Two-dimensional Hubbard model

The single-band Hubbard model $H=-t\sum_{\langle ij\rangle\sigma}c^\dagger_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow}$ on the square lattice is the minimal model of strongly correlated electrons and the prime candidate for capturing cuprate high-Tc superconductivity. Its zero-temperature phase diagram as a function of filling and $U/t$ is the target.

What is known: Half-filled antiferromagnetic/Mott behavior has strong support, with rigorous results under particular lattice, coupling and limiting assumptions. Lieb's total-spin theorem is not a proof of long-range antiferromagnetic order at every coupling, and Nagaoka's one-hole infinite-repulsion theorem is a different limit. At finite doping, sign problems and finite-size/geometry effects complicate numerical extrapolation. Multi-method studies find closely competing stripe and pairing states in specific parameter regimes; these are not a rigorous full phase diagram.

The full doped phase diagram — whether and where the model superconducts, the nature of the pseudogap, and the competition with stripe/charge order — is contested even numerically, and there is no rigorous determination. This is a grand programme, not a single conjecture. Related: B11.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.