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 behavior^5,6. 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.