Phonon Dispersion Relation in Weakly Coupled Bose Gas Beyond Bogoliubov Approximation

We study the quasiparticle dispersion relation for a homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. Inspired by the Bogoliubov method, we rearrange the usual Bose gas Hamiltonian in a form adapted to perturbation theory in the weak-coupling regime, both in the canonical and grand-canonical setting. We compute the first non-trivial corrections to the dispersion relation in both settings. We show that for small momenta they are linear in the absolute value of the momentum. Thus, in this approximation phonons have a well-defined velocity, which we compute.