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.