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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 quadratic (magnon-like) dispersion relation and subdiffusive decay rate. Hydrodynamics itself is unstable below four spatial dimensions. We show that the momentum density is, at leading order, the Goldstone boson for a dipole symmetry which appears spontaneously broken at finite charge density. Unlike an ordinary fluid, the presence or absence of energy conservation qualitatively changes the decay rates of the hydrodynamic modes. This effective field theory naturally couples to curved spacetime and background gauge fields; in the flat spacetime limit, we reproduce the “mixed rank tensor fields” previously coupled to fracton matter.

 

We extend recent work on hydrodynamics with global multipolarsymmetries — known as “fracton hydrodynamics” — to systems in which themultipolar symmetries are gauged. We refer to the latter as “fractonmagnetohydrodynamics”, in analogy to conventional magnetohydrodynamics(MHD), which governs systems with gauged charge conservation. We showthat fracton MHD arises naturally from higher-rank Maxwell’s equationsand in systems with one-form symmetries obeying certain constraints;while we focus on “minimal” higher-rank generalizations of MHD thatrealize diffusion, our methods may also be used to identify other, moreexotic hydrodynamic theories (e.g., with magnetic subdiffusion). Incontrast to semi-microscopic derivations of MHD, our approach elucidatesthe origin of the hydrodynamic modes by identifying the correspondinghigher-form symmetries. Being rooted in symmetries, the hydrodynamicmodes may persist even when the semi-microscopic equations no longerprovide an accurate description of the system.

 

A bstract We develop a hydrodynamic effective field theory on the Schwinger-Keldysh contour for fluids with charge, energy, and momentum conservation, but only discrete rotational symmetry. The consequences of anisotropy on thermodynamics and first-order dissipative hydrodynamics are detailed in some simple examples in two spatial dimensions, but our construction extends to any spatial dimension and any rotation group (discrete or continuous). We find many possible terms in the equations of motion which are compatible with the existence of an entropy current, but not with the ability to couple the fluid to background gauge fields and vielbein.