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Plaquette versus ordinary $d$-wave pairing in the $t'$-Hubbard model on a width 4 cylinder

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 $t&#39;$ (the $t&#39;$- Hubbard model), pairing is strongly influenced by $t&#39;$ . In particular, a recent study on a width-4 cylinder observed quasi-long-rage superconducting order, associated with a negative $t&#39;$ , which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-4 $t&#39;$-Hubbard model. We show that in this specific system, the pairing symmetry with $t&#39;<0$ is not the ordinary $d$-wave one would expect in the 2D limit. Instead we observe a so-called plaquette d-wave pairing. The plaquette d-wave exists only on a width-4 cylinder, and so is not representative of the 2D limit. We find that a negative $t&#39;$ suppresses the conventional d-wave, leading to plaquette pairing. In contrast, a different $t&#39;&#39;$ coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional $d$-wave pairing.