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

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{}ψⱼ{} has random-walk behavior for one-dimensional systems with first-, or first- and third-neighbor random hopping, leading to exp(-{λ} {}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(-{λ}r)] localization. Numerical simulations show anomalous localization in dimensions 1 and 2, with additional periodic structure in some cases.