Localization and absence of Breit-Wigner form for Cauchy random band matrices

Abstract

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 $μ=1/2$ and the characteristic energy scale $α$ is strongly enhanced as compared to the Breit-Wigner width. The unperturbed eigenstates decay according to the non-exponential law $\propto e^{-\sqrt{αt}}$. We analytically determine the localization length by a new method to derive the supersymmetric non-linear $σ$ model for this type of band matrices.