# The Uhlmann connection in fermionic systems undergoing phase transitions

## TL;DR

This paper investigates how the Uhlmann connection detects phase transitions in fermionic systems, including topological insulators, superconductors, and BCS theory, revealing its ability to signal changes in eigenbasis and clarify order parameters.

## Contribution

It demonstrates the role of the Uhlmann connection in identifying phase transitions and clarifies the relevant parameter space for order in mixed states in fermionic systems.

## Key findings

- Uhlmann connection signals eigenbasis changes during phase transitions
- No thermally driven phase transitions in topological insulators and superconductors
- Identifies the parameter space where Uhlmann connection indicates order

## Abstract

We study the behaviour of the Uhlmann connection in systems of fermions undergoing phase transitions. In particular, we analyse some of the paradigmatic cases of topological insulators and superconductors in dimension one, as well as the BCS theory of superconductivity in three dimensions. We show that the Uhlmann connection signals phase transitions in which the eigenbasis of the state of the system changes. Moreover, using the established fidelity approach and the study of the edge states, we show the absence of thermally driven phase transitions in the case of topological insulators and superconductors. We clarify what is the relevant parameter space associated with the Uhlmann connection so that it signals the existence of order in mixed states.