Finite-temperature geometric properties of the Kitaev honeycomb model

We study finite-temperature topological properties 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 fermionization procedure to study the system as a two-band p-wave superconductor described by a Bogoliubov–de Gennes Hamiltonian. This allows us 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 up a gap in the phase of the system characterized by non-Abelian statistics. The resulting model belongs to a symmetry-protected class, so that the Uhlmann number can be analyzed. 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 analyzed by assuming a thermal state. The mean Uhlmann curvature describes a crossover effect as temperature rises. In the trivial phase, a nonmonotonic dependence of the Uhlmann number, as temperature increases, is reported and explained.