Probability distribution and new scaling law for the resistance of a one-dimensional Anderson model

Abstract

The exact probability distributions of the resistance $\rho$, the conductance, and $\ln (1+\rho )$ are calculated for the 1D Anderson model with purely off-diagonal disorder at $E=0\mathrm{}$. Analysis of the distribution yields the surprising results that $〈\rho 〉$ grows exponentially with length, despite previous studies indicating that the state at $E=0\mathrm{}$ is extended, and the typical resistance $\tilde{\rho }$ increases as $\exp \left(L^{\frac{1}{2}}\right)$. The relationship between this behavior and the temperature dependence of the resistance of thin wires is discussed.