Fidelity and Uhlmann connection analysis of topological phase transitions in two dimensions

TL;DR

This paper investigates how fidelity and Uhlmann connection can detect topological phase transitions in two-dimensional free fermion systems, revealing zero-temperature transitions and the smearing of topological features at finite temperature.

Contribution

It introduces a combined use of fidelity and Uhlmann connection to identify topological phase transitions at zero and finite temperatures in 2D fermionic systems, with analytical and numerical validation.

Findings

Fidelity drops sharply at zero-temperature phase transitions.

Uhlmann connection signals rapid eigenbasis change during transitions.

Topological features are smoothed out at finite temperature, with no finite-temperature phase transitions.

Abstract

We study the behaviour of the fidelity and the Uhlmann connection in two-dimensional systems of free fermions that exhibit non-trivial topological behavior. In particular, we use the fidelity and a quantity closely related to the Uhlmann factor in order to detect phase transitions at zero and finite temperature for topological insulators and superconductors. We show that at zero temperature both quantities predict quantum phase transitions: a sudden drop of fidelity indicates an abrupt change of the spectrum of the state, while the behavior of the Uhlmann connection signals equally rapid change in its eigenbasis. At finite temperature, the topological features are gradually smeared out, indicating the absence of finite-temperature phase transitions, which we further confirm by performing a detailed analysis of the edge states. Moreover, we performed both analytical and numerical analysis of the fidelity susceptibility in the thermodynamic limit, providing an explicit quantitative criterion for the existence of phase transitions. The critical behaviour at zero temperature is further analysed through the numerical computation of critical exponents.