Plaquette versus ordinary d -wave pairing in the t′ -Hubbard model on a width-4 cylinder

Abstract

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 ${t}^{\ensuremath{'}}$ (the ${t}^{\ensuremath{'}}$-Hubbard model), pairing is strongly influenced by ${t}^{\ensuremath{'}}$. In particular, a recent study on a width-4 cylinder observed quasi-long-range superconducting order, associated with a negative ${t}^{\ensuremath{'}}$, which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-$4\phantom{\rule{4pt}{0ex}}{t}^{\ensuremath{'}}$-Hubbard model. We show that in this specific system, the pairing symmetry with ${t}^{\ensuremath{'}}<0$ 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 ${t}^{\ensuremath{'}}$ suppresses the conventional $d$-wave, leading to plaquette pairing. In contrast, a different ${t}^{\ensuremath{'}\ensuremath{'}}$ coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional $d$-wave pairing.