Generic power laws in higher-dimensional lattice models with multidirectional hopping: Centre of mass conservation and hyperuniformity

We show that, on a d-dimensional hypercubic lattice with d>1, conserved-mass transport processes, with multidirectional hopping that respects all symmetries of the lattice, exhibit power-law correlations for generic parameter values - even far from phase transition point, if any. The key idea for generating the algebraic decay is the notion of multidirectional hopping, which means that several chunks of mass, or several particles, can hop out simultaneously from a lattice site in multiple directions, generating currents in each direction and consequently breaking detailed balance. Notably, the systems we consider are described by a continuous-time Markov process, are diffusive, lattice-rotation symmetric, spatially homogeneous and thus have no net mass current. Using hydrodynamic and exact microscopic theory, we show that, for spatial dimensions d>1, the steady-state static density-density correlation functions in the thermodynamic limit typically decay as ∼1/r^d+2 at large distance r=|r|. In particular, our theory explains why center-of-mass-conserving dynamics, used to model novel disordered hyperuniform state of matter, result in generic long-ranged correlations.