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 illdefined, 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 Z2. As concrete examples, we numerically investigate two toy models and confirm the bulkedge correspondence even in the presence of complex eigenvalues. The Z2 invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.arXiv:1904.08724v1 [cond-mat.quant-gas]

No abstract