Symmetric Superfluids

We present a complete classification of symmetric superfluids, namely shift-symmetric and Poincar\'{e} invariant scalar field theories that have an enlarged set of classically conserved currents at leading order in derivatives. These theories arise in the decoupling limit of the effective field theory of shift-symmetric, single-clock cosmologies and our results pick out all models with couplings fixed by additional symmetry. Remarkably, in $D\geq 2$ spacetime dimensions there are only two possibilities: the Dirac-Born-Infeld theory and Scaling Superfluids with Lagrangian $(-\partial_{\mu}\phi\partial^{\mu}\phi)^{\alpha}$, for some real $\alpha$. The scaling symmetry present for any $\alpha$ is further enhanced to the full conformal group only for $\alpha=D/2$, and to infinitely many additional generators for $\alpha=1/2$. We discuss the stability of Scaling Superfluids and point out that all coupling constants are determined by the speed of sound.