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arXiv:cond-mat/9609072

Disorder-Induced Critical Phenomena in Hysteresis: A Numerical Scaling Analysis

Olga Perković, Karin A. Dahmen, James P. Sethna

Published Sep 6, 1996• 8 citations
arXivHTMLPDF

Abstract

Experimental systems with a first order phase transition will often exhibit hysteresis when out of equilibrium. If defects are present, the hysteresis loop can have different shapes: with small disorder the hysteresis loop has a macroscopic jump, while for large disorder the hysteresis loop is smooth. The transition between these two shapes is critical, with diverging length scales and power laws. We simulate such a system with the zero temperature random field Ising model, in 2, 3, 4, 5, 7, and 9 dimensions, with systems of up to 1000^3 spins, and find the critical exponents from scaling collapses of several measurements. The numerical results agree well with the analytical predictions from a renormalization group calculation.

DOI
10.48550/arxiv.cond-mat/9609072

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Experimental systems with a first order phase transition will often exhibit hysteresis when out of equilibrium. If defects are present, the hysteresis loop can have different shapes: with small disorder the hysteresis loop has a macroscopic jump, while for large disorder the hysteresis loop is smooth. The transition between these two shapes is critical, with diverging length scales and power laws. We simulate such a system with the zero temperature random field Ising model, in 2, 3, 4, 5, 7, and 9 dimensions, with systems of up to 1000^3 spins, and find the critical exponents from scaling collapses of several measurements. The numerical results agree well with the analytical predictions from a renormalization group calculation.
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A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
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By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.
Cooperative behaviors near the disorder-induced critical point in a random-field Ising model are numerically investigated by analyzing time-dependent magnetization in ordering processes from a special initial condition. We find that the intensity of fluctuations of time-dependent magnetization, chi(t) , attains a maximum value at a time t=tau in a normal phase and that chi(tau) and tau exhibit divergences near the disorder-induced critical point. Furthermore, spin configurations around the time tau are characterized by a length scale, which also exhibits a divergence near the critical point. We estimate the critical exponents that characterize these power-law divergences by using a finite-size scaling method.
Abstract We present a dynamical numerical study on a three-dimensional (3D) Gaussian distribution random-field Ising model in a wide range of disorder strengths by means of the finite-time dynamics method. By validating the Rushbrooke scaling law and analyzing the variation of exponents with disorder amplitudes, the asymptotic critical regime controlled by a hysteresis fixed point is identified with exponents , , and . The phase transition is found to be continuous and there is not a tricritical point in this version of Gaussian distribution. Moreover, our results support that the hysteresis and zero-temperature fixed points belong to a different universality class.