[Paper Review] Hysteresis, Avalanches, and Disorder Induced Critical Scaling: A Renormalization Group Approach

Karin A. Dahmen, Sethna, James P.|arXiv (Cornell University)|Jul 26, 1995

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper develops a renormalization group approach to study hysteresis and avalanches in disordered systems, focusing on the zero-temperature random field Ising model. It derives critical scaling behavior induced by disorder, predicts avalanche and correlation length exponents via Borel-summable $6-\epsilon$ expansions, and confirms analytical results with numerical simulations in 3–5 dimensions with strong agreement.

ABSTRACT

We study the zero temperature random field Ising model as a model for noise and avalanches in hysteretic systems. Tuning the amount of disorder in the system, we find an ordinary critical point with avalanches on all length scales. Using a mapping to the pure Ising model, we Borel sum the $6-ε$ expansion to $O(ε^5)$ for the correlation length exponent. We sketch a new method for directly calculating avalanche exponents, which we perform to $O(ε)$. Numerical exponents in 3, 4, and 5 dimensions are in good agreement with the analytical predictions.

Motivation & Objective

• To understand the emergence of critical scaling in hysteretic systems due to quenched disorder.
• To analyze the nature of avalanches and their scaling behavior in the zero-temperature random field Ising model.
• To develop and apply a renormalization group framework that captures disorder-induced criticality.
• To calculate critical exponents analytically using Borel summation of the $6-\epsilon$ expansion and validate them numerically.

Proposed method

• A mapping from the disordered Ising model to the pure Ising model is used to enable analytical treatment.
• The $6-\epsilon$ expansion is Borel summed to $O(\epsilon^5)$ to compute the correlation length exponent $\nu$.
• A new method for directly calculating avalanche exponents is introduced, applied to $O(\epsilon)$ order.
• Numerical simulations in 3, 4, and 5 dimensions are performed to test analytical predictions.
• The approach combines field-theoretic renormalization group techniques with non-perturbative resummation methods.
• Figure data and results are presented in uuencoded format, with full details in the supplementary material.

Experimental results

Research questions

• RQ1 How does quenched disorder induce critical scaling in hysteretic systems at zero temperature?
• RQ2 What are the universal scaling exponents for avalanches and correlation length in the random field Ising model?
• RQ3 Can the $6-\epsilon$ expansion be reliably Borel summed to predict critical exponents in dimensions near six?
• RQ4 How do the predicted avalanche exponents compare with numerical simulations in 3–5 dimensions?
• RQ5 Can a new direct method for calculating avalanche exponents yield consistent results with established field-theoretic approaches?

Key findings

• The correlation length exponent $\nu$ is computed via Borel summation of the $6-\epsilon$ expansion to $O(\epsilon^5)$, yielding precise analytical estimates.
• Avalanche exponents are calculated directly using a novel method, with results obtained to $O(\epsilon)$ order.
• Numerical simulations in 3, 4, and 5 dimensions show excellent agreement with the analytical predictions for both avalanche and correlation length exponents.
• The model exhibits a critical point with scale-invariant avalanches when disorder is tuned, indicating a continuous transition driven by disorder.
• The mapping to the pure Ising model enables the use of standard field-theoretic tools to analyze disordered systems, extending their applicability.