Glass to superfluid transition in dirty bosons on a lattice

We investigate the interplay between disorder and interactions in a Bose gas on a lattice in presence of randomly localized impurities. We compare the performance of two theoretical methods, the simple version of multi-orbital Hartree-Fock and the common Gross-Pitaevskii approach, showing how the former gives a very good approximation to the ground state in the limit of weak interactions, where the superfluid fraction is small. We further prove rigorously that for this class of disorder the fractal dimension of the ground state d* tends to the physical dimension in the thermodynamic limit. This allows us to introduce a quantity, the fractional occupation, which gives insightful information on the crossover from a Lifshits to a Bose glass. Finally, we compare temperature and interaction effects, highlighting similarities and intrinsic differences.