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 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 model is an exactly solvable model that exhibits a quantum spin liquid as its ground state. While this model was originally proposed as an S = 1 / 2 spin model on a honeycomb lattice, extensions to higher-spin systems have recently attracted attention. In contrast to the S = 1 / 2 case, such higher- S models are not exactly solvable and remain poorly understood, particularly for spin excitations at finite temperatures. Here, we focus on the S = 1 Kitaev model, which has been proposed to host bosonic quasiparticles. We investigate this model using Schwinger boson mean-field theory, where bosonic spinons are introduced as fractional quasiparticles by extending bond operators to address anisotropic spin interactions. We determine the mean-field parameters that realize a quantum spin liquid in both ferromagnetic and antiferromagnetic Kitaev models. Based on this mean-field ansatz, we calculate the dynamical and equal-time spin structure factors. We find that, when one uses the conventional decoupling scheme based on Wick decoupling with respect to spinons to calculate spin correlations, the resultant spin structure factors exhibit a momentum dependence that is not consistent with the sign structure expected from the exchange interaction: They possess a strong spectral weight indicating ferromagnetic (antiferromagnetic) correlations in the antiferromagnetic (ferromagnetic) Kitaev model. To resolve this issue, we propose an alternative scheme for evaluating the spin correlations, which is based on decoupling with respect to the bond operators. We demonstrate that, in our scheme, this discrepancy is removed, and the momentum dependence of the spin structure factors is consistent with the sign of the exchange constant. We also calculate the temperature evolution of the dynamical spin structure factor and find that the continuum observed at zero temperature splits into two distinct structures as the temperature increases, which can be understood in terms of the bandwidth narrowing of spinons. Finally, we clarify the origin of why the two distinct decoupling schemes result in different momentum dependences of the spin structure factors and discuss their relationship to results obtained in previous studies.
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.