Thermodynamics in the vicinity of a relativistic quantum critical point in 2+1 dimensions

We study the thermodynamics of the relativistic quantum O($N$) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N(T^3/c^2)\calF_N(\Delta/T)$ where $c$ is the velocity of the excitations at the QCP and $\Delta$ is a characteristic zero-temperature energy scale. Using both a large-$N$ approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function $\calF_N$. For small values of $N$ ($N\lesssim 10$) we find that $\calF_N(x)$ is nonmonotonous in the quantum critical regime ($|x|\lesssim 1$) with a maximum near $x=0$. The large-$N$ approach -- if properly interpreted -- is a good approximation both in the renormalized classical ($x\lesssim -1$) and quantum disordered ($x\gtrsim 1$) regimes, but fails to describe the nonmonotonous behavior of $\calF_N$ in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio $\Tkt/\rho_s(0)$ is very close to $\pi/2$, implying that the stiffness $\rho_s(\Tkt^-)$ at the transition is only slightly reduced with respect to the zero-temperature stiffness $\rho_s(0)$. Finally, we briefly discuss the experimental determination of the universal function $\calF_2$ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.