A relativistic non-relativistic Goldstone theorem: gapped Goldstones at finite charge density

TL;DR

This paper extends the Goldstone theorem to relativistic theories at finite charge density, revealing that certain spontaneously broken symmetries lead to gapped excitations instead of gapless modes, with exact gap formulas derived.

Contribution

It introduces a relativistic perspective to the Goldstone theorem at finite density, showing that non-commuting broken charges can produce gapped Goldstone modes.

Findings

Gapped Goldstone modes occur when broken charges do not commute.

Exact expressions for the gaps are derived in terms of chemical potential and symmetry algebra.

Relativistic dynamics are crucial for understanding symmetry breaking at finite density.

Abstract

We adapt the Goldstone theorem to study spontaneous symmetry breaking in relativistic theo- ries at finite charge density. It is customary to treat systems at finite density via non-relativistic Hamiltonians. Here we highlight the importance of the underlying relativistic dynamics. This leads to seemingly new results whenever the charge in question is spontaneously broken and does not commute with other broken charges. We find that that the latter interpolate gapped excitations. In contrast, all existing versions of the Goldstone theorem predict the existence of gapless modes. We derive exact non-perturbative expressions for their gaps, in terms of the chemical potential and of the symmetry algebra.