Unusual scaling for two-dimensional avalanches: Curing the faceting and scaling in the lower critical dimension

Abstract

The nonequilibrium random-field Ising model is well studied, yet there are outstanding questions. In two dimensions, power-law scaling approaches fail and the critical disorder is difficult to pin down. Additionally, the presence of faceting on the square lattice creates avalanches that are lattice dependent at small scales. We propose two methods which we find solve these issues. First, we perform large-scale simulations on a Voronoi lattice to mitigate the effects of faceting. Second, the invariant arguments of the universal scaling functions necessary to perform scaling collapses can be directly determined using our recent normal form theory of the renormalization group. This method has proven useful in cleanly capturing the complex behavior which occurs in both the lower and upper critical dimensions of systems and here captures the two-dimensional nonequilibrium random-field Ising model behavior well. The obtained scaling collapses span a range of a factor of 10 in the disorder and a factor of 104 in avalanche cutoff. They are consistent with a critical disorder at zero and with a lower critical dimension for the model equal to 2.