Consistent Field Theory Across the Mott-Insulator to Superfluid Transition

We employ a field-theoretical approach to analyze the Bose-Hubbard model on a lattice, with a focus on the low-energy properties across the Mott insulator (MI) to superfluid (SF) transition. Prior approaches approximated the partition function using cumulant expansions around the MI ground state, which, while accurate in the MI phase, lead to inaccuracies in the SF phase where the MI state is a false ground state. By expanding around the correct mean-field vacuum, we derive the effective field theory (EFT) governing the MI to SF transition. Through this, we reveal the underlying structure of the EFT governing the nucleation of low-energy excitations, particularly the massless and lowest massive modes, offering new insights into their emergence.