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

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 ${\lambda }_{\mathrm{\infty }}=l_{\mathrm{\infty }}\mathrm{/}{l}_{\mathrm{e}\mathrm{l}}$, the ratio between the Anderson localization length $l_{\mathrm{\infty }}$ and the Stark localization length~$l_{\mathrm{e}\mathrm{l}}$. 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 ${\lambda }_N=l_{\mathrm{\infty }}\mathrm{/}N$ and ${\lambda }_{\mathrm{\infty }}=l_{\mathrm{\infty }}\mathrm{/}{l}_{\mathrm{e}\mathrm{l}}$.