  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.
