Spin-Gap Phase in the One-Dimensional t-J-J' Model
  The spin-gap phase of the one-dimensional t-J-J' model is studied by the level-crossing of the singlet and the triplet excitation spectra.   The phase boundary obtained between the Tomonaga-Luttinger and the spin-gap phases is remarkably consistent with the analytical results at the $J,J'\to 0$ and the low-density limits discussed by Ogata et al.   The spin-gap phase has a single domain in the phase diagram even if the spin gap opens at half-filling.   The phase boundary coincides with the $K_{\rho}=1$ line where the Tomonaga-Luttinger liquid behaves as free electrons, in the low-density region.   The relation between our method and the solution of the two-electron problem is also discussed. 
Masaaki Nakamura