We investigate the Kitaev-$J_3$ honeycomb model using variational Monte Carlo calculations combined with a vison-quasiparticle analysis of the parent Kitaev spin liquid (KSL). We provide evidence for deconfined magnetic phases in which zigzag or antiferromagnetic order coexists with remnant $\mathbb{Z}_2$ topological structure inherited from the KSL. The optimized variational wave functions retain multiple linearly independent topological sectors on a torus, whereas those of conventional ordered phases collapse to a single sector. The vison-quasiparticle analysis shows that magnetic order naturally arises from vison-pair condensation while single visons remain gapped, yielding a microscopic mechanism for magnetic ordering without immediate confinement. The resulting phases further host gapless spinons with multiple Majorana cones, offering a possible microscopic scenario for the anomalous low-temperature longitudinal thermal transport reported in magnetically ordered Kitaev materials such as Na$_2$Co$_2$TeO$_6$. Our results reveal a microscopic route to fractionalized magnetism beyond the conventional dichotomy between magnetic order and spin-liquid behavior.
Evidence for Deconfined Magnetic Order in the Kitaev-J3 Model
We investigate the Kitaev-J3 honeycomb model using variational Monte Carlo calculations combined with a vison-quasiparticle analysis of the parent Kitaev spin liquid (KSL). We provide evidence for deconfined magnetic phases in which zigzag or antiferromagnetic order coexists with remnant Z2 topological structure inherited from the KSL. The optimized variational wave functions retain multiple linearly independent topological sectors on a torus, whereas those of conventional ordered phases collapse to a single sector. The vison-quasiparticle analysis shows that magnetic order naturally arises from vison-pair condensation while single visons remain gapped, yielding a microscopic mechanism for magnetic ordering without immediate confinement. The resulting phases further host gapless spinons with multiple Majorana cones, offering a possible microscopic scenario for the anomalous low-temperature longitudinal thermal transport reported in magnetically ordered Kitaev materials such as Na2Co2TeO6. Our results reveal a microscopic route to fractionalized magnetism beyond the conventional dichotomy between magnetic order and spin-liquid behavior.
High-field fate of the Kitaev quantum spin liquid in α-RuCl3
Kitaev quantum spin liquids (KQSLs) host fractionalized excitations described by itinerant Majorana quasiparticles and gapped Z2 fluxes (visons), providing a platform for emergent topological matter. Whether such a state survives under strong magnetic fields, however, remains an open question. The layered honeycomb magnet α-RuCl3 is a leading candidate material: an in-plane field of ∼ 7 T suppresses antiferromagnetic order and induces a quantum-disordered phase exhibiting signatures consistent with Majorana excitations, including an anomalous thermal Hall effect and field-angle-dependent specific heat. At higher fields, the magnetization approaches saturation, suggesting a transition to a spin-polarized state, yet the microscopic evolution between these limits remains unresolved. Here we report high-field specific heat measurements up to 24 T that reveal a distinct crossover at $μ_0H^*\approx$15 T, beyond which the perturbative Kitaev description breaks down. Above H∗, the characteristic six-fold angular modulation of the specific heat collapses and the excitation gap deviates from the predicted H3 scaling. Meanwhile, the gap decreases with increasing field and the in-plane magnetization anisotropy persists up to ∼ 24 T, both in sharp contrast to a trivial spin-polarized state, indicating that KQSL signatures are preserved even at ∼ 90 % of magnetization saturation. These results reveal that the KQSL in α-RuCl3 extends well beyond the perturbative window, persisting as a nonperturbative regime in which the Majorana and vison energy scales merge, before eventually giving way to spin polarization. This thermodynamic roadmap provides a basis for understanding how fractionalized phases evolve under strong magnetic fields.
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.
Understanding the nonequilibrium relaxation dynamics of quantum spin liquids is an important and relatively unexplored frontier, where fractionalized quasiparticles and emergent gauge fields can give rise to dynamical behaviors far beyond those of conventional magnets. Here we study the relaxation dynamics of the Kitaev honeycomb model following a thermal quench, focusing on visons—gapped Z2 flux excitations of the emergent gauge field in the Kitaev spin liquid. The long-time evolution is governed by the diffusion and pair annihilation of visons, with both the activation barrier for vison motion and the effective vison-vison interactions mediated by fractionalized Majorana fermions. The resulting open-system quantum dynamics, induced by coupling to a thermal bath, can be efficiently simulated using an effective kinetic Monte Carlo approach. We uncover a rich set of temperature-dependent dynamical regimes. With decreasing temperature, the relaxation crosses over from diffusion-limited annihilation to terminal-velocity–limited kinetics, and ultimately to pronounced dynamical arrest and freezing. We show that this freezing behavior is closely tied to the formation of long-lived, metastable 3×3 vison crystals, which strongly suppress further annihilation. Remarkably, even in the absence of true long-range crystalline order, the system exhibits a hidden coarsening dynamics of vison superclusters associated with an emergent Z3 symmetry.
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.
Spin-orbital generalization of Kitaev model provides a robust extension to the original Kitaev model. However, real materials often exhibit competing interactions that break exact solvability which can give rise to new phases. Motivated by recent microscopic proposals of coexisting Yao-Lee and Kitaev couplings, we investigate the fate of the ground state when two independent exactly solvable spin liquid Hamiltonians each originally formulated on different lattice geometries are combined on a common lattice environment. We first focus on the hybrid Kitaev's honeycomb and square-lattice model. Using self-consistent mean-field analysis and perturbative calculation, we show that the strong-Kitaev regime yields magnetic order in the spin sector, while the orbital sector retains its topological order. We further analyze the hybridization of the Yao-Lee and square-lattice models and find that the model exhibits a rich evolution of Majorana Dirac bands and Lifshitz transitions. Remarkably, when the Yao-Lee and square-lattice couplings are equal and opposite, the model restores its exact solvability with a single itinerant Majorana flavor. These results demonstrate that hybrid spin liquid platforms may host various emergent phases beyond conventional exactly solvable limits.
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped Z2 spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated ${120}^{\ensuremath{\circ}}$ compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis or lying in the xy plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.