Beyond Kitaev physics in strong spin-orbit coupled magnets.

We review the recent advances and current challenges in the field of strong spin-orbit coupled Kitaev materials, with a particular emphasis on the physics beyond the exactly-solvable Kitaev spin liquid point. To this end, we present a comprehensive overview of the key exchange interactions in candidate materials with a specific focus on systems featuring effectiveJeff=1/2magnetic moments. This includes, but not limited to,5d5iridates,4d5ruthenates and3d7cobaltates. Our exploration covers the microscopic origins of these interactions, along with a systematic attempt to map out the most intriguing correlated regimes of the multi-dimensional parameter space. Our approach is guided by robust symmetry and duality transformations as well as insights from a wide spectrum of analytical and numerical studies. We also survey higher spin Kitaev models and recent exciting results on quasi-one-dimensional models and discuss their relevance to higher-dimensional models. Finally, we highlight some of the key questions in the field as well as future directions.

Vacancy Spectroscopy of Non-Abelian Kitaev Spin Liquids.

Spin vacancies in the non-Abelian Kitaev spin liquid are known to harbor Majorana zero modes, potentially enabling topological quantum computing at elevated temperatures. Here, we study the spectroscopic signatures of such Majorana zero modes in a scanning tunneling setup where a non-Abelian Kitaev spin liquid with a finite density of spin vacancies forms a tunneling barrier between a tip and a substrate. Our key result is a well-defined peak close to zero bias voltage in the derivative of the tunneling conductance whose voltage and intensity both increase with the density of vacancies. This "quasi-zero-voltage peak" is identified as the closest analog of the zero-voltage peak observed in topological superconductors that additionally reflects the fractionalized nature of spin-liquid-based Majorana zero modes. We further highlight a single-fermion Van Hove singularity at a higher voltage that reveals the energy scale of the emergent Majorana fermions in the Kitaev spin liquid. Our proposed signatures are within reach of current experiments on the candidate material α-RuCl_{3}.

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.

Classical Spin Liquid Instability Driven By Off-Diagonal Exchange in Strong Spin-Orbit Magnets.

We show that the off-diagonal exchange anisotropy drives Mott insulators with strong spin-orbit coupling to a classical spin liquid regime, characterized by an infinite number of ground states and Ising variables living on closed or open strings. Depending on the sign of the anisotropy, quantum fluctuations either fail to lift the degeneracy down to very low temperatures, or select noncoplanar magnetic states with unconventional spin correlations. The results apply to all 2D and 3D tricoordinated materials with bond-directional anisotropy and provide a consistent interpretation of the suppression of the x-ray magnetic circular dichroism signal reported recently for ß-Li_{2}IrO_{3} under pressure.

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.

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.

Critical properties of the Kitaev-Heisenberg model.

We study the critical properties of the Kitaev-Heisenberg model on the honeycomb lattice at finite temperatures that might describe the physics of the quasi-two-dimensional compounds, Na(2)IrO(3) and Li(2)IrO(3). The model undergoes two phase transitions as a function of temperature. At low temperature, thermal fluctuations induce magnetic long-range order by the order-by-disorder mechanism. This magnetically ordered state with a spontaneously broken Z(6) symmetry persists up to a certain critical temperature. We find that there is an intermediate phase between the low-temperature, ordered phase and the high-temperature, disordered phase. Finite-sized scaling analysis suggests that the intermediate phase is a critical Kosterlitz-Thouless phase with continuously variable exponents. We argue that the intermediate phase has been observed above the low-temperature, magnetically ordered phase in Na(2)IrO(3), and also, likely exists in Li(2)IrO(3).

Magnetism in parent iron chalcogenides: quantum fluctuations select plaquette order.

We analyze magnetic order in Fe chalcogenide Fe(1+y)Te, the parent compound of the high-temperature superconductor Fe(1+y)Te(1-x)Se(x). Experiments show that magnetic order in this material contains components with momentum Q(1)=(π/2,π/2) and Q(2)=(π/2,-π/2) in the Fe only Brillouin zone. The actual spin order depends on the interplay between these two components. Previous works assumed that the ordered state has a single Q (either Q(1) or Q(2)). In such a state, spins form double stripes along one of the diagonals breaking the rotational C(4) symmetry. We show that quantum fluctuations actually select another order-a double Q plaquette state with equal weight of Q(1) and Q(2) components, which preserves C(4) symmetry. We argue that the order in Fe(1+y)Te is determined by the competition between quantum fluctuations and magnetoelastic coupling.

Orbital disorder induced by charge fluctuations in vanadium spinels.

Motivated by recent experiments on vanadium spinels, AV2O4, that show an increasing degree of electronic delocalization for smaller cation sizes, we study the evolution of orbital ordering (OO) between the strong and intermediate-coupling regimes of a multiorbital Hubbard Hamiltonian. The underlying magnetic ordering of the Mott insulating state leads to a rapid suppression of OO due to enhanced charge fluctuations along ferromagnetic bonds. Orbital double occupancy is rather low at the transition point indicating that the system is in the crossover region between strong and intermediate-coupling regimes when the orbital degrees of freedom become disordered.

Quantum spin liquid in the semiclassical regime.

Quantum spin liquids (QSLs) have been at the forefront of correlated electron research ever since their proposal in 1973, and the realization that they belong to the broader class of intrinsic topological orders. According to received wisdom, QSLs can arise in frustrated magnets with low spin S, where strong quantum fluctuations act to destabilize conventional, magnetically ordered states. Here, we present a Z2 QSL ground state that appears already in the semiclassical, large-S limit. This state has both topological and symmetry-related ground-state degeneracy, and two types of gaps, a "magnetic flux" gap that scales linearly with S and an "electric charge" gap that drops exponentially in S. The magnet is the spin-S version of the spin-1/2 Kitaev honeycomb model, which has been the subject of intense studies in correlated electron systems with strong spin-orbit coupling, and in optical lattice realizations with ultracold atoms.