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

Abstract

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.

Topological Uhlmann phase transitions for a spin-jparticle in a magnetic field

“…where ρ is the system's density matrix with a spectral decomposition k p k |k k| and P is the path-ordering operator [5]. The Uhlmann connection ÂU is given by [18] ÂU = l,k…”

“…The appropriate extension of the geometric phase to mixed states was developed by Uhlmann [13,14] and is thus called the Uhlmann phase. It has been theoretically studied in the context of 1D and 2D topological insulators [15][16][17][18], for example the 1D SSH model [19] and the Qi-Wu-Zhang 2D Chern insulator [20]. A key feature of these systems is the appearance of a critical temperatures above which the Uhlmann phase vanishes regardless of the topological character of the system in the ground state.…”

Uhlmann number in translational invariant systems

“…On the other hand, the fate of these topological ordered phases remains unclear, when a mixed state is the faithful description of the quantum system, either because of thermal equilibrium, or due to out-ofequilibrium conditions. Over the last few years, different attempts have been done to reconcile the above topological criteria with a mixed state configuration [16][17][18][19][20][21][22][23][24][25][26]. The recent success of the Uhlmann approach [27] in describing the topology of 1D Fermionic systems [18,19], remains in higher dimensions [20] not as straightforward [21].…”

Comparison of finite-temperature topological indicators based on Uhlmann connection

“…Ref. 30 also suggests a modified Chern-number formula to extract nonvanishing results from the Uhlmann connection.…”