Elementary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition

We investigate the zero-temperature excitation spectrum of two-dimensional soft-core bosons for a wide range parameters and across the phase transition from a superfluid to a supersolid state. Based on mean field calculations and recent Quantum Monte Carlo results, we demonstrate the applicability of the Bogoliubov-de Gennes equations, even at high interaction strengths where the system forms an insulating cluster crystal. Interestingly, our study reveals that the maximum energy of the longitudinal phonon band in the supersolid phase connects to the maxon energy of the superfluid at the phase transition.