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From a mean-field solution of the Hubbard-Holstein model, we show that a richvariety of different electronic phases can result at the interface between twopolar materials such as LaAlO$_3$/SrTiO$_3$. Depending on the strengths of thevarious 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 a2D metallic or an insulating phase, (ii) spread deeper into the bulk making a3D phase, or (iii) become localized at individual sites forming a Jahn-Tellerpolaronic phase. In the polaronic phase, the Coulomb interaction could lead tounpaired electrons resulting in magnetic Kondo centers. Under appropriateconditions, electronic phase separation may also occur resulting in thecoexistence of metallic and insulating regions at the interface.

Electronic Phases and Phase Separation in the Hubbard-Holstein Model of a Polar Interface