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

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-ε dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-ε expansion to O(${\mathrm{\epsilon }}^5$) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(ε). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perković et al., Phys. Rev. Lett. 75, 4528 (1995)]. © 1996 The American Physical Society.