Kitaev Spin Liquid Phase Research Articles

Anisotropic Thermal Conductivity Oscillations in Relation to the Putative Kitaev Spin Liquid Phase of α-RuCl_{3}.

In the presence of an external magnetic field, the Kitaev model could host either gapped topological anyons or gapless Majorana fermions. In α-RuCl_{3}, the gapped and gapless cases are only separated by a 30° rotation of the in-plane magnetic field vector. The presence or absence of the spectral gap is key for understanding the thermal transport behavior in α-RuCl_{3}. Here, we study the anisotropy of the oscillatory features of thermal conductivity in α-RuCl_{3}. We examine the oscillatory features of thermal conductivities (κ//a, κ//b) with fixed external fields and find distinct behavior for the gapped (B//a) and gapless (B//b) scenarios. Furthermore, we track the evolution of thermal resistivity (λ_{a}) and its oscillatory features with the rotation of in-plane magnetic fields from B//b to B//a. The thermal resistivity λ(B,ϕ) displays distinct rotational symmetries before and after the emergence of the field-induced Kitaev spin liquid phase. These results suggest that oscillatory features of thermal conductivity in α-RuCl_{3} are closely linked to the putative Kitaev spin liquid phase and its excitations.

Electric polarization near vortices in the extended Kitaev model

We formulate a Majorana mean-field theory for the extended JKΓ Kitaev model in a magnetic Zeeman field of arbitrary direction, and apply it for studying spatially inhomogeneous states harboring vortices. This mean-field theory is exact in the pure Kitaev limit and captures the essential physics throughout the Kitaev spin liquid phase. We determine the charge profile around vortices and the corresponding quadrupole tensor. The quadrupole-quadrupole interaction between distant vortices is shown to be either repulsive or attractive, depending on parameters. We predict that electrically biased scanning probe tips enable the creation of vortices at preselected positions. Our results paves the way for the electric manipulation of Ising anyons in Kitaev spin liquids.

Active orbital degree of freedom and potential spin-orbit-entangled moments in the Kitaev magnet candidate BaCo2(AsO4)2

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 high-accuracy crystal and electronic structure of BaCo$_2$(AsO$_4$)$_2$ by employing a combined density functional and dynamical mean-field theory calculations, which correctly capture the Mott-insulating nature of the target system. We show that Co$^{2+}$ ions form a high spin configuration, $S=3/2$, with an active $L_{\rm eff}=1$ orbital degree of freedom, in the absence of spin-orbit coupling. The size of trigonal distortion within CoO$_6$ octahedra is found to be not strong enough to completely quench the orbital degree of freedom, so that the presence of spin-orbit coupling can give rise to the formation of spin-orbit-entangled moments and the Kitaev exchange interaction. Our finding supports recent studies on potential Kitaev magnetism in this compound and other Co-based layered honeycomb systems.

Sound attenuation in the hyperhoneycomb Kitaev spin liquid

Here, the authors propose that phonon dynamics can be exploited to look for the signatures of fractionalization in three-dimensional Kitaev spin liquids. Motivated by recent studies of the hyperhoneycomb Kitaev material $\ensuremath{\beta}$-Li${}_{2}$IrO${}_{3}$, they compute the sound attenuation in the spin-phonon coupled Kitaev model on the hyperhoneycomb lattice, in which Majorana fermions have nodal-line band structure. They show that at low temperatures its angular dependence displays a characteristic pattern, which opens the possibility of detecting the proximity to the Kitaev spin liquid phase in this compound in ultrasound experiments.

