$\mathbb{Z}_2$ topological invariants for mixed states of fermions in time-reversal invariant band structures

TL;DR

This paper introduces a $$2 topological invariant for mixed states of fermions in time-reversal invariant systems, connecting it to known ground state invariants and demonstrating its application in finite-temperature topological insulators.

Contribution

It proposes a new $$2 topological invariant for mixed states with TR symmetry, extending topological classification to finite-temperature fermionic systems.

Findings

The invariant matches the ground state $$2 invariant for the fictitious Hamiltonian.

Application to Kane-Mele model confirms the invariant's validity at finite temperature.

Connects observable correlators to topological invariants in mixed states.

Abstract

The topological classification of fermion systems in mixed states is a long standing quest. For Gaussian states, reminiscent of non-interacting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems which break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in $(1+1)$ and $2$ dimensions. Here we propose a corresponding $\mathbb{Z}_2$ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known $\mathbb{Z}_2$ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic $\mathbb{Z}_2$ topological insulator, the Kane-Mele model.