Comparison of finite-temperature topological indicators based on Uhlmann connection

TL;DR

This paper compares two finite-temperature topological indicators based on the Uhlmann connection for time-reversal invariant topological insulators, highlighting how the Uhlmann phase remains quantized and serves as a robust topological marker at finite temperatures.

Contribution

The paper explicitly constructs and compares the Uhlmann-Wilson loop and Uhlmann phase as finite-temperature topological indicators for $Z_2$ topological insulators, revealing their behaviors with temperature.

Findings

Uhlmann-Wilson loop signature fades with increasing temperature.

Uhlmann phase remains quantized and indicates topological properties at finite temperature.

Phase diagram shows where Uhlmann phase jumps occur.

Abstract

Two indicators of finite-temperature topological properties based on the Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop and the other generalizing the Berry phase to the Uhlmann phase, are constructed explicitly for a time-reversal invariant topological insulators with a $Z_2$ index. While the phases of the eigenvalues of the Wilson loop reflect the $Z_2$ index of the model at zero temperature, it is found that the signature from the Uhlmann-Wilson loop gradually fades away as temperature increases. On the other hand, the Berry phase exhibits quantization due to the underlying holonomy group. The Uhlmann phase retains the quantization at finite temperatures and serves as an indicator of topological properties. A phase diagram showing where jumps of the Uhlmann phase can be found is presented. By modifying the model to allow higher winding numbers, finite-temperature topological regimes sandwiched between trivial regimes at high and low temperatures may emerge.