Does the pure square-lattice Hubbard model superconduct at moderate doping
In plain words
With only nearest-neighbor hopping, the most accurate computer methods find stripes of charge and spin instead of superconductivity, but competing states lie very close in energy.
Precise statement
For the square-lattice Hubbard model with $t' = 0$, $U/t = 6\ \text{to}\ 8$ and hole doping $\delta = 1/8$, determine whether the ground state in the thermodynamic limit has long-range d-wave superconducting order or a stripe state whose pair correlations decay to zero. An answer is a controlled extrapolation of the pairing order parameter from at least two independent methods (DMRG, AFQMC, iPEPS, DMET) with consistent error bars.
What would settle it
Agreement of independent unbiased methods on the extrapolated d-wave order parameter at $\delta=1/8$ with error bars smaller than its claimed value.
Status in the literature
A multi-method study (2020) found filled stripes and no long-range d-wave order for $t' = 0$ at $\delta = 1/8$; competing uniform superconducting states lie close in energy.