Higgs and Goldstone modes in crystalline solids

In crystalline solids, the acoustic phonon can be described either as a Goldstone or as a non-Abelian gauge boson. However, the non-Abelianity of the related gauge group apparently makes the acoustic phonon a frequency-gapped mode, in contradiction with the other description. In a different perspective, overcoming this contradiction, both acoustic and optical phonon—the latter never appearing following the other two approaches—emerge, respectively, as the gapless Goldstone (phase) and the gapped Higgs (amplitude) fluctuation mode of an order parameter arising from the spontaneous breaking of a global symmetry, without invoking the gauge principle. In addition, the Higgs mechanism describes all the phonon–phonon interactions, including a possible perturbation of the acoustic phonon's frequency dispersion relation induced by the eventual optical phonon, a peculiar behavior that is able to produce mini-gaps inside the phonon Brillouin zone.