Ground-state phase diagram of the doped 2D Hubbard model
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
Cite
@misc{cairn-hubbard-model-phase-diagram,
title = {Ground-state phase diagram of the doped 2D Hubbard model},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/hubbard-model-phase-diagram}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
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Current state
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The problem
The question. The single-band Hubbard model on the square lattice is the minimal model proposed for the cuprates. Its doped ground state results from tiny energy differences between competing orders (d-wave superconductivity, stripes, antiferromagnetism). For which values of U/t, t′/t and doping is the ground state superconducting in the thermodynamic limit?
Known status. The Simons Collaboration benchmark (LeBlanc et al., PRX 2015) compared many numerical methods on common parameter points. Qin et al. (PRX 2020), combining constrained-path AFQMC and DMRG, found the pure model (t′ = 0) non-superconducting at moderate-to-strong coupling near optimal hole doping. Xu et al. (Science 2024) found superconductivity in both electron- and hole-doped regimes once t′ is included, with partially filled stripes on the hole-doped side. The review by Arovas, Berg, Kivelson and Raghu (2022) lists controlled limits and open controversies.
What counts as progress
- New ground-state energies and correlation functions at published benchmark parameter points, with code, bond dimension / walker settings and extrapolation procedure, that agree with or improve on existing results (e.g. DMRG, AFQMC, iPEPS, neural-network quantum states).
- Controlled studies of finite-size and boundary-condition effects on pairing correlations.
- Cross-method comparison tables on identical parameters, including documented failures of a method in a regime.
How it is checked. Reviewers re-run the provided code at the stated parameters (or a reduced size) and verify the reported energies and correlators within the quoted error bars; energies are compared with the published benchmarks, where lower variational energies are directly comparable.