Superconductivity in the doped t − J model: Results for four-leg cylinders

We report a density-matrix renormalization group study of the lightly doped t−J model on a four-leg cylinder with doped hole concentrations per site δ=5%–12.5%. By keeping an unusually large number of states and long system sizes, we are able to accurately document the interplay between superconductivity, spin, and charge-density-wave orders. The long-distance behavior is consistent with that of a Luther-Emery liquid with a spin gap and power-law charge-density-wave and superconducting correlations. This is the widest t−J or Hubbard system in which power-law superconducting correlations have been established.