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Combining the complementary capabilities of two of the most powerful moderncomputational methods, we find superconductivity in both the electron- andhole-doped regimes of the two-dimensional Hubbard model (with next nearestneighbor hopping). In the electron-doped regime, superconductivity is weakerand is accompanied by antiferromagnetic N\'eel correlations at low doping. Thestrong superconductivity on the hole-doped side coexists with stripe order,which persists into the overdoped region with weaker hole density modulation.These stripe orders, neither filled as in the pure Hubbard model (no nextnearest neighbor hopping) nor half-filled as seen in previous state-of-the-artcalculations, vary in fillings between 0.6 and 0.8. The resolution of the tinyenergy scales separating competing orders requires exceedingly high accuracycombined with averaging and extrapolating with a wide range of system sizes andboundary conditions. These results validate the applicability of this iconicmodel for describing cuprate high-$T_c$ superconductivity.

Coexistence of superconductivity with partially filled stripes in the Hubbard model