Localization by pseudorandom potentials in one dimension

Abstract

Localization in one dimension in the presence of a pseudorandom potential is investigated. The localization length of the tight-binding model ${\mathrm{V}}_{\mathrm{n}}$${\mathrm{u}}_{\mathrm{n}}$+${\mathrm{u}}_{\mathrm{n}+1}$+${\mathrm{u}}_{\mathrm{n}\mathrm{\ensuremath{-}}1}$=${\mathrm{Eu}}_{\mathrm{n}}$ with ${\mathrm{V}}_{\mathrm{n}}$=\ensuremath{\lambda} cos\ensuremath{\pi}\ensuremath{\alpha}\ensuremath{\Vert}n${\mathrm{\ensuremath{\Vert}}}^{\ensuremath{\nu}}$ is calculated numerically and in perturbation theory for \ensuremath{\lambda}\ensuremath{\ll}1, for generic values of \ensuremath{\alpha} and \ensuremath{\nu}. The similarity between the potential ${\mathrm{V}}_{\mathrm{n}}$ and random potentials increases with \ensuremath{\nu}. It is found that for \ensuremath{\nu}\ensuremath{\ge}2 all the states are localized and the localization length is equal to that of the corresponding random model while for 0<\ensuremath{\nu}\ensuremath{\le}1 there are extended states. The intermediate regime 1<\ensuremath{\nu}<2 is discussed as well.