mott insulators with $t_{2g}^4$ electronic configuration. In these compounds, low-energy magnetic degrees of freedom can be cast in terms of three-flavor "triplon" operators describing the transitions between spin-orbit entangled $J=0$ ionic ground state and excited $J=1$ levels. In contrast to a conventional, flavor-isotropic O(3) singlet-triplet models, spin-orbit entangled triplon interactions are flavor-and-bond selective and thus highly frustrated. In a honeycomb lattice, we find clos…
In the search for quantum spin liquids, candidate materials for the Kitaev model and its extensions have been intensively explored during the past decade, as the models realize the exact quantum spin liquids in the ground state. Thus far, insulating magnets in the low-spin $d^5$ electron configuration under the strong spin-orbit coupling have been studied for realizing the Kitaev-type bond-dependent anisotropic interactions between the spin-orbital entangled Kramers doublets. To extend the candidates, here we investigate the systems in a high-spin $d^7$ electron configuration, whose ground sta…
The quantum spin liquid (QSL) state has been searched intensively in Kitaev-like materials, such as the Iridium oxides $A_2$IrO$_3$ and $α$-RuCl$_3$. The half-filled Kitaev-Hubbard model with bond dependent hopping terms is used to describe the Kitaev-like materials, which is calculated using the state-of-the-art fermionic projected entangled pair states (fPEPS) method. We find a QSL phase near the Mott insulator transition, which has a strong first-order transition to the semi-metal phase with the decrease of Hubbard $U$. We suggest that a promising routine to find the QSL is to find the Irid…
Inelastic neutron scattering of the layered Kitaev ferromagnet Li$_3$Co$_2$SbO$_6$
University of Melbourne（墨尔本大学）；Australian Nuclear Science and Technology Organisation (ANSTO)（澳大利亚核科学与技术组织）；The University of Sydney（悉尼大学）；University of Warwick（华威大学）；DESY（德国电子同步加速器研究所）
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 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 $Li_3Co_2SbO_6$, built of high-spin $\it{d^7}$ ($Co^{2+}$) ions, in contrast to the more typical low-spin $\it{d^5}$ 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 $\it{C_C}2/\…
Elementary excitations in highly entangled states such as quantum spin liquids may exhibit exotic statistics, different from those obeyed by fundamental bosons and fermions. Excitations called non-Abelian anyons are predicted to exist in a Kitaev spin liquid - the ground state of an exactly solvable model proposed by Kitaev almost a decade ago. A smoking-gun signature of such non-Abelian anyons, namely a half-integer quantized thermal Hall conductivity, was recently reported in $α$-RuCl$_3$. While fascinating, a microscopic theory for this phenomenon in $α$-RuCl$_3$ remains elusive because the…
Recent discovery of the half quantized thermal Hall conductivity in $α$-RuCl$_3$, a candidate material for the Kitaev spin liquid, suggests the presence of a highly entangled quantum state in external magnetic fields. This field induced phase appears between the low field zig-zag magnetic order and the high field polarized state. Motivated by this experiment, we study possible field induced quantum phases in theoretical models of the Kitaev magnets, using the two dimensional tensor network approach or infinite tensor product states. We find various quantum ground states in addition to the chir…
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 extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite Projected Entangled-Pair States tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model we confirm the existence of 6 different phases: Néel, stripy, ferromagnetic, zigzag and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase t…
In transition-metal compounds with partially filled $4d$ and $5d$ shells spin-orbit entanglement, electronic correlations, and crystal-field effects conspire to give rise to a variety of novel forms of topological quantum matter. This includes Kitaev materials -- a family of spin-orbit assisted Mott insulators, in which local, spin-orbit entangled $j=1/2$ moments form that are subject to strong bond-directional interactions. On a conceptual level, Kitaev materials attract much interest for their unconventional forms of magnetism, such as spin liquid physics in two- and three-dimensional lattic…
This pedagogical article is aimed to the beginning graduate students interested in broad field of frustrated magnetism. We introduce and present some of the exact results obtained in Kitaev model. The Kitaev model embodies an unusual two spin interactions yet exactly solvable model in two dimension. This exact solvability renders it to realize many emergent many body phenomena such as $Z_2$ gauge field, spin liquid states, spin fractionalization, topological order exactly. First we present the exact solution of Kitaev model using Majorana fermionisation and elaborate in detail the $Z_2$ gauge …
University of Munich（慕尼黑大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Shanghai Jiao Tong University（上海交通大学）
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-$Γ$ 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-in…
The Kitaev quantum spin liquid epitomizes an entangled topological state, for which two flavors of fractionalized low-energy excitations are predicted: the itinerant Majorana fermion and the Z2 gauge flux. Detection of these excitations remains challenging, because of their fractional quantum numbers and non-locality. It was proposed recently that fingerprints of fractional excitations are encoded in the phonon spectra of Kitaev quantum spin liquids through a novel fractional-excitation-phonon coupling. Here, we uncover this effect in $α$-RuCl3 using inelastic X-ray scattering with meV resolut…
We study the 3D Kitaev and Kitaev-Heisenberg models respectively on the hyperhoneycomb and hyperoctagon lattices, both at zero and finite-temperature, in the thermodynamic limit. Our analysis relies on advanced tensor network (TN) simulations based on graph Projected Entangled-Pair States (gPEPS). We map out the TN phase diagrams of the models and characterize their underlying gapped and gapless phases both at zero and finite temperature. In particular, we demonstrate how cooling down the hyperhoneycomb system from high-temperature leads to fractionalization of spins to itinerant Majorana ferm…
