Spontaneous breaking of continuous translational invariance

TL;DR

This paper develops a framework treating translational invariance as a symmetry subject to spontaneous breaking, extending existing theorems and analyzing Nambu-Goldstone modes in a supersolid.

Contribution

It introduces a unified approach to spontaneous symmetry breaking, including translational invariance, and clarifies the nature of Nambu-Goldstone modes in supersolids.

Findings

Extended SSB theorems to translational invariance

Distinct Nambu-Goldstone modes identified in supersolids

Clarified the physical nature of phonons and internal symmetry modes

Abstract

Unbroken continuous translational invariance is often taken as a basic assumption in discussions of spontaneous symmetry breaking (SSB), which singles out SSB of translational invariance itself as an exceptional case. We present a framework which allows us to treat translational invariance on the same footing as other symmetries. It is shown that existing theorems on SSB can be straightforwardly extended to this general case. As a concrete application, we analyze the Nambu-Goldstone modes in a (ferromagnetic) supersolid. We prove on the ground of the general theorems that the Bogoliubov mode stemming from a spontaneously broken internal U(1) symmetry and the longitudinal phonon due to a crystalline order are distinct physical modes.