Commensurability effects in one-dimensional Anderson localization: anomalies in eigenfunction statistics

The one-dimensional (1d) Anderson model (AM) has statistical anomalies at any rational point $f=2a/\lambda_{E}$, where $a$ is the lattice constant and $\lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions $\psi(r)$ at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function $\Phi_{r}(u, \phi)$ ($u$ and $\phi$ have a meaning of the squared amplitude and phase of eigenfunctions, $r$ is the position of the observation point). The descender of the generating function ${\cal P}_{r}(\phi)\equiv\Phi_{r}(u=0,\phi)$ is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we have derived a second-order partial differential equation for the $r$-independent ("zero-mode") component $\Phi(u, \phi)$ at the $E=0$ ($f=\frac{1}{2}$) anomaly. This equation is nonseparable in variables $u$ and $\phi$. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for $\Phi(u, \phi)$ explicitly in quadratures. Using this solution we have computed moments $I_{m}=N<|\psi|^{2m}>$ ($m\geq 1$) for a chain of the length $N \rightarrow \infty$ and found an essential difference between their $m$-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the "extrinsic" localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio ("intrinsic" localization length). This is not the case at the $E=0$ anomaly where the extrinsic localization length is smaller than the intrinsic one.