Uhlmann number in translational invariant systems

TL;DR

This paper introduces the Uhlmann number as a finite-temperature topological invariant for 2D translational invariant fermionic systems, linking it to measurable physical quantities like susceptibility and conductivity.

Contribution

It extends the concept of the Chern number to finite temperatures using the Uhlmann number and connects it to physical response functions.

Findings

Uhlmann number effectively describes topology at finite temperature.

Studied two paradigmatic systems to observe topological changes.

Linked geometrical quantities to measurable physical responses.

Abstract

We define the Uhlmann number as an extension of the Chern number, and we use this quantity to describe the topology of 2D translational invariant Fermionic systems at finite temperature. We consider two paradigmatic systems and we study the changes in their topology through the Uhlmann number. Through the linear response theory we linked two geometrical quantities of the system, the mean Uhlmann curvature and the Uhlmann number, to directly measurable physical quantities, i.e. the dynamical susceptibility and to the dynamical conductivity, respectively.