More on gapped Goldstones at finite density: more gapped Goldstones

It was recently argued that certain relativistic theories at finite density can exhibit an unconventional spectrum of Goldstone excitations, with gapped Goldstones whose gap is exactly calculable in terms of the symmetry algebra. We confirm this result as well as previous ones concerning gapless Goldstones for non-relativistic systems via a coset construction of the low-energy effective field theory. Moreover, our analysis unveils additional gapped Goldstones, naturally as light as the others, but this time with a model-dependent gap. Their exact number cannot be inferred solely from the symmetry breaking pattern either, but rather depends on the details of the symmetry breaking mechanism - a statement that we explicitly verify with a number of examples. Along the way we provide what we believe to be a particularly transparent interpretation of the so-called inverse-Higgs constraints for spontaneously broken spacetime symmetries.