The interface between the two insulating oxides SrTiO$_3$ and LaAlO$_3$ gives rise to a two-dimensional electron system with intriguing transport phenomena, including superconductivity, which are controllable by a gate. I will present experimental results by the Dagan group at Tel Aviv University, which for the first time measured together the Hall response, quantum oscillations, and superconductivity, both on the (001) interface and the much less understood (111) interface. In the (001) interface the superconducting critical temperature, the Hall density, and the frequency of quantum oscillations, vary nonmonotonically and in a correlated fashion with the gate voltage. The (111) interface features a qualitatively distinct behavior, in which the frequency of Shubnikov-de Haas oscillations changes monotonically, while the variation of other properties is nonmonotonic albeit uncorrelated. I will present a theoretical model, incorporating the different symmetries of these interfaces as well as electronic-correlation-induced band competition. I will show that the latter dominates at (001), leading to similar nonmonotonicity in all observables, while the former is more important at (111), giving rise to highly curved Fermi contours, and accounting for all its anomalous transport measurements. These results form the necessary basis for future studies of the origin and behavior of superconductivity, magnetism, and ferroelectricity in these and related interfaces.
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