# Absence of topology in Gaussian mixed states of bosons

## TL;DR

This paper demonstrates that Gaussian states of bosons lack non-trivial topological invariants, implying no quantized topological charge pumping occurs in such systems, unlike fermionic counterparts.

## Contribution

It shows that the topological invariant based on many-body polarization is always trivial for bosonic Gaussian states, contrasting with fermionic systems.

## Key findings

- Bosonic Gaussian states have trivial topological invariants.
- No quantized topological charge pumping occurs in bosonic systems.
- Finite-temperature bosonic lattices do not exhibit topological non-triviality.

## Abstract

In a recent 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.