Inhomogeneous mean-field approach to collective excitations in disordered interacting bosons

We develop an inhomogeneous quantum mean-field approach to the behavior of collective excitations across the superfluid-Mott glass quantum phase transition in two dimensions, complementing recent quantum Monte Carlo simulations [Phys. Rev. Lett. {\bf 125}, 027002 (2020)]. In quadratic (Gaussian) approximation, the Goldstone (phase) and Higgs (amplitude) modes completely decouple. Each is described by a disordered Bogoliubov Hamiltonian which can be solved by an inhomogeneous multi-mode Bogoliubov transformation. We find that the Higgs mode is spatially localized in both phases. The corresponding scalar spectral function shows a broad peak that is noncritical in the sense that its peak frequency does not soften but remains nonzero across the quantum phase transition. In contrast, the lowest-energy Goldstone mode delocalizes in the superfluid phase, leading to a zero-frequency spectral peak. We compare these findings to the results of the quantum Monte Carlo simulations. We also relate them to general results on the localization of bosonic excitations, and we discuss the limits and generality of our approach.