A recipe for topological observables of density matrices

TL;DR

This paper introduces a pragmatic method to define topological invariants for mixed quantum states using expectation values of many-body operators, maintaining a connection to physical observables and extending known invariants like Berry phases.

Contribution

It proposes a new approach to construct topological invariants for mixed states that does not rely on single-particle wavefunctions, applicable to systems with charge conservation.

Findings

Extends $U(1)$ geometric phases to mixed states.

Provides measurement schemes for the invariants.

Constructs invariants for thermal states of quantum Hall systems.

Abstract

Meaningful topological invariants for mixed quantum states are challenging to identify as there is no unique way to define them, and most choices do not directly relate to physical observables. Here, we propose a simple pragmatic approach to construct topological invariants of mixed states while preserving a connection to physical observables, by continuously deforming known topological invariants for pure (ground) states. Our approach relies on expectation values of many-body operators, with no reference to single-particle (e.g., Bloch) wavefunctions. To illustrate it, we examine extensions to mixed states of $U(1)$ geometric (Berry) phases and their corresponding topological invariant (winding or Chern number). We discuss measurement schemes, and provide a detailed construction of invariants for thermal or more general mixed states of quantum systems with (at least) $U(1)$ charge-conservation symmetry, such as quantum Hall insulators.