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arXiv:1811.02092 · Physical review. B./Physical review. B

Vison-Majorana complex zero-energy resonance in the Kitaev spin liquid

Masafumi Udagawa

Published Dec 11, 2018• 25 citations
arXivHTMLPDF

Abstract

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 ${Z}_{2}$ 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, $1/{T}_{1}$. As a result, we found an enhancement of dynamical magnetic response in the vicinity of vacancy, which leads to Friedel-like oscillation in local $1/{T}_{1}$, 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 $1/{T}_{1}$.

DOI
10.1103/physrevb.98.220404

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arXiv:2305.08357•Physical review. B./Physical review. B•Dec 20, 2023•6 citations
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​.
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.
Abstract In this article, we review the theoretical formulation of finite temperature dynamics of Kitaev’s spin liquid. We present the exact analytical solution of the dynamical spin correlation function at the integrable limit of Kitaev’s model, on the basis of (2018 Phys. Rev. B 98 220404). By combining the analytical solution with the equilibrium classical Monte-Carlo scheme, we construct a formulation to access the finite temperature dynamics of Kitaev’s spin liquid exactly, with a reasonable amount of computational cost. This formulation is based on the real-time representation, which enables us to directly access the experimental observables defined in real frequency, without analytical continuation. The real-time scheme is essential to capturing the resonant features of the spectrum accurately, which occurs e.g. in the chiral spin liquid phase with isolated Majorana zero modes. Accordingly, this scheme provides an effective approach to address the nature of fractional excitations in Kitaev’s spin liquid. As an application, we address the detection of zero mode around the site vacancy through the local resonant spectrum and discuss how the character of Kitaev’s spin liquid emerges in its dynamical signature.
In the Kitaev chiral spin liquid, Ising anyons are realized as Z_{2} fluxes binding Majorana zero modes, which, however, are thermal excitations with finite decay rates. On the other hand, a lattice vacancy traps a Z_{2} flux even in the ground state, resulting in the stable realization of a Majorana zero mode in a vacancy. We demonstrate that spin-spin correlation functions between two vacancy sites exhibit long-range correlation arising from the fractionalized character of Majorana zero modes, in spite of the strong decay of bulk spin correlations. Remarkably, this nonlocal spin correlation does not decrease as the distance between two vacancy sites increases, signaling Majorana teleportation. Furthermore, we clarify that the nonlocal correlation can be detected electrically via the measurement of nonlocal conductance between two vacancy sites, which is straightforwardly utilized for the readout of Majorana qubits. These findings pave the way to the measurement-based quantum computation with Ising anyons trapped in vacancies of the Kitaev spin liquid.
Fractionalization is a hallmark of spin-liquid behavior; it typically leads to response functions consisting of continua instead of sharp modes. However, microscopic processes can enable the formation of short-distance bound states of fractionalized excitations, despite asymptotic deconfinement. Here we study such bound-state formation for the Z2{\mathbb{Z}}_{2}Z2​ spin liquid realized in Kitaev's honeycomb compass model, supplemented by a kekul\'e distortion of the lattice. Bound states between flux pairs and Majorana fermions form in the Majorana band gaps. We calculate the dynamic spin susceptibility and show that bound states lead to sharp modes in the magnetic response of the spin liquid, with the momentum dependence of the corresponding spectral weight encoding the internal symmetry of the bound state. As a byproduct, we also show that isolated fluxes may produce Majorana bound states at exactly zero energy. Generalizations and implications of the results are discussed.
The Kitaev model, renowned for its exact solvability and potential to host non-Abelian anyons, remains a focal point in the study of quantum spin liquids and topological phases. While much of the existing literature has employed Majorana fermion techniques to analyze the model, particularly at zero temperature, its finite-temperature behavior has been less thoroughly explored via alternative methods. In this paper, we investigate the finite-temperature properties of the Kitaev model using the spin Green's function formalism. This approach provides a unified framework for computing key physical quantities, such as spin correlations, magnetic susceptibility, and the dynamical spin structure factor, offering valuable insights into the system's thermal dynamics. To solve the equation of motion for the spin Green's function, we truncate the hierarchy of multispin Green's functions using the Tyablikov decoupling approximation, which is particularly accurate at high temperatures. Our results show several similarities with Majorana-based numerical simulations, though notable differences emerge. Specifically, both static and dynamical spin-spin correlation functions capture not only Z2{\mathbb{Z}}_{2}Z2​ flux excitations but also simple spin-flip excitations, with the latter dominating the response. Additionally, without explicitly assuming fractionalization, our results for the spin susceptibility and spin relaxation rate suggest the presence of fermionic degrees of freedom at low temperatures. Unlike Majorana-based approaches, which rely on exact solvability and are inherently limited to the pure Kitaev model, the spin Green's function formalism is well-suited for studying more realistic systems. It can naturally incorporate the effects of magnetic fields, including linear-order terms, and non-Kitaev interactions, making it applicable to a broader range of materials and experimental conditions. By providing an alternative method for analyzing the finite-temperature properties of the Kitaev model, this study complements existing approaches and lays the groundwork for future investigations into real materials and extended models.
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.