  Understanding the role of defects in oxide heterostructures is crucial for
future materials control and functionalization. We hence study the impact of
oxygen vacancies (OVs) at variable concentrations on orbital- and spin exchange
in the LaAlO$_3$/SrTiO$_3$ interface by first principles many-body theory and
real-space model-Hamiltonian techniques. Intricate interplay between Hubbard
$U$ and Hund's coupling $J_{\rm H}$ for OV-induced correlated states is
demonstrated. Orbital polarization towards an effective $e_g$ state with
predominant local antiferromagnetic alignment on Ti sites near OVs is
contrasted with $t_{2g}(xy)$ states with ferromagnetic tendencies in the
defect-free regions. Different magnetic phases are identified, giving rise to
distinct net-moment behavior at low and high OV concentrations. This provides a
theoretical basis for prospective tailored magnetism by defect manipulation in
oxide interfaces.
