Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons

On top of the mean-field analysis of a Bose-Einstein condensate, one typically applies the Bogoliubov theory to analyze quantum fluctuations of the excited modes. Therefore, one has to diagonalize the Bogoliubov Hamiltonian in a symplectic manner. In our article we investigate the topology of these Bogoliubov excitations in inversion-invariant systems of interacting bosons. We analyze how the condensate influences the topology of the Bogoliubov excitations. Analogously to the fermionic case, here we establish a symplectic extension of the polarization characterizing the topology of the Bogoliubov excitations and link it to the eigenvalues of the inversion operator at the inversion-invariant momenta. We also demonstrate an instructive but experimentally feasible example that this quantity is also related to edge states in the excitation spectrum.