[Paper Review] Random-Field Ising Models of Hysteresis

James P. Sethna, Karin A. Dahmen+1 more|arXiv (Cornell University)|Jun 14, 2004

Theoretical and Computational Physics4 citations

TL;DR

This paper presents a comprehensive study of the zero-temperature random-field Ising model as a framework for understanding hysteresis, crackling noise, and avalanche dynamics in magnetic systems. Using renormalization group theory and extensive numerical simulations, it derives universal scaling laws for avalanche size, duration, and correlation functions, showing power-law distributions near critical disorder and validating these against Barkhausen noise experiments, with corrections to scaling analyzed in detail across dimensions.

ABSTRACT

This is a review article of our work on hysteresis, avalanches, and criticality. We provide an extensive introduction to scaling and renormalization--group ideas, and discuss analytical and numerical results for size distributions, correlation functions, magnetization, avalanche durations and average avalanche shapes, and power spectra. We focus here on applications to magnetic Barkhausen noise.

Motivation & Objective

• To establish the zero-temperature random-field Ising model as a universal framework for understanding hysteresis and crackling noise in magnetic materials.
• To derive and validate universal scaling laws for avalanche size, duration, and correlation functions using renormalization group theory.
• To analyze corrections to scaling and finite-size effects in avalanche distributions across different spatial dimensions.
• To connect theoretical predictions with experimental Barkhausen noise data, particularly near critical disorder.
• To explore history-dependent critical behavior, including return-point memory and subloop dynamics.

Proposed method

• Employing large-scale numerical simulations of billion-spin systems to observe avalanche dynamics in the random-field Ising model at zero temperature.
• Applying finite-size scaling and two-variable scaling collapses using scaling forms such as $ F(x,r,h) \sim x^{-\alpha} \tilde{\cal F}(x^{\zeta}r, x^{\zeta\beta\delta}h) $ to collapse data across disorder and field.
• Deriving exponential scaling forms for two-dimensional systems where linear corrections vanish, using $ F(x,r) = x^{-\alpha} \tilde{\cal F}(x e^{-1/2k r^2}) $, to account for non-analytic corrections.
• Using magnetization curves, avalanche size distributions, and energy spectra to measure critical exponents and test scaling relations.
• Analyzing avalanche duration and shape using time-resolved correlation functions and power spectra.
• Testing the robustness of scaling by examining data collapse at various disorders, including near-critical and finite-sweep-rate conditions.

Experimental results

Research questions

• RQ1 How do avalanche size distributions in the random-field Ising model exhibit universal power-law behavior near the critical disorder?
• RQ2 What are the corrections to scaling in avalanche distributions, and how do they depend on dimensionality and field history?
• RQ3 How do the scaling functions for avalanche size, duration, and correlation depend on the control parameters (disorder, field, system size)?
• RQ4 To what extent do the model's predictions for Barkhausen noise match experimental observations in real magnetic materials?
• RQ5 What is the role of return-point memory and subloop dynamics in inducing critical behavior in hysteretic systems?

Key findings

• Avalanche size distributions follow a power law $ D(S) \sim S^{-\tau} $ near the critical disorder $ R_c $, with a scaling collapse spanning over six orders of magnitude at 5% above $ R_c $.
• The integrated avalanche size distribution $ D_{\text{int}}(S) \sim S^{-(\tau + \sigma\beta\delta)} $ at $ R_c $ confirms the existence of a universal power law at criticality.
• Corrections to scaling are subdominant and become significant only for large avalanches $ S > h^{-1/\sigma} $, with the scaling variable $ X = S^{\sigma\beta\delta}h $ determining the onset of corrections.
• In two dimensions, exponential scaling forms such as $ F(x,r) = x^{-\alpha} \tilde{\cal F}(x e^{-1/2k r^2}) $ are required due to vanishing linear terms in the rescaling, enabling successful data collapse.
• The model successfully reproduces key features of Barkhausen noise, including power-law avalanche distributions and universal scaling functions, across a wide range of disorders.
• Finite sweep rates and history-dependent dynamics (e.g., subloops) lead to critical behavior consistent with return-point memory, with scaling functions collapsing across different field histories.