Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap

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 $\omega^2-\Omega_a^2$, where $\omega$ is the harmonic trapping frequency and $\Omega_a$ are angular velocities acting as chemical potentials for angular momentum. The corresponding residue is determined by a single-variable function of $\mu/T$, with $\mu$ the particle-number chemical potential and $T$ the temperature. We then study the large-angular-momentum limit $\Omega_a\to\omega$. In this regime centrifugal effects nearly cancel the trapping potential, and the logarithm of the partition function again exhibits simple poles in $\omega^2-\Omega_a^2$, but with a less universal residue depending separately on $\mu/T$ and $\omega/T$. As explicit examples we analyze superfluid systems realizable in cold-atom experiments, in particular fermions at unitarity confined in a harmonic trap.