The Hubbard model and its extensions are important microscopic models for understanding high-$T_c$ superconductivity in cuprates. In the model with next-nearest-neighbor hopping $t^{\prime }$ (the $t^{\prime }$-Hubbard model), pairing is strongly influenced by $t^{\prime }$. In particular, a recent study on a width-4 cylinder observed quasi-long-range superconducting order, associated with a negative $t^{\prime }$, which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-$4t^{\prime }$-Hubbard model. We show that in this specific system, the pairing symmetry with $t^{\prime }<0$ is not the ordinary $d$-wave one would expect in the 2D limit. Instead we observe a so-called plaquette $d$-wave pairing. We show that the plaquette $d$-wave or its extension is difficult to generalize in other geometries, for example a 4-leg ladder with open boundaries or a width-6 cylinder. We find that a negative $t^{\prime }$ suppresses the conventional $d$-wave, leading to plaquette pairing. In contrast, a different $t^{\prime \prime }$ coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional $d$-wave pairing.

• Received 11 April 2020
• Revised 21 June 2020
• Accepted 23 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.041106