What is a superfluid in field theoretic terms?

Question

I'm wondering how one precisely defines a superfluid in terms of the effective field theory description. In Nicolis's paper http://arxiv.org/abs/1108.2513 there seems to be an extremely simple characterization of the superfluid: it is just a scalar field $\psi$ with a nonlinearly realized $U(1)$ shift symmetry, $\psi\mapsto \psi + a$.

Does this mean I can call any theory with such a shift symmetry a relativistic superfluid? E.g., the simplest action you could write for $\psi$ is just

$$S=\int d^4x \sqrt{-g}\partial_a\psi \partial^a\psi$$

which is just a massless free scalar field. Is it correct to regard this as a superfluid? Or is there some other feature that is needed as well? For definitiveness, I'm just concerned with zero temperature superfluids, so I'm not worried about including the second fluid that is needed for vortices and the like.

This post imported from StackExchange Physics at 2014-08-07 15:39 (UCT), posted by SE-user asperanz