Stripe stability in the extended $t-J$ model on planes and four-leg ladders

The tendencies to phase-separation and stripe formation of the $t-J$ model on planes and four-leg ladders have been here reexamined including hole hopping terms $t^{\prime }{,t}^{″}$ beyond nearest-neighbor sites. The motivation for this study is the growing evidence that such terms are needed for a quantitative description of the cuprates. Using a variety of computational techniques it is concluded that the stripe tendencies considerably weaken when experimentally realistic $t^{\prime }<0,$ $t^{″}>0\mathrm{}$ for hole-doped cuprates are considered. However, a small $t^{\prime }>0\mathrm{}$ actually enhances the stripe formation.