Phase diagrams of Kitaev models for arbitrary magnetic field orientations

The Kitaev model is an exactly solvable quantum spin model within the language of constrained real fermions. In spite of numerous studies for magnetic fields along special orientations, there is a limited amount of knowledge on the complete field-angle characterization, which can provide valuable information on the existence of fractionalized excitations. For this purpose, we first study the pure ferromagnetic and antiferromagnetic Kitaev models for arbitrary external magnetic field directions via a mean-field theory, showing that there are many topological phases with different (or zero) Chern numbers, depending on the magnetic field strength and orientations. However, a realistic description of the candidate Kitaev materials, within the edge-sharing octahedra paradigm, requires additional coupling terms, including a large off-diagonal term $\mathrm{\ensuremath{\Gamma}}$ along with possible anisotropic corrections ${\mathrm{\ensuremath{\Gamma}}}_{p}$. It is therefore not sufficient to rely on the topological properties of the bare Kitaev model as the basis for the observed thermal Hall-conductivity signals, and an understanding of these extended Kitaev models with a complete field response is demanded. Starting from the zero-field phase diagram of $\mathrm{K}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}_{p}$ models, we identify, besides the Kitaev spin liquid phase, antiferromagnetic zigzag, ferromagnetic phases, as well as an unusual Kitaev(-$\mathrm{\ensuremath{\Gamma}}$) spin liquid phase. The magnetic field response of these phases for arbitrary field orientations provides a remarkably rich phase diagram. For an extended parameter range and just above the critical field where the zigzag phase is suppressed, there is an intermediate phase region with suppressed energy gaps and substantial spin fractionalization. To comply our findings with experiments, we also reproduce a large asymmetry in the extent of this intermediate phases specifically for the two different field directions $\ensuremath{\theta}=\ifmmode\pm\else\textpm\fi{}{60}^{o}$ with respect to the normal to the plane of the honeycomb lattice.

Magnetic excitations and interactions in the Kitaev hyperhoneycomb iridate β−Li2IrO3

We present a thorough experimental study of the three-dimensional hyperhoneycomb Kitaev magnet $\beta$-Li$_2$IrO$_3$, using a combination of inelastic neutron scattering (INS), time-domain THz spectroscopy, and heat capacity measurements. The main results include a massive low-temperature reorganization of the INS spectral weight that evolves into a broad peak centered around 12 meV, and a distinctive peak in the THz data at 2.8(1) meV. A detailed comparison to powder-averaged spin-wave theory calculations reveals that the positions of these two features are controlled by the anisotropic $\Gamma$ coupling and the Heisenberg exchange $J$, respectively. The refined microscopic spin model places $\beta$-Li$_2$IrO$_3$ in close proximity to the Kitaev spin liquid phase.

Exchange interactions in d5 Kitaev materials: From Na2IrO3 to α−RuCl3

We present an analytical study of the exchange interactions between pseudospin one-half ${d}^{5}$ ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard $U$, charge-transfer excitations, and cyclic exchange are considered. Hoppings within ${t}_{2g}$ orbitals as well as between ${t}_{2g}$ and ${e}_{g}$ orbitals are included. Special attention is paid to the trigonal crystal field $\mathrm{\ensuremath{\Delta}}$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction $K$ within a wide range of $\mathrm{\ensuremath{\Delta}}$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\mathrm{\ensuremath{\Delta}}$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic $K$ and positive nondiagonal $\mathrm{\ensuremath{\Gamma}}$ values. However, their Heisenberg $J$ terms have opposite signs: AFM $J>0$ in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and FM $J<0$ in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM $J$ is emphasized in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. The differences between the exchange couplings of these two materials originate from the opposite $\mathrm{\ensuremath{\Delta}}$ values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of ${d}^{5}$ spin-orbit Mott insulators.

Instabilities of spin-1 Kitaev spin liquid phase in presence of single-ion anisotropies

We study the spin-one Kitaev model on the honeycomb lattice in the presence of single-ion anisotropies. We consider two types of single ion anisotropies: A ${D}_{111}$ anisotropy which preserves the symmetry between $X, Y$, and $Z$ bonds but violates flux conservation and a ${D}_{100}$ anisotropy that breaks the symmetry between $X, Y$, and $Z$ bonds but preserves flux conservation. We use series expansion methods, degenerate perturbation theory, and exact diagonalization to study these systems. Large positive ${D}_{111}$ anisotropy leads to a simple product ground state with conventional magnonlike excitations, while large negative ${D}_{111}$ leads to a broken symmetry and degenerate ground states. For both signs there is a phase transition at a small $|{D}_{111}|\ensuremath{\approx}0.12$ separating the more conventional phases from the Kitaev spin liquid phase. With large ${D}_{100}$ anisotropy, the ground state is a simple product state, but the model lacks conventional dispersive excitations due to the large number of conservation laws. Large negative ${D}_{100}$ leads to decoupled one-dimensional systems and many degenerate ground states. No evidence of a phase transition is seen in our numerical studies at any finite ${D}_{100}$. Convergence of the series expansion extrapolations all the way to ${D}_{100}=0$ suggests that the nontrivial Kitaev spin liquid is a singular limit of this type of single-ion anisotropy going to zero, which also restores symmetry between the $X, Y$, and $Z$ bonds.

