Absence of topology in Gaussian mixed states of bosons

Abstract: In a paper [Bardyn et al. Phys. Rev. X 8, 011035 (2018)], it was shown that the generalization of the many-body polarization to mixed states can be used to construct a topological invariant which is also applicable to finite-temperature and non-equilibrium Gaussian states of lattice fermions. The many-body polarization defines an ensemble geometric phase (EGP) which is identical to the Zak phase of a fictitious Hamiltonian, whose symmetries determine the topological classification. Here we show that in the case of Gaussian states of bosons the corresponding topological invariant is always trivial. This also applies to finite-temperature states of bosons in lattices with a topologically non-trivial band-structure. As a consequence there is no quantized topological charge pumping for translational invariant bulk states of non-interacting bosons.