Criterion for stability of Goldstone Modes and Fermi Liquid behavior in a metal with broken symmetry

There are few general physical principles that protect the low energy excitations of a quantum phase. Of these, Goldstone's theorem and Landau Fermi liquid theory are the most relevant to solids. We investigate the stability of the resulting gapless excitations - Nambu Goldstone bosons (NGBs) and Landau quasiparticles - when coupled to one another, which is of direct relevance to metals with a broken continuous symmetry. Typically, the coupling between NGBs and Landau quasiparticles vanishes at low energies leaving the gapless modes unaffected. If however the low energy coupling is non-vanishing, non-Fermi liquid behavior and overdamped bosons are expected. Here we prove a general criterion which specifies when the coupling is non-vanishing. It is satisfied by the case of a nematic Fermi fluid, consistent with earlier microscopic calculations. In addition, the criterion identifies a new kind of symmetry breaking - of magnetic translations - where non-vanishing couplings should arise, opening a new route to realizing non-Fermi liquid phases.