Two parameter scaling in the crossover from symmetry class BDI to AI

The transport statistics of the 1D chain and metallic armchair graphene nanoribbons with hopping disorder are studied, with a focus on understanding the cross-over between the zero-energy critical point and the localized regime at large energy. In this cross-over region, transport is found to be described by a 2-parameter scaling with the ratio $s$ of system size to mean-free-path, and the product $r$ of energy and scattering time. This 2-parameter scaling shows excellent data collapse across a wide a variety of system sizes, energies, and disorder strengths. The numerically obtained transport distributions in this regime are found to be well-described by a Nakagami distribution, whose form is controlled up to an overall scaling by the ratio $s / |\ln r|^2$. For sufficiently small values of this parameter, transport appears virtually identical to that of the zero-energy critical point, while at large values, a localized, Gaussian distribution is recovered. For intermediate values, the distribution interpolates smoothly between these two limits.