Probing the Topology of Density Matrices

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 that 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.

Tenfold Way for Quadratic Lindbladians

“…In this section, we demonstrate that the robust spectral features of quadratic Lindbladians discussed in the main text are independent of properties of the steady state, the latter of which was the subject of study in Refs. [20][21][22][23]. Specifically, we show that any system with some topological features in its spectrum can be continuously deformed to a system with the same spectrum, but a trivial (infinite temperature) steady state, while at all times maintaining the relevant spectral gaps, symmetries, and locality of the equations of motion.…”

“…For example, these universal topological properties of the complex excitation spectrum -which have direct physical consequences in spectroscopy -are unconnected to the classification of steady-state density matrices employed in Refs. [20][21][22][23]. Our work highlights the various manifestations of band topology in a very general class of exactly solvable open systems, and provides formalisms which can be applied to understand generic interacting systems in future work.…”

Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states

“…However, Eq. (36) implies that all eigenvalues of ρ −1H √ ρ are real. Thus, we reach a contradiction.…”

Absence of topology in Gaussian mixed states of bosons

“…In a recent paper [18], it was shown that the winding of the many-body polarization introduced by Resta [27] upon a closed path in parameter space is an alternative and useful many-body topological invariant for Gaussian states of fermions. The polarization of a non-degenerate ground-state |ψ corresponding to a filled band of a lattice Hamiltonian with periodic boundary conditions is the phase (in units of 2π) induced by a momentum shift T P = 1 2π Im log ψ T ψ .…”

“…However, φ EGP remains well defined and meaningful for arbitrarily large but finite systems [18] as long as the so-called purity gap of ρ does not close. Furthermore as shown in [18] the EGP of a Gaussian density matrix is reduced to the ground-state Zak phase of a fictitious Hamiltonian in the thermodynamic limit L → ∞. The symmetries of this fictitious Hamiltonian determine the topological classification [12] following the scheme of Altland and Zirnbauer [33][34][35].…”