Phase diagram of the $t-t-U$ chain at half filling

We study the half filled Hubbard chain including next-nearest-neighbor hopping $t$. 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 $Delta_{c}$.