Plaquette versus ordinary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-wave pairing in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>-Hubbard model on a width-4 cylinder
Chia-Min Chung, Mingpu Qin, Shiwei Zhang, Ulrich Schollwöck, Steven R. White
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.