Pseudo-time-reversal-symmetry-protected topological Bogoliubov excitations of Bose-Einstein condensates in optical lattices

TL;DR

This paper demonstrates that Bogoliubov excitations in Bose-Einstein condensates within optical lattices can exhibit topological properties protected by a pseudo-time-reversal symmetry, with a new Krein-space formalism applicable to bosonic topological bands.

Contribution

It introduces a Krein-space theoretical framework for topological bosonic Bogoliubov bands, revealing a pseudo-time-reversal symmetry protecting their topological nature, and provides practical expressions for the associated invariant.

Findings

Topological invariants characterized by a $ ext{Z}_2$ index.

Numerical confirmation of bulk-boundary correspondence in toy models.

Development of a universal Krein-space formalism for symmetry-protected bosonic topological bands.

Abstract

Bogoliubov excitations of Bose-Einstein condensates in optical lattices may possess band topology in analogous to topological insulators in class AII of fermions. Using the language of the Krein-space theory, this topological property is shown to be protected by a pseudo-time-reversal symmetry that is pseudo-antiunitary and squares to $-1$, with the associated bulk topological invariant also being a $\mathbb Z_2$ index. We construct three equivalent expressions for it, relating to the Pfaffian, the pseudo-time-reversal polarization, and most practically, the Wannier center flow, all adopted from the fermionic case, defined here with respect to the pseudo inner product. In the presence of an additional pseudo-unitary and pseudo-Hermitian inversion symmetry, a simpler expression is derived. We then study two toy models feasible on cold atom platforms to numerically confirm the bulk-boundary correspondence. The Krein-space approach developed in this work is a universal formalism to study all kinds of symmetry-protected topological bosonic Bogoliubov bands.