Scaling properties of one-dimensional Anderson models in an electric field: Exponential vs. factorial localization

Abstract

We investigate the scaling properties of eigenstates of a one-dimensional (1D) Anderson model in the presence of a constant electric field. The states show a transition from exponential to factorial localization. For infinite systems this transition can be described by a simple scaling law based on a single parameter $\lambda_{\infty} = l_{\infty}/l_{\rm el}$, the ratio between the Anderson localization length $l_{\infty}$ and the Stark localization length~$l_{\rm el}$. For finite samples, however, the system size $N$ enters the problem as a third parameter. In that case the global structure of eigenstates is uniquely determined by two scaling parameters $\lambda_N=l_\infty/N$ and $\lambda_\infty=l_\infty/l_{\rm el}$.