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

TL;DR

The paper addresses how many Nambu–Goldstone bosons arise in spontaneously broken, non‑Lorentz invariant systems and proposes an exact counting principle linking NG numbers to conserved charge densities. It introduces the conjecture $n_{\mathrm{BS}}-n_{\mathrm{NG}}=\frac{1}{2}\mathrm{rank}\rho$, where $i\rho_{ab}=\lim_{\Omega\to\infty}(1/\Omega)\langle0|[Q_a,Q_b]|0\rangle$, and provides a one‑sided proof of a bound, along with a physical argument for equality in broad settings. The authors further propose a refined NG boson classification into type‑C (charged) and type‑N (neutral), linking robust type‑II behavior to charge commutators and allowing for nontrivial dispersion relations. This approach connects NG counting to ground‑state charge densities, offering a framework for predicting NG spectra in non‑Lorentz invariant systems and guiding future work on nonuniform symmetries and gauged cases.

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.