Finite temperature geometric properties of the Kitaev honeycomb model

TL;DR

This paper investigates finite temperature topological phase transitions in the Kitaev honeycomb model using mean Uhlmann curvature, revealing crossover effects and nonmonotonic behaviors related to thermal effects and band filling.

Contribution

It introduces a Fermionisation approach to analyze the Kitaev model as a two-band superconductor using mean Uhlmann curvature at finite temperatures.

Findings

Mean Uhlmann curvature indicates phase crossover at high temperature.

Nonmonotonic Uhlmann number behavior due to band filling effects.

External magnetic field induces non-Abelian topological phases.

Abstract

We study finite temperature topological phase transitions of the Kitaev's spin honeycomb model in the vortex-free sector with the use of the recently introduced mean Uhlmann curvature. We employ an appropriate Fermionisation procedure to study the system as a two-band p-wave superconductor described by a BdG Hamiltonian. This allows to study relevant quantities such as Berry and mean Uhlmann curvatures in a simple setting. More specifically, we consider the spin honeycomb in the presence of an external magnetic field breaking time reversal symmetry. The introduction of such an external perturbation opens a gap in the phase of the system characterised by non-Abelian statistics, and makes the model to belong to a symmetry protected class, so that the Uhmann number can be analysed. We first consider the Berry curvature on a particular evolution line over the phase diagram. The mean Uhlmann curvature and the Uhlmann number are then analysed considering the system to be in a Gibbs state at finite temperature. Then, we show that the mean Uhlmann curvature describes a cross-over effect of the phases at high temperature. We also find an interesting nonmonotonic behaviour of the Uhlmann number as a function of the temperature in the trivial phase, which is due to the partial filling of the conduction band around Dirac points.