Critical properties of electron eigenstates in incommensurate systems

Abstract
This paper describes some properties of the eigenvalue equation ψn+1 + ψn-1 + 2α cos (2πβn + ∆) ψn = Eψn. This is an example of the more general problem of a Hermitian eigenvalue equation in the form of a difference equation with periodic coefficients.These equations arise in solid state physics; they occur in connection with tight-binding models for electrons in one-dimensional solids with an incommensurate modulation of the structure, and in models for the energy bands of Bloch electrons moving in a plane with a perpendicular magnetic field. The model studied has a critical point when α = 1. Following some earlier work by Azbel (Azbel, M. Ya., Phys. Rev. Lett. 43, 1954 (1979)), an approximate renormalization group transformation is derived. This predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point, which is dependent on the expansion of β as a continued fraction. Also, when β is an irrational number, there is a localization transition from extended states to localized states as α increases through the critical point. This localization transition, which was previously discovered by Aubry & André (Aubry, S. & André, G. Ann. Israel phys. Soc. 3, 133 (1979)) using the Thouless formula for the localization length, is explained by the renormalization group transformation derived here.