Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems

TL;DR

This paper proposes using the Uhlmann geometric phase to identify topological phases in 1D fermion systems at finite temperatures, revealing a temperature-dependent phase transition.

Contribution

It introduces the Uhlmann phase as a novel topological invariant applicable to mixed states and finite temperature conditions in 1D fermion systems.

Findings

Uhlmann phase detects topological phases at finite temperature.

Critical temperature T_c marks a topological phase transition.

At low temperatures, the Uhlmann phase aligns with traditional topological invariants.

Abstract

We introduce the Uhlmann geometric phase as a tool to characterize symmetry-protected topological phases in 1D fermion systems, such as topological insulators and superconductors. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. We illustrate these ideas with some paradigmatic models and find that there exists a critical temperature $T_c$ at which the Uhlmann phase goes discontinuously and abruptly to zero. This stands as a borderline between two different topological phases as a function of the temperature. Furthermore, at small temperatures we recover the usual notion of topological phase in fermion systems.