Thermal Uhlmann-Chern Number: Bridging Pure and Mixed States

• URL: http://arxiv.org/abs/2506.18022v1
• Date: Sun, 22 Jun 2025 13:03:25 GMT
• Title: Thermal Uhlmann-Chern Number: Bridging Pure and Mixed States
• Abstract summary: We introduce the thermal Uhlmann-Chern number, a generalization of the Chern number, to characterize the topological properties of mixed states.<n>We provide a rigorous mathematical proof that the first- and higher-order Uhlmann-Chern numbers converge to the corresponding Chern numbers in the zero-temperature limit.<n>Our results establish a robust bridge between the topological properties of pure and mixed states, offering new insights into finite-temperature topological phases.
• Score: 7.3022646129024595
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Topological properties of quantum systems at finite temperatures, described by mixed states, pose significant challenges due to the triviality of the Uhlmann bundle. We introduce the thermal Uhlmann-Chern number, a generalization of the Chern number, to characterize the topological properties of mixed states. By inserting the density matrix into the Chern character, we introduce the thermal Uhlmann-Chern number, a generalization of the Chern number that reduces to the pure-state value in the zero-temperature limit and vanishes at infinite temperature, providing a framework to study the temperature-dependent evolution of topological features in mixed states. We provide, for the first time, a rigorous mathematical proof that the first- and higher-order Uhlmann-Chern numbers converge to the corresponding Chern numbers in the zero-temperature limit, differing only by a factor of $1/D$ for $D$-fold degenerate ground states. We demonstrate the utility of this framework through applications to a two-level system, the coherent state model, the 2D Haldane model, and a four-band model, highlighting the temperature-dependent behavior of topological invariants. Our results establish a robust bridge between the topological properties of pure and mixed states, offering new insights into finite-temperature topological phases.

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