Kitaev Quantum Spin Research Articles

Concept and realization of Kitaev quantum spin liquids

A decade ago, Alexei Kitaev proposed an exactly solvable $S$ = 1/2 model on a two-dimensional honeycomb lattice, where the spins fractionalize into Majorana fermions and form a topological quantum spin liquid (QSL) in the ground state. It was soon recognized that a family of complex iridium oxides, as well as ruthenium chloride, with honeycomb structure are magnetic insulators and accommodate essential ingredients of the Kitaev model, due to the interplay of electron correlation and spin-orbit coupling. This initiated a race to materialize the Kitaev QSL and to capture the signature of Majorana fermions. In this review, we provide a wide perspective of this rapidly growing field, including theory, materials and experiment. We first summarize the theoretical background of the Kitaev QSL ground state and its materialization using spin-orbital-entangled $J_{\rm eff}$ = 1/2 moments. This is followed by an overview of candidate materials and their magnetic properties, including Na$_2$IrO$_3$, $\alpha$, $\beta$, $\gamma$-Li$_2$IrO$_3$,$\alpha$-RuCl$_3$ and H$_3$LiIr$_2$O$_6$. Finally, we review the latest exciting progress in the search for the Kitaev QSL. In particular, H$_3$LiIr$_2$O$_6$ and $\alpha$-RuCl$_3$ in applied magnetic field show signatures of the QSL state, and $\alpha$-RuCl$_3$ has unusual magnetic excitations and thermal transport properties that are consistent with spin fractionalization.

Field evolution of magnons in α−RuCl3 by high-resolution polarized terahertz spectroscopy

The Kitaev quantum spin liquid (KSL) is a theoretically predicted state of matter whose fractionalized quasiparticles are distinct from bosonic magnons, the fundamental excitation in ordered magnets. The layered honeycomb antiferromagnet $\alpha$-RuCl$_3$ is a KSL candidate material, as it can be driven to a magnetically disordered phase by application of an in-plane magnetic field, with $H_c \sim 7$ T. Here we report a detailed characterization of the magnetic excitation spectrum of this material by high-resolution time-domain terahertz (THz) spectroscopy. We observe two sharp magnon resonances whose frequencies and amplitudes exhibit a discontinuity as a function of applied magnetic field, as well as two broader peaks at higher energy. Below the N\'eel temperature, we find that linear spin wave theory can account for all of these essential features of the spectra when a $C_3$-breaking distortion of the honeycomb lattice and the presence of structural domains are taken into account.

Pressure-driven collapse of the relativistic electronic ground state in a honeycomb iridate

Honeycomb-lattice quantum magnets with strong spin-orbit coupling are promising candidates for realizing a Kitaev quantum spin liquid. Although iridate materials such as Li2IrO3 and Na2IrO3 have been extensively investigated in this context, there is still considerable debate as to whether a localized relativistic wavefunction (Jeff = 1/2) provides a suitable description for the electronic ground state of these materials. To address this question, we have studied the evolution of the structural and electronic properties of α-Li2IrO3 as a function of applied hydrostatic pressure using a combination of x-ray diffraction and x-ray spectroscopy techniques. We observe striking changes even under the application of only small hydrostatic pressure (P ≤ 0.1 GPa): a distortion of the Ir honeycomb lattice (via X-ray diffraction), a dramatic decrease in the strength of spin-orbit coupling effects (via X-ray absorption spectroscopy), and a significant increase in non-cubic crystal electric field splitting (via resonant inelastic X-ray scattering). Our data indicate that α-Li2IrO3 is best described by a Jeff = 1/2 state at ambient pressure, but demonstrate that this state is extremely fragile and collapses under the influence of applied pressure.

Detuning the honeycomb of α−RuCl3 : Pressure-dependent optical studies reveal broken symmetry

The honeycomb Mott insulator $\alpha$-RuCl$_3$ loses its low-temperature magnetic order by pressure. We report clear evidence for a dimerized structure at $P>1$ GPa and observe the breakdown of the relativistic $j_{\rm eff}$ picture in this regime strongly affecting the electronic properties. A pressure-induced Kitaev quantum spin liquid cannot occur in this broken symmetry state. We shed light on the new phase by broad-band infrared spectroscopy of the low-temperature properties of $\alpha$-RuCl$_{3}$ and ab initio density functional theory calculations, both under hydrostatic pressure.

