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

Abstract

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 $P(T)=P(0)+N({T}^{3}/{c}^{2}){\mathcal{F}}_{N}(\ensuremath{\Delta}/T)$, where $c$ is the velocity of the excitations at the QCP and $|\ensuremath{\Delta}|$ 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 ${\mathcal{F}}_{N}$. For small values of $N$ ($N\ensuremath{\lesssim}10$) we find that ${\mathcal{F}}_{N}(x)$ is nonmonotonic in the quantum critical regime ($|x|\ensuremath{\lesssim}1$) with a maximum near $x=0$. The large-$N$ approach---if properly interpreted---is a good approximation both in the renormalized classical ($x\ensuremath{\lesssim}\ensuremath{-}1$) and quantum disordered ($x\ensuremath{\gtrsim}1$) regimes, but fails to describe the nonmonotonic behavior of ${\mathcal{F}}_{N}$ 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 ${T}_{\mathrm{BKT}}/{\ensuremath{\rho}}_{s}(0)$ is very close to $\ensuremath{\pi}/2$, implying that the stiffness ${\ensuremath{\rho}}_{s}({T}_{\mathrm{BKT}}^{\ensuremath{-}})$ at the transition is only slightly reduced with respect to the zero-temperature stiffness ${\ensuremath{\rho}}_{s}(0)$. Finally, we briefly discuss the experimental determination of the universal function ${\mathcal{F}}_{2}$ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.