  We explore many-body entanglement in spinful Fermi gases with short-range
interactions, for metrology purposes. We characterize the emerging quantum
phases via Density-Matrix Renormalization Group simulations and quantify their
entanglement content for metrological usability via the Quantum Fisher
Information (QFI). Our study establishes a method, promoting the QFI to be an
order parameter. Short-range interactions reveal to build up metrologically
promising entanglement in the XY-ferromagnetic and cluster ordering, the
cluster physics being unexplored so far.
