Lyapunov exponent and transfer-matrix spectrum of the random binary alloy

The one-dimensional Anderson model with binary distributed onsite disorder is investigated using the transfer-matrix approach. The random matrix product is transformed into an iterated conformal map and the convergence of the mapping is studied for various values of the concentration of onsite impurities and the energy. For dilute impurity concentrations the convergence of the mapping is strongly energy dependent and exhibits interesting behavior in the complex mapping plane. As the impurity concentration increases the mapping converges to the unit circle which results from the underlying conformal structure. Explicit expressions for the Lyapunov exponent and the eigenvalues of the transfer matrix are obtained in terms of the conformal map. From the Lyapunov exponent, the localization length and the integrated density of states are calculated. In the dilute limit there exist states in which the localizaiton length exceeds the system size. These states correspond to extended electronic states and it is shown that the number of extended states scales with system size as ${\mathit{N}}_{\mathrm{extended}}$∝${\mathit{N}}^{\mathrm{-}2}$. Increasing the impurity concentration beyond a critical value destroys these states and the localization length over the entire energy range becomes smaller than the system size.