Does the two-dimensional Hubbard model superconduct?

The simplest model believed to capture high-temperature superconductivity, and its phase diagram is still contested.

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Problem statement

Consider the repulsive Hubbard model on the two-dimensional square lattice,

H = -t\sum_{\langle i,j\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow},

with $U > 0$ . Determine whether the ground state exhibits $d$ -wave superconducting order away from half filling, and map the phase diagram in the $(U/t, \text{doping}, T)$ space.

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Background

Anderson proposed the Hubbard model as the minimal description of the cuprates in 1987. Nearly four decades of numerical work — quantum Monte Carlo, DMRG, dynamical mean field theory, tensor networks — have produced competing answers, in part because the sign problem restricts unbiased Monte Carlo at low temperature and finite doping.

Known partial results

Multi-method comparisons (the Simons Collaboration benchmark studies) established that in the strongly correlated regime near $1/8$ doping the ground state has stripe order rather than uniform superconductivity, for the pure model.
Adding a next-nearest-neighbour hopping $t'$ changes the answer, and recent DMRG work finds superconducting order in that regime.
Whether the pure model superconducts at any doping remains contested, and the fermion sign problem is NP-hard in general.

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