Collective modes in Bogoliubov Fermi surfaces without gapless Goldstone modes

We study collective modes of a finite temperature Bogoliubov Fermi surface with no symmetries. The Bogoliubov Fermi surface is assumed to form in a superconductor with Coulomb interactions, so that the plasmon is gapped and there is no low lying Goldstone mode in the system. Despite this we find the existence of a gapless second sound, which is a collective mode of energy and momentum fluctuations that is not related to any Goldstone mode. This is different from previously discussed second sound modes in either superfluids or crystals, where the second sound co-exists with and is related to a spontaneously broken symmetry with a gapless Goldstone mode. Thus these modes are an example of collective modes that are solely related to conservation of energy/momentum.