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

We investigate the scaling properties of eigenstates of a one-dimensional 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 ${\lambda }_{\infty }{=l}_{\infty }{/l}_{\mathrm{el}},$ the ratio between the Anderson localization length $l_{\infty }$ and the Stark localization length $l_{\mathrm{el}}.$ 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 ${\lambda }_{\infty }{=l}_{\infty }{/l}_{\mathrm{el}}.$