Title: Topological phases of mixed states and their detection
Abstract: What is left of topology at finite temperatures and can topological protection be extended to systems with dissipation? Motivated by topological charge pumps, I will introduce a classification for topological phases of matter applicable to finite-temperature states as well as stationary states of driven, dissipative systems based on a generalization of the many-body polarization. In contrast to quantized charge transport, the polarization can be used to probe topological properties of non-interacting and interacting closed and open systems alike and remains a meaningful quantity at finite T. For non-interacting fermions it defines an ensemble topological phase (ETP), which in the thermodynamic limit is the Zak or Berry phase of a ficticious Hamiltonian given by the covariance matrix of single-particle correlations . As examples, I discuss a Thouless pump in the steady state of one-dimensional lattices driven by Markovian reservoirs  and the finite-temperature Rice-Mele and Harper-Hofstadter models. The ETP winding is shown to be robust against Hamiltonian perturbations as well as homogeneous dephasing and particle losses. I will also discuss a scheme that maps the covariance matrix to the hamiltonian of an auxiliary system of free fermions at T=0, thus allowing to directly observe the ficticious Hamiltonian. Finally I will show that the same scheme can be used to transfer topological properties from an interacting to a non-interacting system.
 C.E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, S.Diehl (arxiv: 1706.02741)
 D. Linzner, L. Wawer, F. Grusdt, M. Fleischhauer, Phys. Rev. B 94, 201105 (2016)