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Topological order has become a new paradigm to distinguish ground states ofinteracting many-body systems without conventional long-range order. Here wediscuss possible extensions of this concept to density matrices describingstatistical ensembles. For a large class of quasi-thermal states, which can berealized as thermal states of some quasi-local Hamiltonian, we generalizeearlier definitions of density matrix topology to generic many-body systemswith strong correlations. We point out that the robustness of topologicalorder, defined as a pattern of long-range entanglement, depends crucially onthe perturbations under consideration. While it is intrinsically protectedagainst local perturbations of arbitrary strength in an infinite closed quantumsystem, purely local perturbations can destroy topological order in opensystems coupled to baths if the coupling is sufficiently strong. We discuss ourclassification scheme using the finite-temperature quantum Hall states andpoint out that the classical Hall effect can be understood as a finitetemperature topological phase.

Topological order of mixed states in quantum many-body systems