We report a density-matrix renormalization group study of the $t$-$J$ modelwith nearest ($t_1$ & $J_1$) and next-nearest ($t_2$ & $J_2$) interactions on a4-leg cylinder with concentration $\delta=1/8$ of doped holes. We observe anastonishingly complex interplay between uniform $d$-wave superconductivity (SC)and strong spin and charge density wave ordering tendencies (SDW and CDW).Depending on parameters, the CDWs can be commensurate with period 4 or 8. Bycomparing the charge ordering vectors with $2 k_F$, we rule out Fermi surfacenesting-induced density wave order in our model. Magnetic frustration (i.e.$J_2/J_1 \sim 1/2$) significantly quenches SDW correlations with little effecton the CDW. Typically, the SC order is strongly modulated at the CDW orderingvector and exhibits $d$-wave symmetry around the cylinder. There is no evidenceof a near-degenerate tendency to pair-density wave (PDW) ordering, charge 4eSC, or orbital current order.
展开▼
