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arXiv:cond-mat/9507118 · Physical review. B, Condensed matter

Hysteresis, avalanches, and disorder-induced critical scaling: A renormalization-group approach

Karin A. Dahmen, James P. Sethna

Published Jun 1, 1996• 269 citations
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Abstract

Hysteresis loops are often seen in experiments at first-order phase transformations, when the system goes out of equilibrium. They may have a macroscopic jump (roughly as in the supercooling of liquids) or they may be smoothly varying (as seen in most magnets). We have studied the nonequilibrium zero-temperature random-field Ising-model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition we find a diverging length scale, power-law distributions of noise (avalanches), and universal behavior. We expand the critical exponents about mean-field theory in 6-\ensuremath{\epsilon} dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-\ensuremath{\epsilon} expansion to O(${\mathrm{\ensuremath{\epsilon}}}^{5}$) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(\ensuremath{\epsilon}). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perkovi\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 75, 4528 (1995)]. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.14872

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Hysteresis loops are often seen in experiments at first-order phase transformations, when the system goes out of equilibrium. They may have a macroscopic jump (roughly as in the supercooling of liquids) or they may be smoothly varying (as seen in most magnets). We have studied the nonequilibrium zero-temperature random-field Ising-model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition we find a diverging length scale, power-law distributions of noise (avalanches), and universal behavior. We expand the critical exponents about mean-field theory in 6-\ensuremath{\epsilon} dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-\ensuremath{\epsilon} expansion to O(\ensuremathϵ5{\mathrm{\ensuremath{\epsilon}}}^{5}\ensuremathϵ5) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(\ensuremath{\epsilon}). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perkovi\ifmmode \acute{c}\else '{c}\fi{} et al., Phys. Rev. Lett. 75, 4528 (1995)]. \textcopyright{} 1996 The American Physical Society.
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.
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
We explain Barkhausen noise in magnetic systems in terms of avalanches of domains near a plain old critical point in the hysteretic zero-temperature random-field Ising model. The avalanche size distribution has a universal scaling function, making nontrivial predictions of the shape of the distribution up to 50% above the critical point, where two decades of scaling are still observed. We simulate systems with up to 10003{1000}^{3}10003 domains, extract critical exponents in 2, 3, 4, and 5 dimensions, compare with our 2D and $6\ensuremath{-}\ensuremath{\epsilon}$ predictions, and compare to a variety of experiments.
The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions ddd less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d>dDRd>{d}_{\text{DR}}d>dDR​, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
Dynamical phase transitions in the Ising model on hypercubic lattices are considered. Under a linearly swept magnetic field, the hysteresis loop that characterizes the field-driven first-order phase transition is studied carefully. Using the Glauber dynamics, we find that, in the mean-field approximation, the energy dissipation of this phase transition or the hysteresis loop area A of the M-H curve can be scaled with respect to the sweep rate h of magnetic field in the form with a = 2 and b = 2/3. However, b varies when fluctuations and spin correlations are taken into account. Monte Carlo simulation is used to obtain the scaling relation for A in two-, three- and four-dimensional Ising models and we obtain the exponents and respectively. These exponents are obviously different from those obtained by scaling A as for any temperatures in Ising models under a sinusoidal field. Finally we point out that, in the concept of universality, field-driven first-order phase transitions in the Ising model in different dimensions belong to different universal classes due to the spin fluctuation and correlation below the Curie temperature.