The 2023 framework of Honet, Henrard and Meunier reformulates the mean-field approximation of the Hubbard model directly in the many-body (Fock-space) basis used by exact diagonalization (MF-MB), exposing an ambiguity in the mean-field density of states and satellite peaks generated by purely mean-field sector readjustment. The paper's conclusion explicitly leaves as intended future work the rewriting of known symmetry-restoration procedures — projection of broken-symmetry mean-field states onto good spin (and spatial) quantum numbers — in this many-body basis. No later work has formulated, implemented, or benchmarked symmetry restoration within MF-MB. The open problem is to define the projection operators acting on the self-consistent MF-MB states in the N_el and N_el±1 sectors, to implement them on the framework's own small-chain test systems, to quantify how much correlation energy and exact-diagonalization density-of-states structure the restored states recover, and to compare accuracy and numerical cost with symmetry restoration applied to the usual single-particle-basis mean field.

Background

The single-band Hubbard model is the minimal lattice model of correlated electrons: electrons hop between nearest-neighbor sites with amplitude tt and pay an on-site Coulomb penalty UU for double occupation. Exact diagonalization (ED) solves it by diagonalizing the Hamiltonian in the many-body basis — the Fock space of all Slater determinants (antisymmetrized many-electron basis states) over the lattice sites — whose dimension grows combinatorially with system size. The usual mean-field approximation (called MF-U in this literature; essentially unrestricted Hartree–Fock) decouples the interaction into a self-consistent one-body potential and works entirely in the single-electron basis: it is cheap but misses electron correlation, and its self-consistent solutions often break symmetries (spin or spatial) of the exact ground state. Because ED lives in the many-body basis and MF-U in the single-particle one, the two are hard to compare beyond total energies. Honet, Henrard and Meunier (2023) therefore reformulated the mean-field approximation itself in the many-body basis (MF-MB): the self-consistent mean-field solution is represented and iterated directly in the Fock-space basis used by ED, at much higher computational cost, but producing Slater-determinant states that can be compared one-to-one with ED eigenstates. On two- and three-site chains this exposed an ambiguity in the mean-field density of states (DOS): the usual recipe implicitly freezes the NelN_{\mathrm{el}}-electron orbitals when adding or removing an electron (Koopmans' theorem), whereas recomputing the mean field in the Nel±1N_{\mathrm{el}}\pm1 sectors produces extra poles and forces a manual, physically motivated exclusion of some sector ground states; the ED-like DOS definition also generates satellite peaks in a purely mean-field calculation, although satellites are conventionally read as signatures of correlation.

Mean-field states that break a symmetry of the Hamiltonian (for example spin-polarized or antiferromagnetically ordered self-consistent solutions) can be symmetrized after the fact by symmetry restoration: the broken-symmetry determinant is projected onto good quantum numbers with projection operators (projection after variation), or the projection is built into the variational search (variation after projection), partially recovering correlation energy at moderate cost. These techniques originate largely in nuclear structure (projected Hartree–Fock–Bogoliubov and the generator coordinate method) and in studies of finite fermion systems such as quantum dots; the two works the source paper explicitly points to are the reviews by Yannouleas and Landman (2007) and by Sheikh, Dobaczewski, Ring, Robledo and Yannouleas (2021). In the single-particle basis such restorations are well established for lattice Hubbard models. Acting on the MF-MB self-consistent states — that is, formulated inside the many-body basis, sector by sector, where the broken-symmetry determinants of the NelN_{\mathrm{el}} and Nel±1N_{\mathrm{el}}\pm1 sectors and the exclusion rules of the DOS construction live — symmetry restoration has not been formulated, implemented, or assessed; the source paper's conclusion states this verbatim as intended future work.

