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We report a density-matrix renormalization group study of the lightly doped$t$-$J$ model on a 4-leg cylinder with doped hole concentrations per site$\delta=5\%\sim 12.5\%$. By keeping an unusually large number of states andlong system sizes, we are able to accurately document the interplay betweend-wave superconductivity (SC) and spin and charge-density-wave (CDW) orders.The long-distance behavior is consistent with that of a Luther-Emery liquidwith a spin-gap, power-law CDW correlations with wave-length $\lambda=1/2\delta$, and power-law SC correlations. This is the widest (most nearly 2D)such system for which power law SC correlations have been established.

Superconductivity in the doped $t$-$J$ model: results for four-leg cylinders