On Higher-Form Fracton Hydrodynamics with Momentum Conservation

Abstract: Momentum conservation is common in fluids, yet it may be incompatible with the hydrodynamics of theories with exotic conservation laws (such as fracton hydrodynamics) unless symmetries are spontaneously broken. Using the Martin-Siggia-Rose (MSR) effective field theory formalism, we develop a representation-theoretic framework for the hydrodynamics of translationally invariant systems with exotic conservation laws without spontaneous symmetry breaking. Under mild assumptions on the form of hydrodynamics, we show that such theories must satisfy two representation theoretic consistency conditions and a generalized Gauss's law constraint. Higher-form symmetries satisfy these conditions and we derive their ideal hydrodynamics; in three dimensions, the one-form case produces ideal non-relativistic magnetohydrodynamics. Finally, we prove that no other theories are compatible with momentum conservation in SO(3) nor SO($d$), for sufficiently large odd $d$.