Low-Energy Spin Dynamics of the Honeycomb Spin Liquid Beyond the Kitaev Limit

Deciphering competing interactions of Kitaev-Heisenberg-Γ system in clusters: I. Static properties

Recently, the Kitaev-Heisenberg-Γ system has been used to explore various aspects of Kitaev spin liquid physics. Here, we consider a few small clusters of up to twelve sites and study them in detail to unravel many interesting findings due to the competition between all possible signs and various magnitudes of these interactions under the influence of an external magnetic field. When Heisenberg interaction is taken anti-ferromagnetic, one obtains plateaus in correlation functions where, surprisingly, the exact groundstate reduces to the eigenstate of Heisenberg interaction as well. On the other hand, for ferromagnetic Heisenberg interaction, its competition with Kitaev interaction results in non-monotonicity in the correlation functions. We discuss, in detail, the competing effects on low energy spectrum, flux operator, magnetization, susceptibility, and specific heat. Finally, we discuss how our findings could be helpful to explain some of the recent experimental and theoretical findings in materials with Kitaev interactions.

Nonlocal Spin Correlation as a Signature of Ising Anyons Trapped in Vacancies of the Kitaev Spin Liquid

In the Kitaev chiral spin liquid, Ising anyons are realized as Z_{2} fluxes binding Majorana zero modes, which, however, are thermal excitations with finite decay rates. On the other hand, a lattice vacancy traps a Z_{2} flux even in the ground state, resulting in the stable realization of a Majorana zero mode in a vacancy. We demonstrate that spin-spin correlation functions between two vacancy sites exhibit long-range correlation arising from the fractionalized character of Majorana zero modes, in spite of the strong decay of bulk spin correlations. Remarkably, this nonlocal spin correlation does not decrease as the distance between two vacancy sites increases, signaling Majorana teleportation. Furthermore, we clarify that the nonlocal correlation can be detected electrically via the measurement of nonlocal conductance between two vacancy sites, which is straightforwardly utilized for the readout of Majorana qubits. These findings pave the way to the measurement-based quantum computation with Ising anyons trapped in vacancies of the Kitaev spin liquid.

Disorder, Low-Energy Excitations, and Topology in the Kitaev Spin Liquid

The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. However, deviations from the original Kitaev model are expected to appear in real materials. In this Letter, we investigate the fate of Kitaev's spin liquid in the presence of disorder-bond defects or vacancies-for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states with a nonuniversal exponent. We link this power-law distribution of energy scales to weakly coupled droplets inside the bulk, in an uncanny similarity to the Griffiths phase often present in the vicinity of disordered quantum phase transitions. If time-reversal symmetry is broken, we find that power-law singularities are tied to the destruction of the topological phase of the Kitaev model in the presence of bond disorder alone. However, there is a transition from this topologically trivial phase with power-law singularities to a topologically nontrivial one for weak to moderate site dilution. Therefore, diluted Kitaev materials are potential candidates to host Kitaev's chiral spin-liquid phase.

Identification of a Kitaev quantum spin liquid by magnetic field angle dependence

Quantum spin liquids realize massive entanglement and fractional quasiparticles from localized spins, proposed as an avenue for quantum science and technology. In particular, topological quantum computations are suggested in the non-abelian phase of Kitaev quantum spin liquid with Majorana fermions, and detection of Majorana fermions is one of the most outstanding problems in modern condensed matter physics. Here, we propose a concrete way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence. Topologically protected critical lines exist on a plane of magnetic field angles, and their shapes are determined by microscopic spin interactions. A chirality operator plays a key role in demonstrating microscopic dependences of the critical lines. We also show that the chirality operator can be used to evaluate topological properties of the non-abelian Kitaev quantum spin liquid without relying on Majorana fermion descriptions. Experimental criteria for the non-abelian spin liquid state are provided for future experiments.

Non-Abelian phase of Kitaev quantum spin liquid is promising for topological quantum computation. Here, the authors propose a way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence, providing criteria for such a state for future experiments.

