Goldstone bosons and hydrodynamics with dipole and momentum conservation

We develop a Schwinger-Keldysh effective field theory describing the hydrodynamics of a fluid with conserved charge and dipole moments, together with conserved momentum. The resulting hydrodynamic modes are highly unusual, including sound waves with magnon-like propagation and subdiffusive decay rate. We show that the momentum density is approximately the Goldstone boson for a dipole symmetry which appears spontaneously broken at finite charge density. Next, we show how to couple the hydrodynamic field theory to the curved spacetime. We remark that the dipole fluid can be regarded as the infinite mass limit of the Galilean-invariant fluid suggesting applications to many-body flat band systems.