Variational study of the Kitaev-Heisenberg-Gamma model

We compute the low-energy excitation spectrum and the dynamical spin structure factor of the Kitaev-Heisenberg-Gamma model through a variational approach based on the exact fractionalized excitations of the pure Kitaev honeycomb model. This novel approach reveals the physical reason for the asymmetric stability of the Kitaev spin liquid phases around the ferromagnetic and antiferromagnetic Kitaev limits. Moreover, we demonstrate that the fractionalized excitations form bound states in specific regions of each Kitaev spin liquid phase and that certain phase transitions induced by Heisenberg and Gamma interactions are driven by the condensation of such a bound state. Remarkably, this bound state appears as a sharp mode in the dynamical spin structure factor, while its condensation patterns at the appropriate phase transitions provide a simple explanation for the magnetically ordered phases surrounding each Kitaev spin liquid phase.

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.

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.

Kitaev Spin Liquid in 3d Transition Metal Compounds.

We study the exchange interactions and resulting magnetic phases in the honeycomb cobaltates. For a broad range of trigonal crystal fields acting on Co^{2+} 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 Na_{3}Co_{2}SbO_{6} 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 Δ by ∼20 meV, e.g., via strain or pressure control. We argue that, due to the more localized nature of the magnetic electrons in 3d compounds, cobaltates offer the most promising search area for Kitaev model physics.

Pressure-induced dimerization and valence bond crystal formation in the Kitaev-Heisenberg magnet α−RuCl3

Magnetization and high-resolution x-ray diffraction measurements of the Kitaev-Heisenberg material alpha-RuCl3 reveal a pressure-induced crystallographic and magnetic phase transition at a hydrostatic pressure of p=0.2 GPa. This structural transition into a triclinic phase is characterized by a very strong dimerization of the Ru-Ru bonds, accompanied by a collapse of the magnetic susceptibility. Ab initio quantum-chemistry calculations disclose a pressure-induced enhancement of the direct 4d-4d bonding on particular Ru-Ru links, causing a sharp increase of the antiferromagnetic exchange interactions. These combined experimental and computational data show that the Kitaev spin liquid phase in alpha-RuCl3 strongly competes with the crystallization of spin singlets into a valence bond solid.

Erratum: Finite-temperature phase transition to a Kitaev spin liquid phase on a hyperoctagon lattice: A large-scale quantum Monte Carlo study [Phys. Rev. B 96 , 125124 (2017)

Received 30 January 2018DOI:https://doi.org/10.1103/PhysRevB.97.079905©2018 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum spin liquidPhysical SystemsQuantum spin modelsTechniquesKitaev modelQuantum Monte CarloCondensed Matter, Materials & Applied Physics

Thermal properties of spin-S Kitaev-Heisenberg model on a honeycomb lattice

Temperature (T) dependence of heat capacity C(T) in the S=1/2 Kitaev honeycomb model shows a double-peak structure resulting from fractionalization of spins into two kinds of Majorana fermions. Recently it has been discussed that the double-peak structure in C(T) is also observed in magnetic ordered phases of the S=1/2 Kitaev-Heisenberg (KH) model on a honeycomb lattice when the system is located in the vicinity of the Kitaev's spin liquid phase. In addition to the S=1/2 spin case, similar double-peak structure has been confirmed in the KH honeycomb model for classical Heisenberg spins, where spin S is regarded as S→∞. We investigate spin-S dependence of C(T) for the KH honeycomb models by using thermal pure quantum state. We also perform classical Monte Carlo calculations to obtain C(T) for the classical KH model. From obtained results, we find that the origin of the high-temperature peak is different between the quantum spin case with small Ss and the classical Heisenberg spin case. Furthermore, the high-temperature peak in the quantum spin case, which is one of the clues for fractionalization of spins, disappears for S>1.

