Lifshitz Superfluid Hydrodynamics

We construct the first order hydrodynamics of quantum critical points with Lifshitz scaling and a spontaneously broken symmetry. The fluid is described by a combination of two flows, a normal component that carries entropy and a super-flow which has zero viscosity and carries no entropy. We analyze the new transport effects allowed by the lack of boost invariance and constrain them by the local second law of thermodynamics. Imposing time-reversal invariance, we find eight new parity even transport coefficients. The formulation is applicable, in general, to any superfluid/superconductor with an explicit breaking of boost symmetry, in particular to high $T_c$ superconductors. We discuss possible experimental signatures.