# Generalized Berry phase for a bosonic Bogoliubov system with exceptional points

## TL;DR

This paper introduces a method to define a generalized Berry phase for bosonic Bogoliubov systems with exceptional points, enabling topological analysis despite non-Hermitian complexities and complex eigenvalues.

## Contribution

It proposes a systematic way to remove exceptional points from the Brillouin zone by adding an imaginary momentum component, allowing for the definition of topological invariants in non-Hermitian bosonic systems.

## Key findings

- Successfully defines a Z2 Berry phase for inversion-symmetric systems.
- Numerical verification of bulk-edge correspondence in toy models.
- Discusses topological invariants like winding number and Z2 invariant in complex eigenvalue regimes.

## Abstract

We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to remove EPs from the Brillouin zone by introducing an imaginary part of the momentum. We then define the Berry phase for a one-dimensional bosonic Bogoliubov system. Extending the argument for Hermitian systems, the Berry phase for an inversion-symmetric system is shown to be $Z_2$. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The $Z_2$ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.