We present numerical simulations of avalanches and critical phenomena associated with hysteresis loops, modeled using the zero-temperature random-field Ising model. We study the transition between smooth hysteresis loops and loops with a sharp jump in the magnetization, as the disorder in our model is decreased. In a large region near the critical point, we find scaling and critical phenomena, which are well described by the results of an ǫ expansion about six dimensions. We present the results of simulations in 3, 4, and 5 dimensions, with systems with up to a billion spins (1000 3 ).

We explain Barkhausen noise in magnetic systems in terms of avalanches 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 non-trivial 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 $1000^3$ domains, extract critical exponents in 2, 3, 4, and 5 dimensions, compare with our 2d and $6-\epsilon$ predictions, and compare to a variety of experimental Barkhausen measurements.Comment: 12 pages, 2 PostScript figures. Pedagogical introduction with mpeg movies available at http://www.lassp.cornell.edu/LASSP_Science.htm

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, and briefly discuss non-magnetic systems with hysteresis and avalanches.

As computers increase in speed and memory, scientists are inevitably led to simulate more complex systems over larger time and length scales. Although a simple, straightforward algorithm is often the most efficient for small system sizes, especially when the time needed to implement the algorithm is included, the scaling of time and memory with system size becomes crucial for larger simulations.In our studies of hysteresis and avalanches in a simple model of magnetism (the random-field Ising model at zero temperature), we often have found it necessary to do very large simulations. Previous simulations were limited to relatively small systems (up to 900 2 and 128 3 [1], see however [3]). In our simulations we have found that larger systems (up to a billion spins) are crucial to extracting accurate values of the universal critical exponents and understanding important qualitative features of the physics.We have developed two efficient and relatively straightforward algorithms which allow us to simulate these large systems. The first algorithm uses sorted lists and scales as O(N log N ), and asymptotically uses N × (sizeof(double)+sizeof(int)) bytes of memory, where N is the number of spins. The second algorithm, which does not generate the random fields, also scales in time as O(N log N ), but asymptotically needs only one bit of storage per spin, about 96 times less than the first algorithm. Using the latter algorithm, simulations of a billion spins can be run on a workstation with 128MB of RAM in a few hours.In this column we discuss algorithms for simulating the zero-temperature random-field Ising model, which is defined by the energy functionwhere the spins s i = ±1 sit on a D-dimensional hypercubic lattice with periodic boundary conditions. The spins interact ferromagnetically with their z nearest neighbors ; Olga Perković is with McKinsey & Company, olga perkovic@mckinsey.com; Karin Dahmen has just joined the physics faculty of the University of Illinois at Urbana-Champaign; Bruce W. Roberts is working at Starwave Corporation in Seattle, bwr@halcyon.com; James P. Sethna is a professor of physics at Cornell University, sethna@lassp.cornell.edu. Details about his research group can be found at http://www.lassp.cornell.edu/sethna/ with strength J, and experience a uniform external field H(t) and a random local field h i . We choose units such that J = 1. The random field h i is distributed according to the Gaussian distribution ρ(h) of width R:The external field H(t) is increased arbitrarily slowly from −∞ to ∞.The dynamics of our model includes no thermal fluctuations: each spin flips deterministically when it can gain energy by doing so. That is, it flips when its local fieldchanges sign. This change can occur in two ways: a spin can be triggered when one of its neighbors flips (by participating in an avalanche), or a spin can be triggered because of an increase in the external field H(t) (starting a new avalanche). The zero-temperature random-field Ising model was introduced by Robbins and Ji [3] to study flu...

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