Problem Statement

Formulate symmetry restoration for the mean-field approximation of the Hubbard model expressed in the many-body basis (MF-MB). The mandatory floor of the deliverable is spin projection in projection-after-variation form; spatial and particle-number projection, and the variation-after-projection form, are to be carried out wherever the broken-symmetry self-consistent state permits, and the answer must state explicitly, for each of them, whether it applies to the state considered. The formulation must define how the projection operators act on the self-consistent MF-MB states in the NelN_{\mathrm{el}} and Nel±1N_{\mathrm{el}}\pm1 sectors, including the norm and overlap kernels between the non-orthogonal broken-symmetry determinants, and how the projected states enter the density-of-states construction (including its sector-degeneracy exclusion rules). Implement the procedure on the same small Hubbard chains used to define the framework — at minimum the two-site chain at half and quarter filling and the three-site chain, at stated interaction strengths such as U=0.5tU=0.5t — and assess it: how much correlation energy and how much of the ED density-of-states structure (including resolution of the sector-degeneracy/exclusion ambiguity and the fate of the mean-field satellite peaks) does the restored MF-MB state recover relative to unprojected MF-MB, and how do its accuracy and numerical cost compare with symmetry restoration applied to the usual single-particle-basis mean field (MF-U) on the same systems? The completion boundary is a mathematically well-defined formulation together with a quantitative ED-referenced benchmark and cost assessment on these systems; generalization to larger lattices is not required.

References

  1. Antoine Honet, Luc Henrard, Vincent Meunier, "Mean-field approximation of the Fermi–Hubbard model expressed in a many-body basis", AIP Advances 13, 075210 (2023). DOI: 10.1063/5.0153076, https://doi.org/10.1063/5.0153076 (preprint arXiv:2301.03223, https://arxiv.org/abs/2301.03223)
  2. C. Yannouleas, U. Landman, "Symmetry breaking and quantum correlations in finite systems: studies of quantum dots and ultracold Bose gases and related nuclear and chemical methods", Reports on Progress in Physics 70, 2067–2148 (2007). DOI: 10.1088/0034-4885/70/12/R02, https://doi.org/10.1088/0034-4885/70/12/R02
  3. J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, C. Yannouleas, "Symmetry restoration in mean-field approaches", Journal of Physics G: Nuclear and Particle Physics 48 (2021). DOI: 10.1088/1361-6471/ac288a, https://doi.org/10.1088/1361-6471/ac288a
  4. Antoine Honet, Luc Henrard, Vincent Meunier, "Exact and many-body perturbation solutions of the Hubbard model applied to linear chains", AIP Advances 12, 035011 (2022). DOI: 10.1063/5.0082681, https://doi.org/10.1063/5.0082681 (preprint arXiv:2111.13536, https://arxiv.org/abs/2111.13536)
  5. Antoine Honet, Luc Henrard, Vincent Meunier, "Correlation effects on topological end-states in finite-size graphene nanoribbons in the GW approximation", Journal of Physics: Condensed Matter 35, 485703 (2023). DOI: 10.1088/1361-648X/acf35f, https://doi.org/10.1088/1361-648X/acf35f (preprint arXiv:2304.02910 under the earlier title "Effect of electronic correlation on topological end-states in finite-size graphene nanoribbons", https://arxiv.org/abs/2304.02910)
  6. Antoine Honet, Luc Henrard, Vincent Meunier, "Robust correlated magnetic moments in end-modified graphene nanoribbons", Carbon Trends 26, 100377 (2024). DOI: 10.1016/j.cartre.2024.100377, https://doi.org/10.1016/j.cartre.2024.100377 (preprint arXiv:2310.09057, https://arxiv.org/abs/2310.09057)
  7. Mona Abdi, Bandar Astinchap, Erfan Norian, "Dynamical spin susceptibility and magnetic ordering in a square-octagon lattice in the Hubbard model", Scientific Reports 16, 1492 (published online December 2025; volume 16 dated 2026). DOI: 10.1038/s41598-025-31849-7, https://doi.org/10.1038/s41598-025-31849-7
  8. Md. Monirul Islam, "Advancing Electronic Structure Modeling for Next Generation Quantum Materials", Journal of Sustainable Engineering & Renewable Energy 1, 45-53 (2025). DOI: 10.54536/jsere.v1i2.6771, https://doi.org/10.54536/jsere.v1i2.6771