Scaling theory of Anderson localization: A renormalization-group approach

A position-space renormalization-group method, suitable for studying the localization properties of electrons in a disordered system, was developed. Two different approximations to a well-defined exact procedure were used. The first method is a perturbative treatment to lowest order in the intercell couplings. This yields a localization edge in three dimensions, with a fixed point at the band center ($E=0\mathrm{}$) at a critical disorder ${\sigma }_c\simeq 7.0$. In the neighborhood of the fixed point the localization length $L$ is predicted to diverge as $L\sim {(\sigma -{\sigma }_c+\beta E^2)}^{-\nu }$. In two dimensions no fixed point is found, indicating localization even for small randomness, in agreement with Abrahams, Anderson, Licciardello, and Ramakrishnan. The second method is an application of the finite-lattice approximation, in which the intercell hopping between two (or more) cells is treated to infinite order in perturbation theory. To our knowledge, this method has not been previously used for quantum systems. Calculations based on this approximation were carried out in two dimensions only, yielding results that are in agreement with those of the lowest-order approximation.