On the number of Nambu-Goldstone bosons and its relation to charge densities

TL;DR

This paper investigates the relationship between the number of Nambu-Goldstone bosons and broken symmetry generators in quantum systems, proposing a new classification and providing an example where existing inequalities are not saturated.

Contribution

It introduces a novel example where the Nielsen-Chadha inequality is not saturated and proposes a refined counting rule for NG bosons based on charge densities.

Findings

Example system where Nielsen-Chadha inequality is not saturated

New classification scheme for NG bosons

Relation between NG boson count and charge densities

Abstract

The low-energy physics of systems with spontaneous symmetry breaking is governed by the associated Nambu-Goldstone (NG) bosons. While NG bosons in Lorentz-invariant systems are well understood, the precise characterization of their number and dispersion relations in a general quantum many-body system is still an open problem. An inequality relating the number of NG bosons and their dispersion relations to the number of broken symmetry generators was found by Nielsen and Chadha. In this paper, we give a presumably first example of a system in which the Nielsen-Chadha inequality is actually not saturated. We suggest that the number of NG bosons is exactly equal to the number of broken generators minus the number of pairs of broken generators whose commutator has a nonzero vacuum expectation value. This naturally leads us to a proposal for a different classification of NG bosons.