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. Previous measurements on the (001) interface have shown that the superconducting critical temperature, the Hall density, and the frequency of quantum oscillations, vary nonmonotonically and in a correlated fashion with the gate voltage. In this paper we experimentally demonstrate that 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. We develop a theoretical model, incorporating the different symmetries of these interfaces as well as electronic-correlation-induced band competition. We 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.
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