Dipole superfluid hydrodynamics

We construct a theory of hydrodynamic transport for systems with conserved dipole moment, U(1) charge, energy, and momentum. These models have been considered in the context of fractons, since their elementary and isolated charges are immobile by symmetry, and have two known translation-invariant gapless phases: a "p-wave dipole superfluid" phase where the dipole symmetry is spontaneously broken and a "s-wave dipole superfluid" phase where both the U(1) and dipole symmetries are spontaneously broken. We argue on grounds of symmetry and thermodynamics that there is no transitionally-invariant gapless fluid with unbroken dipole symmetry. In this work, we primarily focus on the hydrodynamic description of p-wave dipole superfluids, including leading dissipative corrections. That theory has, in a sense, a dynamical scaling exponent $z=2$, and its spectrum of fluctuations includes novel subdiffusive modes $\omega \sim -i k^4$ in the shear sector and magnon-like sound mode $\omega\sim \pm k^2 -i k^2$. By coupling the fluid to background fields, we find response functions of the various symmetry currents. We also present a preliminary generalization of our work to s-wave dipole superfluids, which resemble $z=1$ fluids and feature sound waves and diffusive shear modes, as in an ordinary fluid. However, the spectrum also contains a magnon-like second-sound mode $\omega\sim \pm k^2 \pm k^4 -i k^4$ with subdiffusive attenuation.