On the proximate Kitaev quantum-spin liquid α-RuCl3: thermodynamics, excitations and continua

This topical review provides an overview over recent thermodynamic, infrared, and THz results on the proximate Kitaev spin-liquid. Quantum-spin liquids are exotic phases characterized by the absence of magnetic ordering even at the lowest temperatures and by the occurrence of fractionalized spin excitations. Among those, Kitaev spin liquids are most fascinating as they belong to the rare class of model systems, that can be solved analytically by decomposing localized spinsS= 1/2 into Majorana fermions. The main aim of this review is to summarize experimental evidence obtained by THz spectroscopy and utilizing heat-capacity experiments, which point to the existence of fractionalized excitations in the spin-liquid state, which in α-RuCl₃exists at temperatures just above the onset of magnetic order or at in-plane magnetic fields just beyond the quantum-critical point where antiferromagnetic order becomes suppressed. Thermodynamic and spectroscopic results are compared to theoretical predictions and model calculations. In addition, we document recent progress in elucidating the sub-gap (<1 eV) electronic structure of the 4d⁵ruthenium electrons to characterize their local electronic configuration. The on-site excitation spectra of thedelectrons below the optical gap can be consistently explained using a spin-orbit coupling constant of ∼170 meV and the concept of multiple spin-orbital excitations. Furthermore, we discuss the phonon spectra of the title compound including rigid-plane shear and compression modes of the single molecular layers. In recent theoretical concepts it has been shown that phonons can couple to Majorana fermions and may play a substantial role in establishing the half-integer thermal quantum Hall effect observed in this material.

Tunneling Spectroscopy of Quantum Spin Liquids

We examine the spectroscopic signatures of tunneling through a Kitaev quantum spin liquid (QSL) barrier in a number of experimentally relevant geometries. We combine contributions from elastic and inelastic tunneling processes and find that spin-flip scattering at the itinerant spinon modes gives rise to a gapped contribution to the tunneling conductance spectrum. We address the spectral modifications that arise in a magnetic field, which is applied to drive the candidate material α-RuCl_{3} into a QSL phase, and we propose a lateral 1D tunnel junction as a viable setup in this regime. The characteristic spin gap is an unambiguous signature of the fractionalized QSL excitations, distinguishing it from magnons or phonons. We discuss the generalization of our results to a wide variety of QSLs with gapped and gapless spin correlators.

Electrical Access to Ising Anyons in Kitaev Spin Liquids

We show that spin-spin correlations in a non-Abelian Kitaev spin liquid are associated with a characteristic inhomogeneous charge density distribution in the vicinity of Z_{2} vortices. This density profile and the corresponding local electric fields are observable, e.g., by means of surface probe techniques. Conversely, by applying bias voltages to several probe tips, one can stabilize Ising anyons (Z_{2} vortices harboring a Majorana zero mode) at designated positions, where we predict a clear Majorana signature in energy absorption spectroscopy.

Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons

We provide an exact study of dynamical correlations for the quantum spin-orbital liquid phases of an SU(2)-symmetric Kitaev honeycomb lattice model. We show that the spin dynamics in this Kugel-Khomskii type model is exactly the density-density correlation function of S=1 fermionic magnons, which could be probed in resonant inelastic x-ray scattering experiments. We predict the characteristic signatures of spin-orbital fractionalization in inelastic scattering experiments and compare them to the ones of the spin-anisotropic Kitaev honeycomb spin liquid. In particular, the resonant inelastic x-ray scattering response shows a characteristic momentum dependence directly related to the dispersion of fermionic excitations. The neutron scattering cross section displays a mixed response of fermionic magnons as well as spin-orbital excitations. The latter has a bandwidth of broad excitations and a vison gap that is three times larger than that of the spin-1=2 Kitaev model.

Majorana-Mediated Spin Transport in Kitaev Quantum Spin Liquids

We study the spin transport through the quantum spin liquid (QSL) by investigating the real-time and real-space dynamics of the Kitaev spin system with zigzag edges using the time-dependent Majorana mean-field theory. After the magnetic-field pulse is introduced to one of the edges, spin moments are excited in the opposite edge region although spin moments are never induced in the Kitaev QSL region. This unusual spin transport originates from the fact that the S=1/2 spins are fractionalized into the itinerant and localized Majorana fermions in the Kitaev system. Although both Majorana fermions are excited by the magnetic pulse, only the itinerant ones flow through the bulk regime without spin excitations, resulting in the spin transport in the Kitaev system despite the presence of a nonzero spin gap. We also demonstrate that this phenomenon can be observed in the system with small Heisenberg interactions using the exact diagonalization.

