Competition between charge-density-wave and superconducting orders on eight-leg square Hubbard cylinders

The issue of whether $d$-wave superconductivity (SC) occurs in the square-lattice Hubbard model with $U$ of order of the bandwidth has been one of the most debated issues to emerge from the study of high temperature SC. Here, we report variational results on eight-leg cylinders with next-nearest-neighbor hopping in the range $-0.5t\le t^{\mathrm{\prime }}\le 0.25t$ with $U=8t$ and $12t$ and doped hole concentrations $\delta =1\mathrm{/}12$ and $1\mathrm{/}8$. For $t^{\mathrm{\prime }}\le 0$, the ground-state appears to be a charge-density wave (CDW) of one sort or another with SC correlations that are extremely short-ranged. In contrast, in some cases, the local magnetic order has a correlation length greater than half the cylinder width - suggestive that magnetic order might also arise in the 2D limit. For $t^{\mathrm{\prime }}>0$, our results depend more strongly on boundary conditions (periodic vs antiperiodic), making it still harder to correctly guess whether SC or CDW correlations dominate in the 2D limit. These results were obtained employing matrix-product states with bond dimensions large enough that energy differences as small as ${10}^{-3}t$ per site can be resolved.