Spread of wave packets in disordered hierarchical lattices

Abstract

We consider the spreading of the wave packet in the generalized Rosenzweig-Porter random matrix ensemble in the region of non-ergodic extended states $1<\gamma<2$. We show that despite non-trivial fractal dimensions $0 < D_{q}=2-\gamma<1$ characterize wave function statistics in this region, the wave packet spreading $\langle r^{2} \rangle \propto t^{\beta}$ is governed by the "diffusion" exponent $\beta=1$ outside the ballistic regime $t>\tau\sim 1$ and $\langle r^{2}\rangle \propto t^{2}$ in the ballistic regime for $t<\tau\sim 1$. This demonstrates that the multifractality exhibits itself only in {\it local} quantities like the wave packet survival probability but not in the large-distance spreading of the wave packet.