Simplest phonons and pseudo-phonons in field theory

We study the emergence of Nambu–Goldstone modes due to broken translation symmetry in field theory. Purely spontaneous breaking yields a massless phonon which develops a mass upon introducing a perturbative explicit breaking. The pseudo-phonon mass agrees with Gell Mann-Oakes-Renner relations. We analyze the simplest possible theories featuring gradient Mexican hats and describing space-dependent order parameters. We comment on homogeneous translation breaking and the connections with holographic Q-lattices.