Intertwined order in a frustrated four-leg t − J cylinder

We report a density-matrix renormalization group study of the t−J model with nearest (t1 and J1) and next-nearest (t2 and J2) interactions on a four-leg cylinder with concentration δ=1/8 of doped holes. We observe an astonishingly 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. By comparing the charge ordering vectors with 2kF, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., J2/J1∼1/2) significantly quenches SDW correlations with little effect on the CDW. Typically, the SC order is strongly modulated at the CDW ordering vector and exhibits d-wave symmetry around the cylinder. There is no evidence of a near-degenerate tendency to pair-density wave (PDW) ordering, charge 4e SC, or orbital current order.