Phase diagram of the $t-t^{\mathrm{\prime }}-U$ chain at half filling

We study the half filled Hubbard chain including next-nearest-neighbor hopping $t^{\mathrm{\prime }}$. The model has three phases: one insulating phase with dominant spin-density-wave correlations at large distances (SDWI), another phase with dominant spin-dimer correlations or dimerized insulator (DI), and a third one in which long distance correlations indicate singlet superconductivity (SS). The boundaries of the SDWI are accurately determined numerically through a crossing of excited energy levels equivalent to the jump in the spin Berry phase. The DI-SS boundary is studied using several indicators like correlation exponent $K_{\rho }$, Drude weight $D_c$, localization parameter $z_L$ and charge gap ${\mathrm{\Delta }}_c$.