Uhlmann number in translational invariant systems

We define the Uhlmann number as an extension of the Chern number, and we use this quantity to describe the topology of 2D translational invariant Fermionic systems at finite temperature. We consider two paradigmatic systems and we study the changes in their topology through the Uhlmann number. Through the linear response theory we link two geometrical quantities of the system, the mean Uhlmann curvature and the Uhlmann number, to directly measurable physical quantities, i.e. the dynamical susceptibility and the dynamical conductivity, respectively. In particular, we derive a non-zero temperature generalisation of the Thouless-Kohmoto-Nightingale-den Nijs formula.