Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap

Abstract

We investigate the equilibrium partition function of non-relativistic conformal field theories in harmonic quantization. We first analyze the hydrodynamic regime and show that, at leading order, the partition function exhibits a universal structure determined by the equation of state: the logarithm of the partition function develops simple poles in $\ddot{I}{‰}^2-\hat{I}{\text{c◯}}_a^2$, where $\ddot{I}‰$ is the harmonic trapping frequency and $\hat{I}{\text{c◯}}_a$ are angular velocities acting as chemical potentials for angular momentum. The corresponding residue is determined by a single-variable function of $\hat{I}¼/T$, with $\hat{I}¼$ the particle-number chemical potential and $T$ the temperature. We then study the large-angular-momentum limit $\hat{I}{\text{c◯}}_a\to \ddot{I}‰$. In this regime centrifugal effects nearly cancel the trapping potential, and the logarithm of the partition function again exhibits simple poles in $\ddot{I}{‰}^2-\hat{I}{\text{c◯}}_a^2$, but with a less universal residue depending separately on $\hat{I}¼/T$ and $\ddot{I}‰/T$. As explicit examples we analyze superfluid systems realizable in cold-atom experiments, in particular fermions at unitarity confined in a harmonic trap.

34 pages