Unusual Thermal Hall Effect in a Kitaev Spin Liquid Candidate α-RuCl_{3}.

The Kitaev quantum spin liquid displays the fractionalization of quantum spins into Majorana fermions. The emergent Majorana edge current is predicted to manifest itself in the form of a finite thermal Hall effect, a feature commonly discussed in topological superconductors. Here we report on thermal Hall conductivity κ_{xy} measurements in α-RuCl_{3}, a candidate Kitaev magnet with the two-dimensional honeycomb lattice. In a spin-liquid (Kitaev paramagnetic) state below the temperature characterized by the Kitaev interaction J_{K}/k_{B}∼80 K, positive κ_{xy} develops gradually upon cooling, demonstrating the presence of highly unusual itinerant excitations. Although the zero-temperature property is masked by the magnetic ordering at T_{N}=7 K, the sign, magnitude, and T dependence of κ_{xy}/T at intermediate temperatures follows the predicted trend of the itinerant Majorana excitations.

Dynamics of a quantum spin liquid beyond integrability: The Kitaev-Heisenberg- Γ model in an augmented parton mean-field theory

We present an augmented parton mean-field theory which (i) reproduces the $exact$ ground state, spectrum, and dynamics of the quantum spin liquid phase of Kitaev's honeycomb model; and (ii) is amenable to the inclusion of integrability breaking terms, allowing a perturbation theory from a controlled starting point. Thus, we exemplarily study dynamical spin correlations of the honeycomb Kitaev quantum spin liquid within the $K-J-\Gamma$ model which includes Heisenberg and symmetric-anisotropic (pseudo-dipolar) interactions. This allows us to trace changes of the correlations in the regime of slowly moving fluxes, where the theory captures the dominant deviations when integrability is lost. These include an asymmetric shift together with a broadening of the dominant peak in the response as a function of frequency; the generation of further-neighbour correlations and their structure in real- and spin-space; and a resulting loss of an approximate rotational symmetry of the structure factor in reciprocal space. We discuss the limitations of this approach, and also view the neutron scattering experiments on the putative proximate quantum spin liquid material, $\alpha$-RuCl$_3$, in the light of the results from this extended parton theory.

Local Moment Instability of Os in Honeycomb Li2.15Os0.85O3

Compounds with honeycomb structures occupied by strong spin orbit coupled (SOC) moments are considered to be candidate Kitaev quantum spin liquids. Here we present the first example of Os on a honeycomb structure, Li2.15(3)Os0.85(3)O3 (C2/c, a = 5.09 Å, b = 8.81 Å, c = 9.83 Å, β = 99.3°). Neutron diffraction shows large site disorder in the honeycomb layer and X-ray absorption spectroscopy indicates a valence state of Os (4.7 ± 0.2), consistent with the nominal concentration. We observe a transport band gap of Δ = 243 ± 23 meV, a large van Vleck susceptibility, and an effective moment of 0.85 μB, much lower than expected from 70% Os(+5). No evidence of long range order is found above 0.10 K but a spin glass-like peak in ac-susceptibility is observed at 0.5 K. The specific heat displays an impurity spin contribution in addition to a power law ∝T(0.63±0.06). Applied density functional theory (DFT) leads to a reduced moment, suggesting incipient itineracy of the valence electrons, and finding evidence that Li over stoichiometry leads to Os(4+)−Os(5+) mixed valence. This local picture is discussed in light of the site disorder and a possible underlying quantum spin liquid state.

