Unusual properties of midband states in systems with off-diagonal disorder

TLDR

Off-diagonal disorder causes anomalous localization at the band center, unlike diagonal disorder. The study aims to show that the key distinction lies between bipartite and nonbipartite lattices rather than between diagonal and off‑diagonal disorder. The authors prove that bipartite lattices have zero‑energy eigenfunctions vanishing on one sublattice and analyze the random‑walk behavior of their amplitudes, finding that off‑diagonal disorder yields exp(−λ√r) localization while adding diagonal disorder biases the walk and restores ordinary exponential localization. Simulations confirm anomalous localization in one‑ and two‑dimensional systems, sometimes with additional periodic structure.

Abstract

It is known that off-diagonal disorder results in anomalous localization at the band center, whereas diagonal disorder does not. We show that the important distinction is not between diagonal and off-diagonal disorder, but between bipartite and nonbipartite lattices. We prove that bipartite lattices in any dimension (and some generalizations that are not bipartite) have zero energy (i.e., band-center) eigenfunctions that vanish on one sublattice. We show that ln\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{j}}$\ensuremath{\Vert} has random-walk behavior for one-dimensional systems with first-, or first- and third-neighbor random hopping, leading to exp(-\ensuremath{\lambda} \ensuremath{\surd}r) localization of the zero-energy eigenfunction. Addition of diagonal disorder leads to a biased random walk. First- and second-neighbor random hopping with no diagonal disorder leads to ordinary exponential [exp(-\ensuremath{\lambda}r)] localization. Numerical simulations show anomalous localization in dimensions 1 and 2, with additional periodic structure in some cases.