Critical behavior in non-Abelian deterministic directed sandpile

We propose a new deterministic directed sandpile model on the two-dimensional square lattice. The avalanche is initiated by adding a grain to a randomly chosen site on the top row, and at each unstable site, all grains are transferred deterministically to downstream neighbors along a direction specified by arrow at this site. The model is local and conservative, but not Abelian. We apply the moment analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. We find that this non-Abelian model defines a new universality class with respect to the Abelian deterministic directed sandpile model for the avalanches. In addition, we find this symmetry breaking introduces an obvious large scale structures consisting of fractal network of occupied sites, whose density vanishes as a power law with distance into the sandpile.