Motivated by growing evidence for the significance of the third-nearest-neighbor Heisenberg (
J3) interaction in candidate Kitaev materials, we investigate the dynamical properties of the Kitaev spin liquid (KSL) under a
J3 perturbation, focusing on its spin dynamical structure factor (DSF) and Raman scattering. Within a self-consistent parton mean-field plus random-phase approximation framework, we find that
J3 induces coherent, paramagnon-like collective modes that coexist with a high-energy Majorana continuum in the spin DSF. The softening of these modes with increasing
∣J3∣ signals a quantum phase transition to magnetic order. Remarkably, magnetic ordering sets in at a common critical
J3 for both ferromagnetic (
K<0) and antiferromagnetic (
K>0) Kitaev models, with the resulting ordered states forming exact dual pairs under a four-sublattice duality transformation that maps
(K,J3)→(−K,J3). An external magnetic field further softens the preexisting paramagnon modes, thereby enhancing magnetic order. Perturbative Raman calculations show that while the Kitaev-like Raman vertex probes only itinerant matter Majorana fermions, the response from the
J3-like vertex features both matter Majoranas and visons. Four-vison excitations produce a sharp peak accompanied by a two-fermion continuum, whereas two-vison excitations yield a continuum closely resembling the single-matter-fermion density of states. These results provide a unified perspective on the dynamical signatures of
J3-perturbed KSL and are helpful for interpreting experimental spectra in candidate Kitaev materials with sizable
J3 interactions.
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 a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter fermions and localized
Z2 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. 117, 157203 (2016)], we calculate the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$, the NMR relaxation rate
1/T1, and the magnetic susceptibility $\ensuremath{\chi}$ while changing the anisotropy in the exchange coupling constants, by using the dynamical mean-field theory based on a Majorana fermion representation. We describe the details of the methodology including the continuous-time quantum Monte Carlo method for computing dynamical spin correlations and the maximum entropy method for analytic continuation. We confirm that the combined method provides accurate results in a wide
T range including the region where the spins are fractionalized. We find that also in the anisotropic cases the system exhibits peculiar behaviors below the high-
T crossover whose temperature is comparable to the average of the exchange constants: $S(\mathbf{q},\ensuremath{\omega})$ shows an inelastic response at the energy scale of the averaged exchange constant,
1/T1 continues to grow even though the equal-time spin correlations are saturated and almost
T independent, and $\ensuremath{\chi}$ deviates from the Curie-Weiss behavior. In particular, when the exchange interaction in one direction is stronger than the other two, the dynamical quantities exhibit qualitatively different
T dependences from the isotropic case at low
T, reflecting the opposite parity between the flux-free ground state and the flux-excited state, and a larger energy cost for flipping a spin in the strong interaction direction. On the other hand, when the exchange anisotropy is in the opposite way, the results are qualitatively similar to those in the isotropic case. All these behaviors manifest 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 the
Z2 gauge fields. Our results could stimulate further experimental and theoretical analyses of candidate materials for the Kitaev quantum spin liquids.
Motivated by recent synthesis of the hyperhoneycomb material $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, we study the dynamical structure factor (DSF) of the corresponding 3D Kitaev quantum spin-liquid (QSL), whose fractionalized degrees of freedom are Majorana fermions and emergent flux loops. The properties of this 3D model are known to differ in important ways from those of its 2D counterpart---it has a finite-temperature phase transition, as well as distinct features in the Raman response. We show, however, that the qualitative behavior of the DSF is broadly dimension-independent. Characteristics of the 3D DSF include a response gap even in the gapless QSL phase and an energy dependence deriving from the Majorana fermion density of states. Since the majority of the response is from states containing a single Majorana excitation, our results suggest inelastic neutron scattering as the spectroscopy of choice to illuminate the physics of Majorana fermions in Kitaev QSLs.
Kitaev's model of spins interacting on a honeycomb lattice describes a quantum spin liquid, where an emergent static
Z2 gauge field is coupled to Majorana fermions. In the presence of an external magnetic field and for a range of interaction strengths, the system behaves as a gapped, non-Abelian quantum spin liquid. In this phase, the vortex excitations of the emergent
Z2 gauge field have Majorana zero modes bound to them. Motivated by recent experimental progress in measuring and characterizing real materials that could exhibit spin-liquid behavior, we analytically calculate the dynamical spin structure factor in the non-Abelian phase of the Kitaev's honeycomb model. In particular, we treat the case of quenched disorder in the vortex configurations. Our calculations reveal a peak in the low-energy dynamical structure factor that is a signature of the spin-liquid behavior. We map the effective Hamiltonian to that of a chiral
p-wave superconductor by using the Jordan-Wigner transformation. Subsequently, we analytically calculate the wave functions of the Majorana zero modes, the energy splitting for finite separation of the vortices and, finally, the dynamical structure factor in presence of quenched disorder.
We investigate the generic features of the low energy dynamical spin structure factor of the Kitaev honeycomb quantum spin liquid perturbed away from its exact soluble limit by generic symmetry-allowed exchange couplings. We find that the spin gap persists in the Kitaev-Heisenberg model, but generally vanishes provided more generic symmetry-allowed interactions exist. We formulate the generic expansion of the spin operator in terms of fractionalized Majorana fermion operators according to the symmetry enriched topological order of the Kitaev spin liquid, described by its projective symmetry group. The dynamical spin structure factor displays power-law scaling bounded by Dirac cones in the vicinity of the Γ, K, and K^{'} points of the Brillouin zone, rather than the spin gap found for the exactly soluble point.
Quantum spin liquids (QSLs) host a variety of fractionalized particles. In Kitaev's paradigmatic honeycomb model a spin-fractionalizes intoflux due to emergentgauge field and matter Majorana fermions. Although these excitations have well-defined dynamics in the integrable limit, their direct experimental identification is notoriously challenging: realistic materials inevitably host additional symmetry-allowed interactions that break integrability and hybridize gauge and matter sectors, while magnetic fields, which are often required to suppress competing order and stabilize a putative QSL regime, further entangle the responses of different fractionalized quasiparticles and may even drive the system into field-induced spin-liquid phases that are not adiabatically connected to the integrable limit. A prominent example is the quantum Majorana metal, in which the distinct dynamics of fractionalized Majorana fermions can become directly visible in scattering. This review highlights recent progress on these related questions: in which field-stabilized QSL regimes and nearby emergent phases, and under what conditions, can the response of a specific fractionalized quasiparticle be isolated and positively understood, thereby clarifying the existence and the experimental scope of putative spin liquids? We review the progress on these questions across Abelian, non-Abelian, and emergent quantum phases under magnetic field that are not perturbatively connected to the integrable limit. We connect these field-induced dynamical phenomena to concrete experimental observables, relevant for neutron scattering, resonant inelastic x-ray scattering, and pump-probe spectroscopy, techniques that are capable of identifying the nature of different magnetic-field-induced phases and resolving specific types of fractionalized particles, including Majoranas andfluxes.