Certified two-sided error bars for 2D Hubbard ground states
In plain words
For the two-dimensional Hubbard model, a basic model of electrons in a crystal, different simulation methods give competing ground states whose energies differ by less than one percent. Methods that give guaranteed upper and lower bounds tight enough to decide between them do not exist.
Precise statement
At $U/t = 8$ and filling $n = 0.875$ with $t' = 0$, DMRG, constrained-path auxiliary-field Monte Carlo, infinite projected entangled-pair states and density-matrix embedding agree on $E/N$ to about $0.01\,t$, while candidate stripe and uniform d-wave states differ by energies of order $1e-3\,t$ (Zheng et al., Science 2017). Construct methods giving rigorous lower bounds (e.g. reduced-density-matrix semidefinite programs) and variational upper bounds for $E/N$ and for long-distance pair correlations in the thermodynamic limit, with gap below $1e-3\,t$. An answer is a certified interval and the scaling of its width with computational cost.
What would settle it
A computation producing upper and lower bounds on E/N for the infinite lattice separated by less than 1e-3 t, together with bounds on pair correlations.