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

Abstract

The non-equilibrium 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. Secondly, 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 2D NE-RFIM behavior well. The obtained scaling collapses span over a range of a factor of ten in the disorder and a factor of 10 4 in avalanche cutoff. They are consistent with a critical disorder at zero and with a lower critical dimension for the model equal to two.We study the avalanche size distribution in the two-dimensional nucleated non-equilibrium random-field Ising model (NE-RFIM), simulated on a Voronoi lattice to bypass faceting, and analyzed using the scaling predictions of the nonlinear renormalization-group flows predicted for the lower critical dimension. We find excellent agreement over a large critical region, addressing several outstanding issues in the field.The NE-RFIM is perhaps the best-understood model of crackling noise [1], exhibiting power-law distributions of avalanche sizes at a critical disorder r c representing the standard deviation of the strength of the random field at each site. The model transitions from a 'downspin' state to an 'up-spin' state as an external field H increases. Above the critical disorder r c , this transition is composed of avalanches of spins of size limited by a typical cutoff Σ + (r); below the critical disorder a finite fraction of the spins flip in a single event, with precursors and aftershock sizes limited by Σ − (r). This model, albeit simple, contains the necessary ingredients to describe hysteretic and avalanche behaviors in a diverse set of systems. Barkhausen noise in magnets [2] decision making in socio-economics [3], absorption and desorption in superfluids [4,5] as well as the effects of nematicity in high T c superconductors [6-8] can each be understood in terms of 'crackling noise' naturally described by the NE-RFIM.Although the NE-RFIM itself has been around in various forms since the 1970s [9], there are still a number of current questions and issues:• Is it in the same universality class as the equilibrium RFIM model [10]? It has long been debated whether the equilibrium and non-equilibrium versions of the model are in the same universality class. This question of universality has been approached in a number of ways which have suggested the same class for the two models [11][12][13][14][15][16]. Recent wor...