In general, a low-energy effective theory inherits continuous symmetries of the microscopic model. For this reason, the effective theory of superfluidity of non-relativistic particles should necessarily be invariant under Galilean relativity. Galilean invariance was used long ago by people like London, Landau and Popov for construction of the effective theory of superfluidity.
You are right, the low-energy degree of freedom in a superfluid- the U(1) Goldstone boson has a linear dispersion relation and unfortunately from that some people make a conclusion that the effective theory of superfluidity must be Lorentz invariant. Remember, however that the phonon is charge neutral collective degree of freedom and thus its linear dispersion is not in conflict with Galilean invariance. In other words from the form of the dispersion relation alone one can not conclude whether the theory is Lorentz or Galilean-invariant. What matters is the transformation property of the effective action!
I nice paper where the Galilean symmetry (and its local extension) of the effective theory of superfluidity is exploited is arXiv:cond-mat/0509786.
This post imported from StackExchange Physics at 2014-04-04 16:44 (UCT), posted by SE-user Sergej Moroz