A fast approach to Anderson localization for even-$N$ Dyson insulator

Dyson insulators with random hoppings in a lattice approach localization faster compared to the usual Anderson insulators with site disorder. For even-$N$ lattice sites the Dyson insulators mimic topological insulators with a pseudo-gap at the band center and the energy-level statistics obtained via the $P(S)$ distribution is of an intermediate type close to the Anderson localized Poisson limit. For odd-$N$ level-repulsion and Wigner statistics appears as in the quasi-metallic regime of $2D$ Anderson insulators, plus a single $E=0$ mode protected by chiral symmetry. The distribution of the participation ratio and the multifractal dimensions of the midband state are computed. In $1D$ the Dyson state is localized and in $2D$ is fractal. Our results might be relevant for recent experimental studies of chiral localization in photonic waveguide arrays.