Density of States in Coupled Chains with Off-Diagonal Disorder

We compute the density of states $\rho \left(\varepsilon \right)$ in $N$ coupled chains with random hopping. At zero energy, $\rho \left(\varepsilon \right)$ shows a singularity that strongly depends on the parity of $N$. For odd $N$, $\rho \left(\varepsilon \right)\propto 1/|\varepsilon {ln}^3\varepsilon |$, with and without time-reversal symmetry. For even $N$, $\rho \left(\varepsilon \right)\propto |\ln \varepsilon |$ in the presence of time-reversal symmetry, while there is a pseudogap, $\rho \left(\varepsilon \right)\propto |\varepsilon \ln \varepsilon |$, in the absence of time-reversal symmetry.