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 $ -0.5 t \leq t'\leq 0.25 t$ with $U = 8t$ and $12t$ and doped hole concentrations $\delta=1/12$ and $1/8$. For $t'\leq 0$, we find that SC correlations are extremely short-ranged and the ground-state is a charge-density wave (CDW) of one sort or another. In some ranges of parameters, local magnetic order is found with a correlation length greater than half the cylinder width - suggestive that magnetic order might arise in the 2D limit. For $t'>0$, our results depend more strongly on boundary conditions (periodic vs antiperiodic), and admit the possibility of a SC phase 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.