Thermal Uhlmann-Chern Number: Bridging Pure and Mixed States

TL;DR

This paper introduces the thermal Uhlmann-Chern number, a new topological invariant for mixed quantum states at finite temperature, bridging the gap between pure and mixed state topological properties with rigorous proofs and practical models.

Contribution

It proposes the thermal Uhlmann-Chern number, generalizing the Chern number to mixed states, and provides mathematical proof of its convergence to pure-state invariants at zero temperature.

Findings

Thermal Uhlmann-Chern number reduces to the Chern number at zero temperature.

It vanishes at infinite temperature, reflecting the loss of topological order.

Applications demonstrate temperature-dependent topological behavior in various models.

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.