Z2 topological invariants for mixed states of fermions in time-reversal invariant band structures

The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting 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 that 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 two dimensions. Here we propose a corresponding Z(2) topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known 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 Z(2) topological insulator, the Kane-Mele model.

Automated summary

The topological classification of fermion systems in mixed states has been a long-standing quest. Observable many-body correlators in mixed states preserve the quantized nature and naturally connect to known topological invariants in the ground state. A Z(2) topological invariant has been proposed for systems with time-reversal symmetry, which is identical to well-known Z(2) invariants for the ground state.