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

Abstract

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.