The dynamics of a driven harmonic oscillator coupled to pairwise interacting Ising spins in random fields

TL;DR

This paper explores the complex dynamics, including chaos and hysteresis, of a driven harmonic oscillator coupled to interacting Ising spins, extending previous models to include pairwise interactions and analyzing the thermodynamic limit.

Contribution

It introduces a model with pairwise interacting spins coupled to a harmonic oscillator, connecting to the Preisach model, and investigates chaotic behavior and hysteresis in large spin systems.

Findings

Interacting spins can exhibit elementary hysteresis.

The thermodynamic limit introduces a hysteretic force modeled by a generalized play operator.

Chaotic behavior is characterized by bifurcation diagrams, Lyapunov exponents, and fractal dimensions.

Abstract

In general we are interested in dynamical systems coupled to complex hysteresis. Therefore as a first step we investigated recently the dynamics of a periodically driven damped harmonic oscillator coupled to independent Ising spins in a random field. Although such a system does not produce hysteresis, we showed how to characterize the dynamics of such a piecewise-smooth system, especially in the case of a large number of spins [P. Zech, A. Otto, and G. Radons, Phys. Rev. E 101, 042217 (2020)]. In this paper we extend our model to spin dimers, thus pairwise interacting spins. We show in which cases two interacting spins can show elementary hysteresis and we give a connection to the Preisach model, which allows us to consider an infinite number of spin-pairs. This thermodynamic limit leads us to a dynamical system with an additional hysteretic force in the form of a generalized play operator. By using methods from general chaos theory, piecewise-smooth system theory and statistics we investigate the chaotic behavior of the dynamical system for a few spins and also in case of a larger number of spins by calculating bifurcation diagrams, Lyapunov exponents, fractal dimensions and self-averaging properties. We find that the fractal dimensions and the magnetization are in general not self-averaging quantities. We show, how the dynamical properties of the piecewise-smooth system for a large number of spins differs from the system in its thermodynamic limit.