We analytically calculate the local density of states for Cauchy random band matrices with strongly fluctuating diagonal elements. The Breit-Wigner form for ordinary band matrices is replaced by a Levy distribution of index $\mu=1/2$ and the characteristic energy scale $\alpha$ is strongly enhanced as compared to the Breit-Wigner width. The unperturbed eigenstates decay according to the non-exponential law $\propto e^{ -\sqrt{ \alpha t}}$. We analytically determine the localization length by a new method to derive the supersymmetric non-linear $\sigma$ model for this type of band matrices.
Categories: cond-mat.mes-hall cond-mat.dis-nn