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

We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently dis- tributed 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 rho(E) similar to 1/vertical bar E ln(3) vertical bar E parallel to and of the localization length xi(E) similar to vertical bar ln vertical bar E parallel to, 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 similar to exp[-root r/r(0)]. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small t is of the form similar to 1/[t ln(lambda+1)(1/t)], characterized by a single, continuously tunable parameter lambda > 0. We find, using a combination of analytic and numerical methods, that while the universal result applies for lambda > 2, it no longer holds in the interval 0 < lambda < 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 lambda; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below lambda = 1. For < 2, the fall-off of the E = 0 state at large distances also deviates from the universal result and is of the form similar to exp[-(r/r(0))(1/lambda), which decays faster than an exponential for lambda < 1.