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Topological order, mixed states and open systems. (English) Zbl 1436.81020

Summary: The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states have received attention in the literature only quite recently. In particular, it is still unclear whether the generalisation of the Aharonov-Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse, from a general viewpoint, topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.

MSC:

81P16 Quantum state spaces, operational and probabilistic concepts
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81S22 Open systems, reduced dynamics, master equations, decoherence
81R05 Finite-dimensional groups and algebras motivated by physics and their representations