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

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.

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• Many-Body Open Quantum Systems
• Topological phase transitions at finite temperature
• Role of Bath-Induced Many-Body Interactions in the Dissipative Phases of the Su-Schrieffer-Heeger Model
• Probing the Topology of Fermionic Gaussian Mixed States with {U(1)} symmetry by Full Counting Statistics
• Proxy ensemble geometric phase and proxy index of time-reversal invariant topological insulators at finite temperatures
• Finite-temperature topological invariant for higher-order topological insulators
• Symmetry classification correspondence between quadratic Lindbladians and their steady states