  From a mean-field solution of the Hubbard-Holstein model, we show that a rich
variety of different electronic phases can result at the interface between two
polar materials such as LaAlO$_3$/SrTiO$_3$. Depending on the strengths of the
various competing interactions, viz., the electronic kinetic energy,
electron-phonon interaction, Coulomb energy, and electronic screening strength,
the electrons could (i) either be strongly confined to the interface forming a
2D metallic or an insulating phase, (ii) spread deeper into the bulk making a
3D phase, or (iii) become localized at individual sites forming a Jahn-Teller
polaronic phase. In the polaronic phase, the Coulomb interaction could lead to
unpaired electrons resulting in magnetic Kondo centers. Under appropriate
conditions, electronic phase separation may also occur resulting in the
coexistence of metallic and insulating regions at the interface.
