Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

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 a 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 nontopological 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.

Figure 1

The nontopological (NT) thermal Uhlmann Chern number (solid symbols) from Eq. (16) and ${\sigma }_{xy}/e^2$ (hollow symbols) from Eq. (19) of the two-band model as functions of temperature. The red and black symbols correspond to $m=1.3$ and $m=2.3$, respectively.

Figure 2

The second NT thermal Uhlmann Chern number of the four-band model, Eq. (36), as a function of temperature. The red triangles and black dots correspond to $m=1.3$ and $m=4.3$, respectively.