Oscillating hysteresis in the q-neighbor Ising model

TL;DR

This paper introduces a modified q-neighbor Ising model demonstrating oscillatory hysteresis behavior depending on the parameter q, revealing complex phase transition phenomena and cautioning against reliance solely on simulations for non-equilibrium systems.

Contribution

The study provides an analytical and simulation-based investigation of hysteresis oscillations in a mean-field q-neighbor Ising model, highlighting novel behaviors for different q values.

Findings

Hysteresis oscillates for q>3, expanding for even and shrinking for odd q.

Phase transition is continuous at q=3 and discontinuous for larger q.

Analytical solutions reveal stable, unstable, and metastable states in the model.

Abstract

We modify the kinetic Ising model with Metropolis dynamics, allowing each spin to interact only with $q$ spins randomly chosen from the whole system, which corresponds to the topology of a complete graph. We show that the model with $q \ge 3$ exhibits a phase transition between ferromagnetic and paramagnetic phases at temperature $T^*$, which linearly increases with $q$. Moreover, we show that for $q=3$ the phase transition is continuous and discontinuous for larger values of $q$. For $q>3$ the hysteresis exhibits oscillatory behavior -- expanding for even values of $q$ and shrinking for odd values of $q$. If only simulation results were taken into account, this phenomenon could be mistakenly interpreted as switching from discontinuous to continuous phase transitions or even as evidence of the so-called mixed phase transitions. Due to the mean-field like nature of the model we are able to calculate analytically not only the stationary value of the order parameter but also precisely determine the hysteresis and the effective potential showing stable, unstable and metastable steady states. The main message is that in case of non-equilibrium systems the hysteresis can behave in an odd way and computer simulations alone may mistakenly lead to incorrect conclusions.