There are no Goldstone bosons on the Bethe lattice

TL;DR

This paper demonstrates that on the Bethe lattice, symmetry breaking phase transitions do not produce Goldstone bosons, as local correlations remain massive even in the broken phase, due to spectral properties of the graph Laplacian.

Contribution

It reveals the absence of Goldstone bosons on the Bethe lattice during symmetry breaking, linking this to spectral properties of expander graphs.

Findings

Global modes' gap vanishes at criticality

Local correlation functions remain massive

No Goldstone bosons in the spectrum

Abstract

We discuss symmetry breaking quantum phase transitions on the oft studied Bethe lattice in the context of the ferromagnetic scalar spherical model or, equivalently, the infinite $N_f$ limit of ferromagnetic models with $O(N_f)$ symmetry. We show that the approach to quantum criticality is characterized by the vanishing of a gap to just the global modes so that {\it all} local correlation functions continue to exhibit massive behavior. This behavior persists into the broken symmetry phase even as the order parameter develops an expectation value and thus there are no massless Goldstone bosons in the spectrum. We relate this feature to a spectral property of the graph Laplacian shared by the set of `expander' graphs, and argue that our results apply to symmetry breaking transitions on such graphs quite generally.