Singular behavior of tight-binding chains with off-diagonal disorder

Abstract

We prove that the density of states of the one-dimensional tight-binding Hamiltonian with off-diagonal disorder is singular at the center of the band, $E=0\mathrm{}$, for every probability distribution of the hopping matrix elements $V$. The asymptotic form is $\rho \mathrm{}\left(E\right)\mathrm{}\simeq \frac{2{\sigma }^2}{{\left|E\right(\ln E^2\left)\right|}^3}$, with ${\sigma }^2\equiv 〈{\left(\ln V^2\right)}^2〉-{〈\ln V^2〉}^2$. The localization length goes to infinity as $L\left(E\right)\mathrm{}\simeq \frac{2\left|\ln E^2\right|}{{\sigma }^2}$. We also give a procedure to handle the problem numerically near the singularity, and we present some sample calculations.