Solving the Schrödinger equation on percolation structures by means of the equation of motion method

Abstract

We investigate the dynamical properties of the quantum percolation model at the critical percolation threshold pc in two dimensions by means of the spreading of a wave packet. We find that for t ∞ the mean square displacement R2(t) converges to R2(t) ≃ 92. In order to obtain a microscopic insight we investigate the averaged squared wave function 〈|ψ(r,t)|2〉 for fixed time t and find that up to a crossover distance r× its spatial decay is given by a stretched exponential with exponent u ≃ 0.58. The exponent u agrees with the exponent dψ obtained recently for the corresponding time independent localization problem. Furthermore we find that the averaged moments 〈|ψ(r,t)|2q〉 for fixed time t scale in a multifractal way 〈|ψ(r,t)|2q〉 ∼ 〈|ψ(r,t)|2〉τ(q) with τ(q) ∼ qδ and δ ≃ 1.25 ± 0.05.