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.
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.
A Majorana fermion is a fermionic particle that is its own antiparticle. Since the theoretical discovery in 1937, the exotic particle has long been searched in particle physics. In the last few decades, however, it has attracted renewed interest in condensed matter physics, where it can be realized as an elementary excitation (quasiparticle) in quantum states of matter. In this review, we discuss another platform for Majorana fermions, the quantum spin liquid, in which interacting magnetic moments remain disordered down to the lowest temperature under strong quantum fluctuations. They are characterized by topological entanglement and fractional excitations, whose possible application to topological quantum computation is recently discussed intensively. As a prime candidate for such exotic states, we here focus on the Kitaev magnets, a subgroup of the spin-orbit Mott insulators. After a brief overview of the Kitaev model and the fractionalization of spins in the exact ground state, we review recent explosive development in this rapidly growing field, with a focus on numerical solutions of the Kitaev model at finite temperatures and the comparison with experiments. The key concept is thermal fractionalization --- two types of fractional excitations manifest themselves at largely different temperatures. This leads to distinct thermodynamics and spin dynamics in a variety of experimentally measurable quantities. We discuss such peculiar behaviors as the signatures of fractional quasiparticles, in careful comparison with the available experimental data for the candidate materials of the Kitaev magnets. Our review gives an overview of the current status of the identification of Majorana fermions in the Kitaev magnets, which would serve as a basis for further experimental and theoretical studies toward the manipulation of the exotic particles for topological quantum computation.
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.
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.
The Kitaev quantum spin liquid possesses two fractional quasiparticles, itinerant Majorana fermions and localized visons. It provides a promising platform for realizing a Majorana zero mode trapped by a vison excitation. This local mode behaves as a non-Abelian anyon capable of applications to quantum computation. However, creating, observing, and manipulating visons remain challenging even in the pristine Kitaev model. Here, we propose a theory to control visons enabled by a time-dependent local magnetic field in the Kitaev spin liquid. Examining the time evolution of the magnetic state, we demonstrate that a vison follows a locally applied field sweeping in the system. We clarify that one can move a vison accompanied by a Majorana zero mode by choosing the velocity and shape of the local field appropriately. In particular, the controllability of visons using local fields shows nonlinear behavior for its strength, which originates from interactions between Majorana fermions and visons. The present results suggest that itinerant Majorana fermions other than zero modes play a crucial role in vison transport. Our finding will offer a guideline for controlling Majorana zero modes in the Kitaev quantum spin liquid.
The fractionalization of quantum numbers in interacting quantum many-body systems is a central motif in condensed-matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here, we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a Z_{2} gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synthesized hyperhoneycomb structure of β-Li_{2}IrO_{3} when applying a magnetic field. We characterize the topologically protected bulk and surface features of this state, which we dub a Weyl spin liquid, including thermodynamic and transport signatures.
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.