Largest Lyapunov exponents for lattices of interacting classical spins

TL;DR

This study explores chaos in classical spin lattices by numerically analyzing Lyapunov exponents across various dimensions, revealing the rarity of integrable models and characterizing chaos growth near the Ising limit.

Contribution

It provides a comprehensive numerical and analytical investigation of chaos onset in classical spin lattices, highlighting the scarcity of integrable cases and the behavior of Lyapunov exponents.

Findings

Integrable nearest-neighbor Hamiltonians are absent in infinite lattices except for the Ising case.

Lyapunov exponents grow as a power law near the Ising point.

Lyapunov exponents are weakly sensitive to anisotropy away from the Ising limit.

Abstract

We investigate how generic the onset of chaos in interacting many-body classical systems is in the context of lattices of classical spins with nearest neighbor anisotropic couplings. Seven large lattices in different spatial dimensions were considered. For each lattice, more than 2000 largest Lyapunov exponents for randomly sampled Hamiltonians were numerically computed. Our results strongly suggest the absence of integrable nearest-neighbor Hamiltonians for the infinite lattices except for the trivial Ising case. In the vicinity of the Ising case, the largest Lyapunov exponents exhibit a power-law growth, while further away they become rather weakly sensitive to the Hamiltonian anisotropy. We also provide an analytical derivation of these results.