Commensurability effects in one-dimensional Anderson localization: Anomalies in eigenfunction statistics Original Research Article

The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any rational point image, where a is the lattice constant and λE is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions ψ(r) at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function Φr(u, ϕ) (u and ϕ have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at image with q > 2. The descender of the generating function image is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states.