Coexistence of superconductivity with partially filled stripes in the Hubbard model

Combining the complementary capabilities of two of the most powerful modern computational methods, we find superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model (with next nearest neighbor hopping). In the electron-doped regime, superconductivity is weaker and is accompanied by antiferromagnetic N\'eel correlations at low doping. The strong 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 next nearest neighbor hopping) nor half-filled as seen in previous state-of-the-art calculations, vary in fillings between 0.6 and 0.8. The resolution of the tiny energy scales separating competing orders requires exceedingly high accuracy combined with averaging and extrapolating with a wide range of system sizes and boundary conditions. These results validate the applicability of this iconic model for describing cuprate high-$T_c$ superconductivity.