We have studied the ground state properties of Hubbard model on long six-leg square cylinders with doped hole concentration $5.55\%\le \delta \le 12.5\%$ using density-matrix renormalization group. Our state-of-the-art DMRG study with a large number of states convincingly shows that the nature of the ground state is remarkably sensitive to the presence of next-nearest-neighbor electron hopping $t^{\prime }$. In the positive $t^{\prime }$ side, we find a robust $d$-wave superconducting (SC) phase characterized by coexisting quasi-long-range SC and charge density wave (CDW) correlations. Without $t^{\prime }$ the ground state forms an insulating stripe phase with long-range CDW order. In stark contrast to four-leg cylinders, our results show that the lightly doped Hubbard model on six-leg cylinders remains insulating in the negative $t^{\prime }$ side where the SC correlations decay exponentially with short correlation lengths. In the larger negative $t^{\prime }$ side, the doped holes form a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin-spin correlations.

• Received 13 April 2023
• Accepted 17 January 2024

DOI:https://doi.org/10.1103/PhysRevB.109.085121