Universal thermodynamics of the superfluid to Mott insulator transition

The superfluid--Mott-insulator transition of a Bose gas in an optical lattice, when it occurs at constant density, belongs to the universality class of the quantum XY model. We discuss the thermodynamics of the two-dimensional quantum O(N) model for $N\geq 2$ in the vicinity of its zero-temperature quantum critical point, and in particular the universal scaling function ${\cal F}_N$ which determines the pressure $P(T)$. We show that the large-$N$ approach is unable to predict the (non-monotonuous) shape of ${\cal F}_N$ for $N < 10$, but ${\cal F}_N$ can be computed from a renormalization-group approach. Finally, we discuss the experimental determination of the scaling function ${\cal F}_2$ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.