Topological phases of mixed states and their detection
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)