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 (t(1) and J(1)) and next-nearest (t(2) and J(2)) interactions on a four-leg cylinder with concentration delta = 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 2k(F), we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., J(2)/J(1) similar to 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.