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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 distributed random hopping and no on-site potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to sub-lattice symmetry, for each realization). For non-singular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $ρ(E) \sim 1/|E \ln^3|E||$ and of the localization length $ξ(E) \sim |\ln|E||$, 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, sub-exponential decay at large distances $\sim \exp [ -\sqrt{r/r_0} ]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\sim 1/ [t \ln^{λ+1}(1/t) ]$, 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 non-universal 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 $\sim \exp [-(r/r_0)^{1/λ}]$, which decays faster than an exponential for $λ< 1$.