Droplet-Edge Operators in Nonrelativistic Conformal Field Theories

TL;DR

This paper analyzes the edge operators in nonrelativistic conformal field theories at large charge, refining the effective field theory approach and identifying universal contributions to the ground state energy.

Contribution

It provides a systematic operator analysis of edge contributions and counterterms, revealing new universal terms and clarifying the structure of large-charge expansions.

Findings

Reproduces known NLO ground state energy using dimensional regularization.

Identifies a new edge contribution to operator dimensions of order Q^{2/3 - 1/d}.

Establishes universality of the order Q^0 term in 2D.

Abstract

We consider the large-charge expansion of the charged ground state of a Schrodinger-invariant, nonrelativistic conformal field theory in a harmonic trap, in general dimension d. In the existing literature, the energy in the trap has been computed to next-to-leading order (NLO) at large charge Q, which comes from the classical contribution of two higher-derivative terms in the effective field theory. In this note, we explain the structure of operators localized at the edge of the droplet, where the density drops to zero. We list all operators contributing to the ground-state energy with nonnegative powers of Q in the large-Q expansion. As a test, we use dimensional regularization to reproduce the calculation of the NLO ground state energy by Kravec and Pal , and we recover the same universal coefficient for the logarithmic term as in that work. We refine the derivation by presenting a systematic operator analysis of the possible edge counterterms, showing that different choices of cutoff procedures must yield the same renormalized result up to an enumerable list of Wilson coefficients for conformally invariant local counterterms at the droplet edge. We also demonstrate the existence of a previously unnoticed edge contribution to the ground-state operator dimension of order Q^{{2\over 3} - {1\over d}} in d spatial dimensions. Finally, we show there is no bulk or edge counterterm scaling as Q^0 in two spatial dimensions, which establishes the universality of the order Q^0 term in large-Q expansion of the lowest charged operator dimension in d=2.