Localization in self-affine energy landscapes

We discuss the localization behavior of quantum particles in a one-dimensional Anderson model with self-affine random potentials, characterized by a Hurst exponent $H>\mathrm{}0.\mathrm{}$ Depending on H and energy E, a new type of “strong” localization can occur, where all states are localized in a way different from the regular Anderson localized states. Using scaling arguments, we derive an analytical expression for the phase diagram and test it by numerical calculations. Finally, we consider a somewhat related model where the variance of the potential fluctuations is kept fixed for all system sizes L and a transition between localized and apparently extended states has been reported.