Interlayer coupling effect on a bilayer Kitaev model

We investigate a bilayer Kitaev model, where two honeycomb layers are coupled by the Heisenberg interactions, to discuss effects of an interlayer coupling against the Kitaev quantum spin liquids (QSLs). In this model, there exists a local conserved quantity, which results in no long-range spin correlations in the system. Using the exact diagonalization, bond operator mean-field theory, and cluster expansion techniques, we study ground state properties in the system. The obtained results suggest the existence of a first-order quantum phase transition between the Kitaev QSL and singlet-dimer states. We find that one-triplet excitation from the singlet-dimer ground state is localized owing to the existence of the local conserved quantity. To examine finite-temperature properties, we make use of the thermal pure quantum state approach. We clarify that double-peak structure in the specific heat inherent in the Kitaev QSL is maintained even above the quantum phase transition. The present results suggest that the Kitaev QSL is stable against the interlayer interference. Magnetic properties of multilayer Kitaev models are also addressed.

Excitations in the field-induced quantum spin liquid state of \u03b1-RuCl3

The celebrated Kitaev quantum spin liquid (QSL) is the paradigmatic example of a topological magnet with emergent excitations in the form of Majorana Fermions and gauge fluxes. Upon breaking of time-reversal symmetry, for example in an external magnetic field, these fractionalized quasiparticles acquire non-Abelian exchange statistics, an important ingredient for topologically protected quantum computing. Consequently, there has been enormous interest in exploring possible material realizations of Kitaev physics and several candidate materials have been put forward, recently including α-RuCl3. In the absence of a magnetic field this material orders at a finite temperature and exhibits low-energy spin wave excitations. However, at moderate energies, the spectrum is unconventional and the response shows evidence for fractional excitations. Here we use time-of-flight inelastic neutron scattering to show that the application of a sufficiently large magnetic field in the honeycomb plane suppresses the magnetic order and the spin waves, leaving a gapped continuum spectrum of magnetic excitations. Our comparisons of the scattering to the available calculations for a Kitaev QSL show that they are consistent with the magnetic field induced QSL phase.

Destabilization of Magnetic Order in a Dilute Kitaev Spin Liquid Candidate.

The insulating honeycomb magnet α-RuCl_{3} exhibits fractionalized excitations that signal its proximity to a Kitaev quantum spin liquid state; however, at T=0, fragile long-range magnetic order arises from non-Kitaev terms in the Hamiltonian. Spin vacancies in the form of Ir^{3+} substituted for Ru are found to destabilize this long-range order. Neutron diffraction and bulk characterization of Ru_{1-x}Ir_{x}Cl_{3} show that the magnetic ordering temperature is suppressed with increasing x, and evidence of zizag magnetic order is absent for x>0.3. Inelastic neutron scattering demonstrates that the signature of fractionalized excitations is maintained over the full range of x investigated. The depleted lattice without magnetic order thus hosts a spin-liquid-like ground state that may indicate the relevance of Kitaev physics in the magnetically dilute limit of RuCl_{3}.

Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of α−RuCl3

α-RuCl_{3} is a leading candidate material for the observation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that α-RuCl_{3} undergoes a quantum phase transition to a QSL in a magnetic field of 7.5T applied in the ab plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of α-RuCl_{3}.

Majorana spin liquids, topology, and superconductivity in ladders

We theoretically address spin chain analogs of the Kitaev quantum spin model on the honeycomb lattice. The emergent quantum spin liquid phases or Anderson resonating valence bond (RVB) states can be understood, as an effective model, in terms of p-wave superconductivity and Majorana fermions. We derive a generalized phase diagram for the two-leg ladder system with tunable interaction strengths between chains allowing us to vary the shape of the lattice (from square to honeycomb ribbon or brickwall ladder). We evaluate the winding number associated with possible emergent (topological) gapless modes at the edges. In the Az phase, as a result of the emergent Z2 gauge fields and pi-flux ground state, one may build spin-1/2 (loop) qubit operators by analogy to the toric code. In addition, we show how the intermediate gapless B phase evolves in the generalized ladder model. For the brickwall ladder, the $B$ phase is reduced to one line, which is analyzed through perturbation theory in a rung tensor product states representation and bosonization. Finally, we show that doping with a few holes can result in the formation of hole pairs and leads to a mapping with the Su-Schrieffer-Heeger model in polyacetylene; a superconducting-insulating quantum phase transition for these hole pairs is accessible, as well as related topological properties.

