  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 linked 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 to the dynamical
conductivity, respectively.
