Hidden quasi-conservation laws in fracton hydrodynamics*

We show that the simplest universality classes of fracton hydrodynamics, including isotropic theories of charge and dipole conservation, can exhibit hidden "quasi-conservation laws", in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. Firstly, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of "harmonic multipole charges" within linear response; we calculate the decay/growth of these charges due to dangerously irrelevant nonlinearities. Secondly, we consider a model with xy and x²-y² quadrupole conservation, in addition to dipole conservation, which is described by isotropic fourth-order subdiffusion, yet has dangerously irrelevant sixth-order corrections necessary to relax the harmonic multipole charges. We confirm our predictions for the anomalously slow decay of the harmonic conserved charges in each setting by using numerical simulations, both of the nonlinear hydrodynamic differential equations, and in quantum automaton circuits on a square lattice.