Avalanches: the crackle of a disordered magnet

Each spin points up or down and feels three things: its four neighbours, a uniform outside field H, and a random field $h_i$ of its own, fixed once and for all and drawn from a bell curve of width R. That width is the disorder. A spin points along the total:

There is no temperature: this is the random-field Ising model at T = 0, as studied by James Sethna, Karin Dahmen and colleagues from 1993. Every spin starts down, and H is raised just far enough to turn the single most eager spin. Turning it adds +2 to each neighbour’s total, which may turn some of them, which may turn their neighbours: an avalanche. When it stops, H is raised to the next most eager spin, and so on. The glowing spins on screen are the ones that just turned, so you can watch each avalanche spread one generation at a time.

• Little disorder: the spins are all much alike. Nothing happens for a long while, then the first spin to go tips its neighbours, which tip theirs, and one avalanche turns almost the whole magnet. The hysteresis loop has a single vertical cliff.
• A lot of disorder: each spin mostly follows its own random field, so avalanches stay small and the loop is smooth.
• In between, at a critical disorder R_c, avalanches come in every size, and the number of avalanches of size s falls off as a power of s: a straight line on the log–log chart.

This is a critical point like the others in the lab, reached by tuning disorder instead of temperature. In three dimensions R_c ≈ 2.16 (Perković, Dahmen and Sethna, 1995). In two dimensions the question was argued over for years; simulations on very large grids put it near 0.54 (Spasojević, Janićević and Knežević, 2011). The grids here are small, and on a small grid avalanches that would keep going are stopped by the edges, so the crossover shows up nearer R ≈ 1.

Turn on Sound. Each avalanche becomes a click, louder and lower the bigger it is. With little disorder you hear a few ticks and one bang; at R near 1, a crackle with every size of event mixed together; with a lot of disorder, a fine hiss. In 1919 Heinrich Barkhausen heard exactly this crackle by winding a coil round a piece of iron and connecting it to a telephone earpiece while he magnetised the iron. The jumps of real domain walls are still called Barkhausen noise.

• Choose Crackling, set the chart to Avalanche sizes, and let the field run up and down a few times. How straight is the line, and over how many powers of ten?
• Lower R in steps of 0.1 and watch the largest avalanche jump from a few hundred spins to nearly all of them.
• Watch a big avalanche closely at a low speed. It does not spread as a disc: it follows the spins whose random fields happen to make them easy to tip.
• Compare the rising and falling halves of a loop. On average they mirror each other, but the avalanches on the way down are different ones from those on the way up.

• Ising model Up-or-down magnets that line up below one exact temperature, and lose it above.
• Clock and XY models Arrows that can point any way. No magnet forms, yet vortices pair up below T_BKT.
• Potts model q colours instead of two. Past four, the transition stops being gradual.
• Percolation and forest fires Plant trees at random and light one edge. Past 59.27% the fire gets across.
• Hopfield memory An Ising model whose couplings store pictures, and roll a damaged one back.

New to all this? The lab’s introduction explains order parameters, critical points and how the simulations work.

• J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts and J. D. Shore (1993), Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett..
• J. P. Sethna, K. A. Dahmen and C. R. Myers (2001), Crackling noise, Nature.
• O. Perković, K. Dahmen and J. P. Sethna (1995), Avalanches, Barkhausen Noise, and Plain Old Criticality, Phys. Rev. Lett..
• D. Spasojević, S. Janićević and M. Knežević (2011), Numerical Evidence for Critical Behavior of the Two-Dimensional Nonequilibrium Zero-Temperature Random Field Ising Model, Phys. Rev. Lett..