Unbounded Quantum Diffusion and a New Class of Level Statistics

Abstract

We point out a new class of level statistics where the level spacing distribution follows inverse power laws p(s) ∼ s −β with 1 < β < 2. It is characteristic of hierarchical level clustering rather than level repulsion and appears to be universal for systems exhibiting unbounded quantum diffusion on 1d-lattices, σ 2(t) ∼ t 2δ with β = δ + 1. A realization of this class with β = 3/2 is a model of Bloch electrons in a magnetic field (Harper’s equation), where we find a purely diffusive spread of wave packets (δ = 1/2) without the quantum limitations known from chaotic systems like the kicked rotator. In the Fibonacci chain model the spread of wave packets shows anomalous diffusion with 0 < δ < 1 and gives rise to exponents β = δ + 1 that can be different from 3/2. We also study how a Cantor spectrum is affected by the onset of classical chaos. While the spectrum undergoes visible changes, its level spacing distribution is unaffected on small scales. In the time domain there is a crossover between two diffusive regimes characterized by a classical and a quantum mechanical diffusion coefficient.