  Long-range hoppings in quantum disordered systems are known to yield quantum
multifractality, whose features can go beyond the characteristic properties
associated with an Anderson transition. Indeed, critical dynamics of long-range
quantum systems can exhibit anomalous dynamical behaviours distinct from those
at the Anderson transition in finite dimensions. In this paper, we propose a
phenomenological model of wave packet expansion in long-range hopping systems.
We consider both their multifractal properties and the algebraic fat tails
induced by the long-range hoppings. Using this model, we analytically derive
the dynamics of moments and Inverse Participation Ratios of the time-evolving
wave packets, in connection with the multifractal dimension of the system. To
validate our predictions, we perform numerical simulations of a Floquet model
that is analogous to the power law random banded matrix ensemble. Unlike the
Anderson transition in finite dimensions, the dynamics of such systems cannot
be adequately described by a single parameter scaling law that solely depends
on time. Instead, it becomes crucial to establish scaling laws involving both
the finite-size and the time. Explicit scaling laws for the observables under
consideration are presented. Our findings are of considerable interest towards
applications in the fields of many-body localization and Anderson localization
on random graphs, where long-range effects arise due to the inherent topology
of the Hilbert space.
