Hysteresis in magnets

TL;DR

This review consolidates theoretical and computational insights into hysteresis in magnetic systems across Ising-like and continuous-spin models, emphasizing how driven dynamics, disorder, and nucleation govern loop shapes and energy dissipation. It links early foundational frameworks (LLG and Preisach) to modern analyses of dynamical phase transitions and avalanche statistics, including exact Bethe-lattice results for disordered ferromagnets. A central theme is the area of the hysteresis loop, $A$, and its non-universal scaling with driving amplitude $h_0$ and frequency $\omega$, encapsulated by exponents $\alpha$ and $\beta$ and by regime-dependent asymptotics. The work highlights Barkhausen noise as a signature of avalanche processes, showing both mean-field and Bethe-lattice perspectives yield characteristic power-law distributions, such as $G_s(h)\sim s^{-3/2}$ near discontinuities and cycle-averaged tails $\sim s^{-5/2}$, thereby connecting microscopic spin dynamics to macroscopic hysteresis phenomenology with practical implications for magnetic materials and noise analysis.

Abstract

We provide an overview of studies of hysteresis in models of magnets. We discuss the shape of the hysteresis loop, dynamical symmetry breaking, and the dependence of the area of the loop on the amplitude and frequency of the driving field. We also discuss Barkhausen noise in the hysteresis loops, where the wide distribution of sizes of magnetization jumps may be modeled by the random-field Ising model. We discuss the distribution of sizes of these jumps in the random field Ising model on the Bethe lattice.

Figures (13)

• Figure 1: Different types of hysteresis loops. (a) Paramagnetic, (b) ferromagnetic symmetric, (c) ferromagnetic asymmetric, (d) 'wasp-waisted' loop in antiferroelectric mixed crystal of betaine phosphate-arsenate ($BP_{0.9}As_{0.1}$) as a function of the electric field. Reprinted figure (d) with permission from kim [https://doi.org/10.1103/PhysRevB.55.R11933]. Copyright (1997) by the American Physical Society.
• Figure 2: Plot of the hysteresis for a single spin driven by the oscillating magnetic field $h(t) = h_0\sin(\omega t)$, where the corresponding time-dependent magnetization is given by Eq. (\ref{['eq:M2-1spin']}). The time dependence of $h(t)$ and $M(t)$ is shown in the corresponding insets. (a) low frequency--high field amplitude, (b) high frequency--high field amplitude, (c) low frequency--low field amplitude, and (d) high frequency--low field amplitude.
• Figure 3: Qualitative behavior of the hysteresis loop in the $d$-dimensional Ising model at very small frequencies. The loops are nearly rectangular, and the area decreases with decreasing frequency.
• Figure 4: Non-equilibrium phase diagram of the Ising model in an oscillating field in the $h-T$ plane, for fixed values of $h_0$ and $\omega$. Reprinted figure with permission from tome [https://doi.org/10.1103/PhysRevA.41.4251]. Copyright (1990) by the American Physical Society.
• Figure 5: Hysteresis loop from a multi-domain sample showing magnetization jumps. Figure reproduced from the SS’s PhD thesis sabhapandit:phd.