  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 had received only quite recently attention in the literature. 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 a periodic symmetry and their evolution under the influence of an
environment preserving its crystalline symmetries.
