  We study the low-energy excitations of a bosonic lattice gas with
cavity-mediated interactions. By performing two successive Hubbard-Stratonovich
transformations, we derive an effective field theory to study the
strongly-coupling regime. Taking into account the quantum fluctuation, we
report the unusual effect of the superradiant cavity light induced density
imbalance, which has been shown to have a negligible effect on the single
particle excitation in the previous studies. Instead, we show that such
negligible fluctuation of density imbalance dramatically changes the behavior
of the low-energy excitation and results in a free switching between two types
of Goldstone modes in its single particle excitation, i.e., type I and type II
with odd and even power energy-momentum dispersion, respectively. Our proposal
would open a new horizon for manipulating Goldstone modes from bridging the
cavity light and strongly interacting quantum matters.
