Extended Mean-Field Theory for the 2D Hubbard Model in Degenerate Dilute Electron Gases: Fluctuations, Superconducting Dome, and Interaction Mechanisms in Strontium Titanate

Strontium titanate ($\mathrm{SrTiO_3, STO}$) dome-shaped superconducting transition temperature as a function of chemical potential, consistent with STO experiments, and shows that tunable s-wave and d-wave symmetries are modulated by doping. Superconducting fluctuations validate the mean-field approximation at low temperatures but destroy pairing at higher temperatures. The charge-density-wave order competes with superconductivity, enhances the effective electron mass inversely with the chemical potential, and increases with the interaction strength $U$ and the temperature $T$. SDW order is rare and fragile, while an additional magnetic term induces subtle band splitting. These findings suggest e-e contributions to STO's transport anomalies and provide criteria to distinguish e-e from e-ph origins, offering insights for engineering higher $T_c$ in dilute systems.