Spread of wave packets in disordered hierarchical lattices

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 $⟨r^2⟩\propto t^{\beta }$ is governed by the “diffusion” exponent $\beta =1$ outside the ballistic regime $t>\tau \sim 1$ and $⟨r^2⟩\propto t^2$ 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.