Phonons as Goldstone Bosons

TL;DR

The paper treats phonons as Goldstone bosons of spontaneously broken translation symmetry in solids and develops an effective field theory to describe their low-energy dynamics. Using a derivative expansion and a covariant formulation based on body-fixed coordinates, it shows that current algebra and Ward identities necessarily induce phonon-phonon interactions, constraining the Lagrangian to three independent couplings $L_1,L_2,L_3$ at leading nontrivial order. Through explicit matching of conservation laws and current commutators, the author derives expressions that relate higher-order couplings $l_1$–$l_6$ to the $L_i$, while $l_7$–$l_9$ vanish, mirroring the structure of low-energy $\pi\pi$ scattering. The result demonstrates that nonlinear propagation of sound is an intrinsic consequence of the symmetry structure of solids, reinforcing the effective field theory perspective in condensed matter.

Abstract

The implications of the hidden, spontaneously broken symmetry for the properties of the sound waves of a solid are analyzed. Although the discussion does not go beyond standard wisdom, it presents some of the known results from a different perspective. In particular, I argue that, as a consequence of the hidden symmetry, the equations of motion for a sound wave necessarily contain nonlinear terms, describing phonon-phonon scattering and emphasize the analogy with the low energy theorems for pion-pion scattering.