Equilibrium Partition Function of Non-Relativistic CFTs in Harmonic Trap

TL;DR

This paper analyzes the equilibrium partition function of non-relativistic conformal field theories in harmonic traps, revealing universal structures and pole behaviors related to angular velocities and chemical potentials, with applications to cold-atom systems.

Contribution

It provides a detailed analysis of the partition function's structure in non-relativistic CFTs under harmonic confinement, including universal features and behavior in large-angular-momentum limits.

Findings

Partition function exhibits simple poles in -\u03a9_a^2, with residues determined by (/T).

Universal structure in the hydrodynamic regime at leading order.

Analysis of fermions at unitarity in harmonic traps relevant to cold-atom experiments.

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 $\omega^2-\Omega_a^2$, 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.