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.

arXiv comment7 pages, 2 figures, revised version

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• Finite temperature geometric properties of the Kitaev honeycomb model
• Haldane Model at finite temperature
• Topological order, mixed states and open systems
• Uhlmann quench and geometric dynamic quantum phase transition of mixed states
• Off-resonant light-induced topological phase transition and thermoelectric transport in semi-Dirac materials
• Uhlmann holonomy against Lindblad dynamics of topological systems at finite temperatures
• Proxy ensemble geometric phase and proxy index of time-reversal invariant topological insulators at finite temperatures
• Geometric phases of Topological Systems under Quench Process
• The Uhlmann Phase Winding in Bose-Einstein Condensates at Finite Temperature
• Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems
• Comparison of finite-temperature topological indicators based on Uhlmann connection