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In a recent paper [Bardyn et al. Phys. Rev. X 8, 011035 (2018)], it was shownthat the generalization of the many-body polarization to mixed states can beused to construct a topological invariant which is also applicable tofinite-temperature and non-equilibrium Gaussian states of lattice fermions. Themany-body polarization defines an ensemble geometric phase (EGP) which isidentical to the Zak phase of a fictitious Hamiltonian, whose symmetriesdetermine the topological classification. Here we show that in the case ofGaussian states of bosons the corresponding topological invariant is alwaystrivial. This also applies to finite-temperature states of bosons in latticeswith a topologically non-trivial band-structure. As a consequence there is noquantized topological charge pumping for translational invariant bulk states ofnon-interacting bosons.

Absence of topology in Gaussian mixed states of bosons