Quantum versus Classical Dynamics in Spin Models: Chains, Ladders, and Square Lattices

TL;DR

This study compares quantum and classical spin dynamics across various geometries, revealing surprisingly good agreement even for small quantum spins, suggesting classical simulations can effectively model quantum many-body dynamics.

Contribution

It demonstrates that classical or semi-classical simulations can accurately approximate quantum spin dynamics in complex geometries, challenging the expectation of significant differences.

Findings

Quantum and classical dynamics agree well across geometries

Best agreement in nonintegrable models in higher dimensions

Classical simulations are effective for small quantum spins

Abstract

We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\"odinger equation in the quantum case, or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-$1/2$ systems with up to $N = 36$ lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semi-classical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number $S = 1/2$ is small and far away from the classical limit $S \to \infty$.