Operator spectrum of nonrelativistic CFTs at large charge

TL;DR

This paper refines the large-charge expansion for nonrelativistic conformal field theories by systematically addressing divergences at the cloud edge, introducing counterterms that yield finite operator dimensions and impacting ultracold atom physics.

Contribution

It extends the large-charge expansion framework by analyzing divergences and constructing counterterms, providing more accurate operator dimension calculations in nonrelativistic CFTs.

Findings

Existence of appropriate edge counterterms for finiteness

Refined corrections to the Thomas-Fermi approximation

Implications for ultracold atom experiments

Abstract

We extend and clarify the large-charge expansion of the conformal dimension $\Delta_Q$ of the lowest operator of charge $Q$ in nonrelativistic CFTs using the state-operator correspondence. The latter requires coupling the theory to an external harmonic trap that confines the particles to a spherical cloud, at the edge of which the effective theory breaks down and leads to divergences. Only recently has this issue been overcome by constructing appropriate counterterms at the edge of the cloud in arXiv:2010.07967. In this note, we extend these results by systematically analyzing the degree of divergence of operators in the effective action and show that there always exist appropriate edge counterterms that make the final contributions to $\Delta_Q$ finite. On the other side of the correspondence, this also provides new corrections to the Thomas-Fermi approximation of the unitary Fermi gas, and we comment on their relevance for ultracold atom physics.