Electron Localization in One-Dimensional Incommensurate Potentials

Abstract

A one-dimensional potential V(x) consisting of two periodic components whose periods are incommensurate can give rise to localized electron states. Conventional numerical methods to calculate the localized eigenstates, either of the Schrodinger equation or the tight-binding models (TBM), are plagued by a numerical instability originating in a second solution which diverges for |x|→∞. We present a practical numerical algorithm for an arbitrary TBM which is free of this difficulty and which provides the localized eigenstates and their energies to ultra-high precision. The underlying method incorporates a generalization of a classic theorem of Pincherele for the existence of minimal solutions to three-term recurrence relations. We present numerical results for the Aubry model for ∆>1, where ∆=V_I/(2V_H) is a dimensionless coupling constant, V_I is the strength of the incommensurate component of the one-electron potential and V_H is the nearest-neighbor hopping matrix element. We propose that the well-known transition from localized to extended eigenstates which occurs as ∆ is reduced towards unity is accompanied by an incipient infinite degeneracy of the localized states. Our numerical results for the energy difference of any pair of nearly-degenerate localized states is well described by a power law, (∆-1)^Y, but with a non-universal exponent Y.