Simplest phonons and pseudo-phonons in field theory

Understand

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.