Weak integrability breaking in classical integrable lattice models with discrete and continuous time

Recently, systematic constructions of special "weak" perturbations to integrable models were introduced in the context of quantum integrable systems like the spin-½ Heisenberg model. These perturbations, which allow quasi-conserved quantities, are expected to induce thermalization on timescales far exceeding those predicted by Fermi's golden rule. While these expectations are based on strongly suggestive arguments, numerical simulations would be useful to gain a deeper understanding of this phenomenon. However, studying long-time dynamics in quantum models is computationally challenging. Classical integrable models provide an alternative avenue for exploring transport properties and long-time behavior under such perturbations. In this work, we show how to systematically construct weak perturbations for classical integrable lattice models, including their discrete time "Trotterized" counterparts. Breaking integrability is expected to restore conventional diffusive spin transport, but recent computational and experimental studies have observed certain robustness under isotropic perturbations. In the non-integrable classical Heisenberg model, for example, superdiffusive spin transport has been observed for unusually long times under generic isotropic perturbations. Still, the origin of superdiffusive spin transport and the nature of the crossover to diffusion are poorly understood. Our construction can be used to investigate the extent to which "weak" perturbations contribute to such anomalous transport phenomena and can serve as a benchmark in studying thermalization in perturbed integrable models.