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arXiv:2412.17774 · npj Quantum Materials

Thermal quench dynamics of visons in gapless Kitaev spin liquid

Yang Yang, Gia-Wei Chern

Published Jun 6, 2026• 2 citations
arXivHTMLPDF

Abstract

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 $${{\mathbb{Z}}}_{2}$$ 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 $$\sqrt{3}\times \sqrt{3}$$ 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 $${{\mathbb{Z}}}_{3}$$ symmetry.

DOI
10.1038/s41535-026-00908-1

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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{{\mathbb{Z}}}_{2}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\sqrt{3}\times \sqrt{3}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{{\mathbb{Z}}}_{3}Z3​ symmetry.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.
A vison is an excitation of the Kitaev spin liquid which carries a Z 2 gauge flux.While immobile in the pure Kitaev model, it becomes a dynamical degree of freedom in the presence of perturbations.We study an isolated vison in the isotropic Kitaev model perturbed by a small external magnetic field h, an off-diagonal exchange interaction Γ, and a Heisenberg coupling J.In the ferromagnetic Kitaev model, the dressed vison obtains a dispersion linear in Γ and h and a fully universal low-T mobility, μ ¼ 6ℏv 2 m =ðk B TÞ 2 , where v m is the velocity of Majorana fermions.In contrast, in the antiferromagnetic (AFM) Kitaev model interference effects suppress coherent propagation and an incoherent Majorana-assisted hopping leads to a T-independent mobility.The motion of a single vison due to Heisenberg interactions is strongly suppressed for both signs of the Kitaev coupling.Vison bands in AFM Kitaev models can be topological and may lead to characteristic features in the thermal Hall effects in Kitaev materials.Furthermore, we argue that vison diffusion leads to universal signatures in quench experiments.
The Kitaev quantum spin liquid provides an ideal playground for the spin fractionalization to Majorana and vison excitations. The fingerprints manifest themselves in spin dynamics, which are accessible via many experimental techniques. However, extrinsic disorder effects are inevitable in real materials. Here, the authors compute finite-temperature spin dynamics in disordered Kitaev spin liquids and clarify that disorder seriously affects the vison excitations rather than Majorana quasiparticles. Furthermore, the softening of the vison excitations and Majorana zero modes emerge due to disorder, which will help identify Kitaev magnets.
Evidence for Deconfined Magnetic Order in the Kitaev-J3J_3J3​ Model
We investigate the Kitaev-J3J_3J3​ 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\mathbb{Z}_2Z2​ 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 Na2_22​Co2_22​TeO6_66​. Our results reveal a microscopic route to fractionalized magnetism beyond the conventional dichotomy between magnetic order and spin-liquid behavior.
The long-sought quantum spin liquid is a quantum-entangled magnetic state leading to the fractionalization of spin degrees of freedom. Quasiparticles emergent from the fractionalization affect not only the ground-state properties but also thermodynamic behavior in a peculiar manner. Here we investigate how the spin dynamics evolves from the high-temperature paramagnet to the quantum spin-liquid ground state, for the Kitaev spin model describing the fractionalization into itinerant matter fermions and localized Z2{Z}_{2}Z2​ fluxes. Beyond the previous studies [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016); J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Phys. Rev. B 96, 024438 (2017)], in which the mean-field nature of the cluster dynamical mean-field theory prevented us from studying low-temperature properties, we develop a numerical technique by applying the continuous-time quantum Monte Carlo (CTQMC) method to statistical samples generated by the quantum Monte Carlo (QMC) method in a Majorana fermion representation. This QMC+CTQMC method is fully unbiased and enables us to investigate the low-temperature spin dynamics dominated by thermally excited Z2{Z}_{2}Z2​ fluxes, including the unconventional phase transition caused by Z2{Z}_{2}Z2​ flux loops in three dimensions, which was unreachable by the previous methods. We apply this technique to the Kitaev model in both two and three dimensions. Our results clearly distinguish two cases: while the dynamics changes smoothly through the crossover in the two-dimensional honeycomb case, it exhibits singular behaviors at the phase transition in the three-dimensional hyperhoneycomb case. We show that the low-temperature spin dynamics is a sensitive probe for thermally fluctuating Z2{Z}_{2}Z2​ fluxes that behave very differently between two and three dimensions.
Abstract This paper reviews the nature of two quasiparticles, the Majorana fermion and vison, emergent in the Kitaev model, mainly from the theoretical point of view. We demonstrate how the fractionalization of a quantum spin into these two quasiparticles occurs and clarify their properties in the presence and absence of magnetic fields. In addition to discussions on the itinerant nature inherent to Majorana fermions, the quantum dynamics of visons induced by an applied magnetic field is revealed. Fractional quasiparticles manifest themselves in temperature evolutions of thermodynamic quantities, excitation spectra, and transport properties. The formulations for them and calculation results are shown in detail. Based on the results, we present how the signatures of fractional quasiparticles appear in physical quantities. The paper also mentions the recent developments of the experimental and theoretical works of Kitaev-related systems and presents outlooks of studies on these systems.