Static and Dynamical Spin Correlations in the Kitaev Model at Finite Temperatures via Green's Function Equation of Motion

TL;DR

This paper explores the finite-temperature behavior of the Kitaev model using Green's function methods, revealing insights into spin correlations and excitations that complement existing Majorana-based studies.

Contribution

It introduces a Green's function equation of motion approach to analyze the Kitaev model at finite temperatures, providing new insights into its thermal spin dynamics.

Findings

Spin correlations capture flux and spin-flip excitations.

Results align with some numerical simulations but show differences.

Evidence of fermionic degrees of freedom at low temperatures.

Abstract

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 approaches. In this paper, we investigate the finite-temperature properties of the Kitaev model using the spin Green's function formalism. This approach enables the computation of key physical quantities such as spin correlations, magnetic susceptibility, and the dynamical spin structure factor, offering crucial insights into the system's thermal dynamics. In solving the equation of motion for the spin Green's function, we truncate the hierarchy of multi-spin Green's functions using a decoupling approximation, which proves to be 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 $\mathbb{Z}_2$ flux excitations but also simple spin-flip excitations, with the latter overshadowing the former. Interestingly, without explicitly assuming fractionalization, our results for the spin susceptibility and spin relaxation rate still suggest the presence of fermionic degrees of freedom at low temperatures. This study provides a complementary approach to understanding the thermal properties of the Kitaev model, which could be relevant for future experiments and theoretical investigations.