Metastable states and T=0 hysteresis in the random-field Ising model on random graphs

TL;DR

This paper investigates the metastable states and hysteresis behavior of the random-field Ising model on random graphs, revealing the connection between metastable state distribution and hysteresis loop shape at zero temperature.

Contribution

It provides explicit calculations of annealed and quenched complexities for the model and establishes the link between the quenched complexity zero contour and the hysteresis loop for z=2.

Findings

Quenched complexity contour matches the saturation hysteresis loop for z=2.

Annealed complexity is not relevant for describing hysteresis properties.

Explicit calculations of complexities for z=2,3,4 are provided.

Abstract

We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of an external magnetic field $H$. We compute the number of single-spin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H-m plane (focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between $-\infty$ and $+\infty$ at T=0. The annealed complexity $\Sigma_A(m,H)$ is calculated for z=2,3,4 and the quenched complexity $\Sigma_Q(m,H)$ for z=2. We prove explicitly for z=2 that the contour $\Sigma_Q(m,H)=0$ coincides with the saturation loop. On the other hand, we show that $\Sigma_A(m,H)$ is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.