Spreading of Perturbations in Long-Range Interacting Classical Lattice Models

TL;DR

This paper establishes Lieb-Robinson-type bounds for classical long-range lattice models, showing that perturbations influence the system within a logarithmic causal region with algebraic decay outside, supported by numerical evidence.

Contribution

It introduces refined bounds for classical long-range interactions that accurately describe the causal region, extending Lieb-Robinson bounds to a broader class of models.

Findings

Perturbations are confined within a logarithmic-shaped causal region.

Refined bounds better capture the shape of the causal influence.

Numerical results confirm the theoretical bounds in classical XY chains.

Abstract

Lieb-Robinson-type bounds are reported for a large class of classical Hamiltonian lattice models. By a suitable rescaling of energy or time, such bounds can be constructed for interactions of arbitrarily long range. The bound quantifies the dependence of the system's dynamics on a perturbation of the initial state. The effect of the perturbation is found to be effectively restricted to the interior of a causal region of logarithmic shape, with only small, algebraically decaying effects in the exterior. A refined bound, sharper than conventional Lieb-Robinson bounds, is required to correctly capture the shape of the causal region, as confirmed by numerical results for classical long-range $XY$ chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.