  Recent advances in cooling techniques make now possible the experimental
study of quantum phase transitions, which are transitions near absolute zero
temperature accessed by varying a control parameter. A paradigmatic example is
the superfluid-Mott transition of interacting bosons on a periodic lattice.
From the relativistic Ginzburg-Landau action of this superfluid-Mott transition
we derive the elementary excitations of the bosonic system, which contain in
the superfluid phase a gapped Higgs mode and a gappless Goldstone mode. We show
that this energy spectrum is in good agreement with the available experimental
data and we use it to extract, with the help of dimensional regularization,
meaningful analytical formulas for the beyond-mean-field equation of state in
two and three spatial dimensions. We find that, while the mean-field equation
of state always gives a second-order quantum phase transition, the inclusion of
Gaussian quantum fluctuations can induce a first-order quantum phase
transition. This prediction is a strong benchmark for next future experiments
on quantum phase transitions.
