Exact results on the hydrodynamics of certain kinetically-constrained hopping processes

We consider a model of interacting random walkers on a triangular chain and triangular lattice, where a particle can move only if the other two sites of the triangle are unoccupied – a kinetically-constrained hopping process (KCHP) recently introduced in the context of non-linear diffusion cascades. Using a classical-to-quantum mapping – where the rate matrix of the stochastic KCHP corresponds to a spin Hamiltonian, and the equilibrium probability distribution to the quantum ground state – we develop a systematic perturbation theory to calculate the diffusion constant; the hydrodynamics of the KCHPs is determined by the low-energy properties of the spin Hamiltonian, which we analyse with the standard Holstein-Primakoff spin-wave expansion. For the triangular hopping we consider, we show that $\mathit{\text{non-interacting}}$ spin-wave theory predicts the $\mathit{\text{exact}}$ diffusion constant. We conjecture this holds for all KCHPs with (i) hard-core occupancy, (ii) parity-symmetry, and (iii) where the hopping processes are given by three-site gates – that is, where hopping between two sites is conditioned on the occupancy of a third. We further show that there are corrections to the diffusion constant when the KCHP is described by $\mathit{\text{four}}$-site gates, which we calculate at leading order in the semi-classical $1\mathrm{/}S$ expansion. We support all these conclusions with numerical simulations.