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

TL;DR

This paper derives exact and approximate results for the hydrodynamics of kinetically-constrained hopping processes using a quantum mapping, revealing conditions under which simple spin-wave theory yields exact diffusion constants.

Contribution

It introduces a systematic perturbation approach to analyze the hydrodynamics of KCHPs via a quantum-spin mapping, identifying when non-interacting spin-wave theory provides exact results.

Findings

Non-interacting spin-wave theory predicts exact diffusion constants for certain KCHPs.

Corrections to diffusion constants are found for KCHPs with four-site gates.

Numerical simulations support the theoretical predictions.

Abstract

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 \textit{non-interacting} spin-wave theory predicts the \textit{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 \textit{four}-site gates, which we calculate at leading order in the semi-classical $1/S$ expansion. We support all these conclusions with numerical simulations.