Static and dynamical spin correlations in the Kitaev model at finite temperatures via Green's function equation of motion

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 ${\mathbb{Z}}_{2}$ 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.