One Proximate Kitaev Spin Liquid in the K-J-Γ Model on the Honeycomb Lattice

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 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 field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.

Heisenberg-Kitaev physics in magnetic fields

Magnetic insulators in the regime of strong spin-orbit coupling exhibit intriguing behaviors in external magnetic fields, reflecting the frustrated nature of their effective interactions. We review the recent advances in understanding the field responses of materials that are described by models with strongly bond-dependent spin exchange interactions, such as Kitaev's celebrated honeycomb model and its extensions. We discuss the field-induced phases and the complex magnetization processes found in these theories and compare with experimental results in the layered Mott insulators [Formula: see text]-RuCl₃ and Na₂IrO₃, which are believed to realize this fascinating physics.

Vison Crystals in an Extended Kitaev Model on the Honeycomb Lattice

We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and, hence, the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric π-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature Monte Carlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.

Bond-Disordered Spin Liquid and the Honeycomb Iridate H_{3}LiIr_{2}O_{6}: Abundant Low-Energy Density of States from Random Majorana Hopping

The 5d-electron honeycomb compound H_{3}LiIr_{2}O_{6} [K. Kitagawa et al., Nature (London) 554, 341 (2018)NATUAS0028-083610.1038/nature25482] exhibits an apparent quantum spin liquid state. In this intercalated spin-orbital compound, a remarkable pileup of low-energy states was experimentally observed in specific heat and spin relaxation. We show that a bond-disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite hopping problem of Majorana fermions in a random flux background. This has a divergent low-energy density of states of the required power-law form N(E)∝E^{-ν} with a drifting exponent which takes on the value ν≈1/2 for relatively strong bond disorder. Breaking time-reversal symmetry removes the divergence of the density of states, as does applying a magnetic field in experiment. We discuss the implication of our scenario, both for future experiments and from a broader perspective.

Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic Field

Motivated by recent experimental observations in α-RuCl_{3}, we study the K-Γ model on the honeycomb lattice in an external magnetic field. By a slave-particle representation and variational Monte Carlo calculations, we reproduce the phase transition from zigzag magnetic order to a field-induced disordered phase. The nature of this state depends crucially on the field orientation. For particular field directions in the honeycomb plane, we find a gapless Dirac spin liquid, in agreement with recent experiments on α-RuCl_{3}. For a range of out-of-plane fields, we predict the existence of a Kalmeyer-Laughlin-type chiral spin liquid, which would show an integer-quantized thermal Hall effect.

Novel bufferless photosynthetic microbial fuel cell (PMFCs) for enhanced electrochemical performance

Photosynthetic microbial fuel cells (PMFCs) are novel bioelectrochemical transducers that employ microalgae to generate oxygen, organic metabolites and electrons. Conventional PMFCs employ non-eco-friendly membranes, catalysts and phosphate buffer solution. Eliminating the membrane, buffer and catalyst can make the MFC a practical possibility. Therefore, single chambered (SPMFC) were constructed and operated at different recirculation flow rates (0, 40 and 240 ml/min) under bufferless conditions. Furthermore, maximum power density of 4.06 mW/m², current density of 46.34 mA/m² and open circuit potential of 0.43 V and low internal resistance of 611.8 Ω were obtained at 40 ml/min. Based on the results it was decided that SPMFC was better for operation at 40 ml/min. Therefore, these findings provided progressive insights for future pilot and industrial scale studies of PMFCs.

Spinon Magnetic Resonance of Quantum Spin Liquids

We describe electron spin resonance in a quantum spin liquid with significant spin-orbit coupling. We find that the resonance directly probes spinon continuum, which makes it an efficient and informative probe of exotic excitations of the spin liquid. Specifically, we consider spinon resonance of three different spinon mean-field Hamiltonians, obtained with the help of projective symmetry group analysis, which model a putative quantum spin liquid state of the triangular rare-earth antiferromagnet YbMgGaO_{4}. The band of absorption is found to be very broad and exhibit strong van Hove singularities of single spinon spectrum as well as pronounced polarization dependence.

Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of α-RuCl_{3}

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 7.5 T applied in the ab plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of α-RuCl_{3}.

Antiferromagnetic Resonance and Terahertz Continuum in α-RuCl_{3}

We report measurements of optical absorption in the zigzag antiferromagnet α-RuCl_{3} as a function of temperature T, magnetic field B, and photon energy ℏω in the range ∼0.3-8.3 meV, using time-domain terahertz spectroscopy. Polarized measurements show that threefold rotational symmetry is broken in the honeycomb plane from 2 to 300 K. We find a sharp absorption peak at 2.56 meV upon cooling below the Néel temperature of 7 K at B=0 that we identify as the magnetic-dipole excitation of a zero-wave-vector magnon, or antiferromagnetic resonance (AFMR). With the application of B, the AFMR broadens and shifts to a lower frequency as long-range magnetic order is lost in a manner consistent with transitioning to a spin-disordered phase. From a direct, internally calibrated measurement of the AFMR spectral weight, we place an upper bound on the contribution to the dc susceptibility from a magnetic excitation continuum.

Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in α-RuCl_{3}

We report on terahertz spectroscopy of quantum spin dynamics in α-RuCl_{3}, a system proximate to the Kitaev honeycomb model, as a function of temperature and magnetic field. We follow the evolution of an extended magnetic continuum below the structural phase transition at T_{s2}=62  K. With the onset of a long-range magnetic order at T_{N}=6.5  K, spectral weight is transferred to a well-defined magnetic excitation at ℏω_{1}=2.48  meV, which is accompanied by a higher-energy band at ℏω_{2}=6.48  meV. Both excitations soften in a magnetic field, signaling a quantum phase transition close to B_{c}=7  T, where a broad continuum dominates the dynamical response. Above B_{c}, the long-range order is suppressed, and on top of the continuum, emergent magnetic excitations evolve. These excitations follow clear selection rules and exhibit distinct field dependencies, characterizing the dynamical properties of a possibly field-induced quantum spin liquid.

Dynamics of the Kitaev-Heisenberg Model

We introduce a matrix-product state based method to efficiently obtain dynamical response functions for two-dimensional microscopic Hamiltonians. We apply this method to different phases of the Kitaev-Heisenberg model and identify characteristic dynamical features. In the ordered phases proximate to the spin liquid, we find significant broad high-energy features beyond spin-wave theory, which resemble those of the Kitaev model. This establishes the concept of a proximate spin liquid, which was recently invoked in the context of inelastic neutron scattering experiments on α-RuCl_{3}. Our results provide an example of a natural path for proximate spin liquid features to arise at high energies above a conventionally ordered state, as the diffuse remnants of spin-wave bands intersect to yield a broad peak at the Brillouin zone center.

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.

Probing Spinon Nodal Structures in Three-Dimensional Kitaev Spin Liquids

We propose that resonant inelastic x-ray scattering (RIXS) is an effective probe of the fractionalized excitations in three-dimensional (3D) Kitaev spin liquids. While the non-spin-conserving RIXS responses are dominated by the gauge-flux excitations and reproduce the inelastic-neutron-scattering response, the spin-conserving (SC) RIXS response picks up the Majorana-fermion excitations and detects whether they are gapless at Weyl points, nodal lines, or Fermi surfaces. As a signature of symmetry fractionalization, the SC RIXS response is suppressed around the Γ point. On a technical level, we calculate the exact SC RIXS responses of the Kitaev models on the hyperhoneycomb, stripyhoneycomb, hyperhexagon, and hyperoctagon lattices, arguing that our main results also apply to generic 3D Kitaev spin liquids beyond these exactly solvable models.

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 15 T, 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.

Quantum spin liquids: a review

Quantum spin liquids may be considered 'quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.

Resonant Inelastic X-Ray Scattering Response of the Kitaev Honeycomb Model

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 measured by inelastic neutron scattering. RIXS can, therefore, pick up the fractionalized excitations of the Kitaev spin liquid separately, making it a sensitive probe to detect spin-liquid character in potential material incarnations of the Kitaev honeycomb model.