Intertwined Order in a Frustrated 4-leg $t - J$ Cylinder

TL;DR

This study uses density-matrix renormalization group methods to explore the complex interplay of superconductivity, spin, and charge orderings in a frustrated 4-leg $t$-$J$ cylinder model with doping, revealing diverse ordering phenomena and their dependencies.

Contribution

It provides a detailed numerical analysis of competing orders in a frustrated $t$-$J$ model, highlighting the effects of next-nearest interactions and magnetic frustration on charge, spin, and superconducting states.

Findings

Charge density waves can have periods of 4 or 8.

Superconductivity exhibits $d$-wave symmetry and is modulated at charge order vectors.

No evidence found for pair-density wave or charge 4e superconductivity.

Abstract

We report a density-matrix renormalization group study of the $t$-$J$ model with nearest ($t_1$ & $J_1$) and next-nearest ($t_2$ & $J_2$) interactions on a 4-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 $2 k_F$, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e. $J_2/J_1 \sim 1/2$) 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.