Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping

Abstract

We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states ρ(E)∼1/|Eln3|E|| and of the localization length ζ(E)∼|ln|E||, near the band center E=0. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at E=0 shows a universal, subexponential decay at large distances ∼exp[-r/r0]. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small t is of the form ∼1/[tlnλ+1(1/t)], characterized by a single, continuously tunable parameter λ>0. We find, using a combination of analytic and numerical methods, that while the universal result applies for λ>2, it no longer holds in the interval 0<λ<2. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter λ; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below λ=1. For λ<2, the fall-off of the E=0 state at large distances also deviates from the universal result and is of the form ∼exp[-(r/r0)1/λ], which decays faster than an exponential for λ<1.