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arXiv:2104.10549 · Physical review. B./Physical review. B

Spin dynamics in the Kitaev model with disorder: Quantum Monte Carlo study of dynamical spin structure factor, magnetic susceptibility, and NMR relaxation rate

Joji Nasu, Yukitoshi Motome

Published Jul 8, 2021• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.104.035116

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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.
Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte Carlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the quantum spin liquid. We find that dynamical spin correlations show peculiar temperature and frequency dependence even below the temperature where static correlations saturate. The results provide the experimentally accessible symptoms of the fluctuating fractionalized spins evincing the quantum spin liquids.
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.
We study the spin-SSS Kitaev model in the classical ($S\ensuremath{\rightarrow}\ensuremath{\infty}$) limit using Monte Carlo simulations combined with semiclassical spin dynamics. We discuss differences and similarities in the dynamical structure factors of the spin-1/21/21/2 and the classical Kitaev liquids. The quantum behavior is restricted to low temperatures where a gap protects visons from decohering the system. Once this quantum gap is breached, at low temperatures compared to the coupling constant, significant entropic disorder decoheres the Majorana fermions and the system is described quantitatively by classical dynamics. The low-temperature and low-energy spectrum of the classical model exhibits a finite-energy peak, which is the precursor of the one produced by the Majorana modes of the S=1/2S=1/2S=1/2 model. The classical peak is spectrally narrowed compared to the quantum result and can be explained by magnon excitations within fluctuating one-dimensional manifolds (loops). Hence the difference from the classical limit to the quantum limit can be understood by the fractionalization of magnons propagating in one-dimensional manifolds. Moreover, we show that the momentum-space distribution of the low-energy spectral weight of the S=1/2S=1/2S=1/2 model follows the momentum-space distribution of zero modes of the classical model.
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.
We study the effect of site dilution in the Kitaev model. We derive an analytical solution of the dynamical spin correlation functions for arbitrary configurations of Z2{Z}_{2}Z2​ fluxes. By incorporating this solution into a classical Monte Carlo scheme, we address how a site vacancy affects the experimental observables, such as the static spin susceptibility and the spin-lattice relaxation rate, 1/T11/{T}_{1}1/T1​. As a result, we found an enhancement of dynamical magnetic response in the vicinity of vacancy, which leads to Friedel-like oscillation in local 1/T11/{T}_{1}1/T1​, in contrast to limited influences on the static susceptibility. Furthermore, we found a sharp zero-energy peak in the magnetic excitation spectrum, which is attributed to the Vison-Majorana zero mode trapped near the site vacancy. This zero mode can be interpreted as a fractionalized spin hole into an Ising triplet with differentiated magnetic axes, which leads to the characteristic temperature and field-orientational dependence of 1/T11/{T}_{1}1/T1​.
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.