Correlated states in \u03b2-Li2IrO3 driven by applied magnetic fields

Magnetic honeycomb iridates are thought to show strongly spin-anisotropic exchange interactions which, when highly frustrated, lead to an exotic state of matter known as the Kitaev quantum spin liquid. However, in all known examples these materials magnetically order at finite temperatures, the scale of which may imply weak frustration. Here we show that the application of a relatively small magnetic field drives the three-dimensional magnet β-Li2IrO3 from its incommensurate ground state into a quantum correlated paramagnet. Interestingly, this paramagnetic state admixes a zig-zag spin mode analogous to the zig-zag order seen in other Mott-Kitaev compounds. The rapid onset of the field-induced correlated state implies the exchange interactions are delicately balanced, leading to strong frustration and a near degeneracy of different ground states.

Pressure tuning of bond-directional exchange interactions and magnetic frustration in the hyperhoneycomb iridate β−Li2IrO3

We explore the response of Ir $5d$ orbitals to pressure in $\beta$-$\mathrm{Li_2IrO_3}$, a hyperhoneycomb iridate in proximity to a Kitaev quantum spin liquid (QSL) ground state. X-ray absorption spectroscopy reveals a reconstruction of the electronic ground state below 2 GPa, the same pressure range where x-ray magnetic circular dichroism shows an apparent collapse of magnetic order. The electronic reconstruction, which manifests a reduction in the effective spin-orbit (SO) interaction in $5d$ orbitals, pushes $\beta$-$\mathrm{Li_2IrO_3}$ further away from the pure $J_{\rm eff}=1/2$ limit. Although lattice symmetry is preserved across the electronic transition, x-ray diffraction shows a highly anisotropic compression of the hyperhoneycomb lattice which affects the balance of bond-directional Ir-Ir exchange interactions driven by spin-orbit coupling at Ir sites. An enhancement of symmetric anisotropic exchange over Kitaev and Heisenberg exchange interactions seen in theoretical calculations that use precisely this anisotropic Ir-Ir bond compression provides one possible route to realization of a QSL state in this hyperhoneycomb iridate at high pressures.

Thermal Transport in the Kitaev Model.

In conventional insulating magnets, heat is carried by magnons and phonons. In contrast, when the magnets harbor a quantum spin liquid state, emergent quasiparticles from the fractionalization of quantum spins can carry heat. Here, we investigate unconventional thermal transport yielded by such exotic carriers, in both longitudinal and transverse components, for the Kitaev model, whose ground state is exactly shown to be a quantum spin liquid with fractional excitations described as itinerant Majorana fermions and localized Z_{2} fluxes. We find that the longitudinal thermal conductivity exhibits a single peak at a high temperature, while the nonzero frequency component has a peak at a low temperature, reflecting the spin fractionalization. On the other hand, we show that the transverse thermal conductivity is induced by the magnetic field in a wide temperature range up to the energy scale of the bare exchange coupling; while increasing temperature, the transverse response divided by temperature decreases from the quantized value expected for the topologically nontrivial ground state and shows nonmonotonic temperature dependence. These characteristic behaviors provide experimentally accessible evidence of fractional excitations in the proximity to the Kitaev quantum spin liquid.

Majorana fermions in the Kitaev quantum spin system α-RuCl3

α-RuCl3 has recently attracted great interest as a possible experimental realization of the Kitaev model. Neutron scattering measurements of a single crystal of this material reveal signatures of Majorana excitations, consistent with Kitaev’s predictions. Geometrical constraints to the electronic degrees of freedom within condensed-matter systems often give rise to topological quantum states of matter such as fractional quantum Hall states, topological insulators, and Weyl semimetals1,2,3. In magnetism, theoretical studies predict an entangled magnetic quantum state with topological ordering and fractionalized spin excitations, the quantum spin liquid4. In particular, the so-called Kitaev spin model5, consisting of a network of spins on a honeycomb lattice, is predicted to host Majorana fermions as its excitations. By means of a combination of specific heat measurements and inelastic neutron scattering experiments, we demonstrate the emergence of Majorana fermions in single crystals of α-RuCl3, an experimental realization of the Kitaev spin lattice. The specific heat data unveils a two-stage release of magnetic entropy that is characteristic of localized and itinerant Majorana fermions. The neutron scattering results corroborate this picture by revealing quasielastic excitations at low energies around the Brillouin zone centre and an hour-glass-like magnetic continuum at high energies. Our results confirm the presence of Majorana fermions in the Kitaev quantum spin liquid and provide an opportunity to build a unified conceptual framework for investigating fractionalized excitations in condensed matter1,6,7,8.

Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Recent experiments reveal that the honeycomb ruthenium trichloride {\alpha}-RuCl3 is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response, due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., {\alpha}-RuCl3) with non-negligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal J1-K1-{\Gamma}1-J3-K3 model for {\alpha}-RuCl3 and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3 stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange {\Gamma}1 plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. Exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange {\Gamma}1 and the third NN Heisenberg interaction J3. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride {\alpha}-RuCl3.

Majorana dynamical mean-field study of spin dynamics at finite temperatures in the honeycomb Kitaev model

A prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom. Fractionalized excitations have their own dynamics in different energy scales, and hence, affect finite-temperature ($T$) properties in a peculiar manner even in the paramagnetic state harboring the quantum spin liquid state. We here present a comprehensive theoretical study of the spin dynamics in a wide $T$ range for the Kitaev model on a honeycomb lattice, whose ground state is such a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter fermions and localized gauge fluxes, which results in two crossovers at very different $T$ scales. Extending the previous study for the isotropic coupling case [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. ${\textbf 117}$, 157203 (2016)], we calculate the dynamical spin structure factor, the NMR relaxation rate, and the magnetic susceptibility while changing the anisotropy in the exchange coupling constants, by using the dynamical mean-field theory and the continuous-time quantum Monte Carlo method based on a Majorana fermion representation. We find that the system exhibits peculiar behaviors below the high-$T$ crossover whose temperature is comparable to the average of the exchange constants, reflecting the spin fractionalization in the paramagnetic region. Among them, the dichotomy between the static and dynamical spin correlations is unusual behavior hardly seen in conventional magnets. We discuss the relation between the dichotomy and the spatial configuration of gauge fluxes. Our results could stimulate further experimental and theoretical analyses of candidate materials for the Kitaev quantum spin liquids.

Evidence for a Field-Induced Quantum Spin Liquid in α-RuCl_{3}.

We report a ^{35}Cl nuclear magnetic resonance study in the honeycomb lattice α-RuCl_{3}, a material that has been suggested to potentially realize a Kitaev quantum spin liquid (QSL) ground state. Our results provide direct evidence that α-RuCl_{3} exhibits a magnetic-field-induced QSL. For fields larger than ∼10 T, a spin gap opens up while resonance lines remain sharp, evidencing that spins are quantum disordered and locally fluctuating. The spin gap increases linearly with an increasing magnetic field, reaching ∼50 K at 15T, and is nearly isotropic with respect to the field direction. The unusual rapid increase of the spin gap with increasing field and its isotropic nature are incompatible with conventional magnetic ordering and, in particular, exclude that the ground state is a fully polarized ferromagnet. The presence of such a field-induced gapped QSL phase has indeed been predicted in the Kitaev model.

Majorana Landau-level Raman spectroscopy

The unambiguous experimental detection of quantum spin liquids and, in particular, of the long-sought Kitaev quantum spin liquid (KQSL) with its Majorana fermion excitations remains an outstanding challenge. One of the major obstacles is the absence of signatures that definitively characterize this phase. Here we propose the Landau levels known to form in the Majorana excitation spectrum of the KQSL when certain strain fields are applied as a direct signature of Majorana fermions with Dirac-like dispersion. In particular, we show that the Majorana Landau level quantization of strained films of the KQSL can be directly probed by Raman spectroscopy. Such experiments are feasible in thin films of $\alpha$-RuCl$_3$, which are a promising place to search for the KQSL.