# Spontaneous Breaking of Non-Relativistic Scale Symmetry

## TL;DR

This paper investigates how spontaneous breaking of scale invariance occurs in Galilean invariant field theories, revealing conditions for gapless modes and their dispersion relations, and discussing implications for RG flow reversibility.

## Contribution

It provides a detailed analysis of the conditions under which spontaneous symmetry breaking leads to gapless modes in non-relativistic theories, highlighting the roles of particle number and boost symmetries.

## Key findings

- Presence of a gapless Nambu-Goldstone mode when both scale and particle number symmetries are broken.
- The dispersion relation of the mode is linear with nonzero chemical potential, quadratic otherwise.
- Discussion on the reversibility of RG flows in these symmetry-breaking scenarios.

## Abstract

We analyze the mechanism of spontaneous symmetry breaking of scale invariance in Galilean invariant field theories. We show that the existence of a dynamic gapless dilaton mode depends on whether the $U(1)$ particle number or the Galilean boost symmetry are spontaneously broken. When both scale and particle number symmetries are spontaneously broken there is one propagating gapless Nambu-Goldstone mode. Its dispersion relation is linear if the chemical potential is nonzero and quadratic otherwise. We discuss the reversibility of RG flows in such theories.