  The metal-insulator transition (MIT), which includes Anderson localization
and Mott insulators as extreme regimes, has received renewed interest as the
many-body effects often constitute a limitation for the handling of quantum
interference. This resulted in the field dubbed many-body localization (MBL),
intensively studied theoretically and experimentally as understanding the
appearance of equilibration and thermalization becomes relevant in dealing with
finite systems. Here, we propose a new observable to study this transition in a
spin chain under the ``disorder'' of a Harper-Hofstadter-Aubry-Andr\'e on-site
potential. This quantity, which we call zeroth-order gradient entanglement
(ZOGE) is extracted from the fundamental Fourier mode of a family of
out-of-time-ordered correlators (OTOCs). These are just Loschmidt Echoes, where
a field gradient is applied before the time reversal. In the absence of
many-body interactions, the ZOGE coincides with the inverse participation ratio
of a Fermionic wave function. By adding an Ising interaction to an XY
Hamiltonian, one can explore the MBL phase diagram of the system. Close to the
critical region, the excitation dynamics is consistent with a diffusion law.
However, for weak disorder, quantum diffusion prevails while for strong
disorder the excitation dynamics is sub-diffusive.
