Phonon spectra, quantum geometry, and the Goldstone theorem

Phonons are essential quasiparticles of all crystals and play a key role in fundamental properties such as thermal transport and superconductivity. Acoustic phonons can be interpreted as Goldstone modes that emerge due to the spontaneous breaking of translational symmetry. In this article, we investigate the quantum geometric contribution to the phonon spectrum in the absence of Holstein phonons. Using graphene as a case study, we decompose the dynamical matrix into distinct terms that exhibit different dependencies on the electron energy and wave function. We then examine the role of quantum geometry in shaping the phonon spectrum of the material, and we find that removing the nontrivial quantum geometric contribution from the dynamical matrix causes the acoustic phonon modes to behave in a nonanalytic fashion.