Distribution of avalanche sizes in the hysteretic response of the random-field ising model on a bethe lattice at zero temperature

Abstract

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution Prob(s) of avalanche sizes s as the external field increases from -∞ to ∞. We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number z=3, and show that in these cases, Prob(s) decreases exponentially with s for large s for all h on the hysteresis loop. We find that for z≥4 and for small disorder, the magnetization shows a first-order discontinuity for several continuous and unimodal distributions of the random fields. The avalanche distribution Prob(s) varies as s ^-3/2 for large s near the discontinuity.