  The Berry phase is a geometric phase of a pure state when the system is
adiabatically transported along a loop in its parameter space. The concept of
geometric phase has been generalized to mixed states by the so called Uhlmann
phase. However, the Uhlmann phase is constructed from the Uhlmann connection
that possesses a well defined global section. This property implies that the
Uhlmann connection is topologically trivial and as a consequence, the
corresponding Chern character vanishes. We propose modified Chern character
whose integral gives the thermal Uhlmann Chern number, which is related to the
winding number of the mapping defined by the Hamiltonian. Therefore, the
thermal Uhlmann Chern number reflects the topological properties of the
underlying Hamiltonian of a mixed state. By including the temperature
dependence in the volume integral, we also introduce the non-topological
thermal Uhlmann Chern number which varies with temperature but is not quantized
at finite temperatures. We illustrate the applications to a two-band model and
a degenerate four-band model.
