| Summary: | We study analytically and numerically the dynamics of the generalized
Rosenzweig-Porter model, which is known to possess three distinct phases:
ergodic, multifractal and localized phases. Our focus is on the survival
probability $R(t)$, the probability of finding the initial state after time
$t$. In particular, if the system is initially prepared in a highly-excited
non-stationary state (wave packet) confined in space and containing a fixed
fraction of all eigenstates, we show that $R(t)$ can be used as a dynamical
indicator to distinguish these three phases. Three main aspects are identified
in different phases. The ergodic phase is characterized by the standard
power-law decay of $R(t)$ with periodic oscillations in time, surviving in the
thermodynamic limit, with frequency equals to the energy bandwidth of the wave
packet. In multifractal extended phase the survival probability shows an
exponential decay but the decay rate vanishes in the thermodynamic limit in a
non-trivial manner determined by the fractal dimension of wave functions.
Localized phase is characterized by the saturation value of $R(t\to\infty)=k$,
finite in the thermodynamic limit $N\rightarrow\infty$, which approaches
$k=R(t\to 0)$ in this limit.
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