One Dimensional Chain with Long Range Hopping

The one-dimensional (1D) tight binding model with random nearest neighbor hopping is known to have a singularity of the density of states and of the localization length at the band center. We study numerically the effects of random long range (power-law) hopping with an ensemble averaged magnitude $\expectation{|t_{ij}|} \propto |i-j|^{-\sigma}$ in the 1D chain, while maintaining the particle-hole symmetry present in the nearest neighbor model. We find, in agreement with results of position space renormalization group techniques applied to the random XY spin chain with power-law interactions, that there is a change of behavior when the power-law exponent $\sigma$ becomes smaller than 2.