Abstract
The mixedness of a quantum state is usually seen as an adversary to topological quantization of observables. For example, exact quantization of the charge transported in a so-called Thouless adiabatic pump is lifted at any finite temperature in symmetry-protected topological insulators. Here, we show that certain directly observable many-body correlators preserve the integrity of topological invariants for mixed Gaussian quantum states in one dimension. Our approach relies on the expectation value of the many-body momentum-translation operator, and leads to a physical observable --- the "ensemble geometric phase" (EGP) --- which represents a bona fide geometric phase for mixed quantum states, in the thermodynamic limit. In cyclic protocols, the EGP provides a topologically quantized observable which detects encircled spectral singularities ("purity-gap" closing points) of density matrices. While we identify the many-body nature of the EGP as a key ingredient, we propose a conceptually simple, interferometric setup to directly measure the latter in experiments with mesoscopic ensembles of ultracold atoms.
Highlights
Topology has emerged as an important paradigm in the classification of ground states in many-particle quantum systems
III, we investigate the geometric nature of the ensemble geometric phase” (EGP) in the context of noninteracting systems, focusing on mixed states generically described by a Gaussian density matrix ρ ∼ e−G. (We consider translationally invariant lattice systems, for convenience.) All
We have shown that density matrices describing mixed fermionic Gaussian states in one dimension encode topological information in a way that enables a direct interpretation in terms of a physical observable
Summary
Topology has emerged as an important paradigm in the classification of ground states in many-particle quantum systems. The homotopy classes characterizing the map α ↦ jψαi can be described in terms of a UðnÞ Berry connection or gauge field ðAiÞss0 ≡ ihψ α;sj∂αi ψ α;si, where n is the number of bands (indexed by s) composing the ground state In this picture, topological invariants can be understood as Chern classes of this gauge field (or quantities that depend on the latter [13]). Where a direct connection to an observable exists, high levels of stability, e.g., with regard to disorder or particle interactions, are to be expected All these concepts are beautifully exemplified in the Rice-Mele model—a paradigmatic model for noninteracting topological insulators in 1D [14] The Zak phase and the corresponding topological invariant are related to physical observables, i.e., to the zero-temperature (ground-state) polarization, and to the associated current flow
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