The Anderson Metal — Insulator Transition: Incommensurate Versus Disordered Systems

Abstract

Much work has been devoted to the study of quantum transport or wave propagation in disordered systems in connection with the phenomenon of Anderson localisation 1 and the assosiated metal insulator transition2. The electron properties and lattice dynamics of modulated structures and quasicrystals have also been considered3. In this context Aubry and Andre4 studied the discrete Harper’s equation used to describe the quantum theory of an electron confined in a plane with a periodic potential in the plane and a uniform perpendicular magnetic field. These systems are intermediate between periodic and random and they often exhibit an Anderson localization transition with a rich complex scaling behavior5,6. From a duality property it was shown4 that the Anderson transition occurs by breaking of anaiyticity at a critical value of the incommensurate potential strength. The critical properties of incommensurate systems are mostly understood4. The Anderson transition in a random potential is more difficult. Here, I shall demonstrate the extra difficulties, emphasize the common characteristics and point out the differences of the two transitions. I shall also present some recent results by using related techniques.