Finite-temperature phase transition to a Kitaev spin liquid phase on a hyperoctagon lattice: A large-scale quantum Monte Carlo study

The quantum spin liquid is an enigmatic quantum state in insulating magnets, in which conventional long-range order is suppressed by strong quantum fluctuations. Recently, an unconventional phase transition was reported between the low-temperature quantum spin liquid and the high-temperature paramagnet in the Kitaev model on a three-dimensional hyperhoneycomb lattice. Here, we show that a similar ``liquid-gas'' transition takes place in another three-dimensional lattice, the hyperoctagon lattice. We investigate the critical phenomena by adopting the Green-function based Monte Carlo technique with the kernel polynomial method, which enables systematic analysis of up to 2048 sites. The critical temperature is lower than that in the hyperhoneycomb case, reflecting the smaller flux gap. We also discuss the transition on the basis of an effective model in the anisotropic limit.

Crystal structure and magnetism inα−RuCl3: Anab initiostudy

$\alpha$-RuCl$_3$ has been proposed recently as an excellent playground for exploring Kitaev physics on a two-dimensional (2D) honeycomb lattice. However, structural clarification of the compound has not been completed, which is crucial in understanding the physics of this system. Here, using {\it ab-initio} electronic structure calculations, we study a full three dimensional (3D) structure of $\alpha$-RuCl$_3$ including the effects of spin-orbit coupling (SOC) and electronic correlations. Three major results are as follows; i) SOC suppresses dimerization of Ru atoms, which exists in other Ru compounds such as isostructural Li$_2$RuO$_3$, and making the honeycomb closer to an ideal one. ii) The nearest-neighbor Kitaev exchange interaction between the $j_{\rm eff}$=1/2 pseudospin depends strongly on the Ru-Ru distance and the Cl position, originating from the nature of the edge-sharing geometry. iii) The optimized 3D structure without electronic correlations has $P{\bar 3}1m$ space group symmetry independent of SOC, but including electronic correlation changes the optimized 3D structure to either $C2/m$ or $Cmc2_1$ within 0.1 meV per formula unit (f.u.) energy difference. The reported $P3_112$ structure is also close in energy. The interlayer spin exchange coupling is a few percent of in-plane spin exchange terms, confirming $\alpha$-RuCl$_3$ is close to a 2D system. We further suggest how to increase the Kitaev term via tensile strain, which sheds new light in realizing Kitaev spin liquid phase in this system.

Dynamical properties of the honeycomb-lattice iridatesNa2IrO3

We investigate the dynamical properties of ${\rm Na_2IrO_3}$. For five effective models proposed for ${\rm Na_2IrO_3}$, we numerically calculate dynamical structure factors (DSFs) with an exact diagonalization method. An effective model obtained from $ab$ $initio$ calculations explains inelastic neutron scattering experiments adequately. We further calculate excitation modes based on linearized spin-wave theory. The spin-wave excitation of the effective models obtained by $ab$ $initio$ calculations disagrees with the low-lying excitation of DSFs. We attribute this discrepancy to the location of ${\rm Na_2IrO_3}$ in a parameter space close to the phase boundary with the Kitaev spin-liquid phase.

Spiral magnetic phase in Li-dopedNa2IrO3

The doping of the honeycomb lattice compound Na2IrO3 with cations having smaller ionic radius such as lithium was supposed to drive the system into a Kitaev spin liquid phase. Even at the smallest doping level the long range magnetic order, present in the parent compound, got suppressed as evidenced by susceptibility and heat capacity measurements. At low doping levels, the long range ordered magnetic phase is accompanied by the formation of a glassy state with a transition temperature of T-F similar to 5 K as determined by dc and ac magnetic susceptibility. The magnetic excitation spectrum, measured by inelastic neutron scattering, shows a significant change with doping. Our inelastic neutron scattering data can be well described by linear spin wave theory, revealing the formation of a spiral magnetic phase.

Density-matrix renormalization group study of the extended Kitaev-Heisenberg model

We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using two-dimensional density-matrix renormalization group method. Calculating the gound-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a 120-degree antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model and find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs but those in the magnetically ordered phases are non-degenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.