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

TL;DR

This paper introduces a rapid method to analyze Anderson localization in Dyson insulators with even number of lattice sites, revealing unique spectral and localization properties that differ from traditional Anderson insulators, with potential experimental relevance.

Contribution

The study provides a novel approach to understanding localization in Dyson insulators, highlighting differences in spectral statistics and localization behavior based on lattice parity and dimensionality.

Findings

Dyson insulators with even N mimic topological insulators with a pseudo-gap.

Energy-level statistics are intermediate between extended and localized regimes.

In 2D, the midband state exhibits fractal characteristics.

Abstract

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.