Cesca and Simenel (Phys. Rev. C 108, 054307 (2023)) benchmarked Hartree-Fock (HF) with broken and restored rotational symmetry against quasi-exact variational Monte Carlo (VMC) ground states of identical fermions on a one-dimensional quantum ring, measuring how much correlation energy symmetry breaking and angular-momentum projection recover. They explicitly left open three linked generalizations: (i) other continuous symmetries, in particular the global U(1) gauge symmetry associated with particle-number conservation, whose breaking (pairing) and restoration (particle-number projection) underlies nuclear superfluidity; (ii) extension to higher spatial dimensions, i.e. two-dimensional spherium and three-dimensional glomium, to quantify how projection efficacy depends on dimensionality; and (iii) evaluation with interactions that include dynamical correlations (energy density functionals or renormalized shell-model effective interactions), to measure the practical benefit of projection in realistic nuclear-structure calculations. As of 2026, the problem remains open.

Background

In the mean-field (Hartree-Fock, HF) description of a finite many-fermion system, the variational wave function is a single Slater determinant of single-particle orbitals. This approximation misses correlation energy, the difference between the HF energy and the exact ground-state energy. Part of it — so-called static correlations — can be captured by allowing the HF solution to break a symmetry of the Hamiltonian (for example rotational invariance), which lets the determinant localize particles in an intrinsic frame; the remaining part, dynamical correlations, reflects short-range relative motion and is not representable by any single determinant. In nuclear structure, dynamical correlations are usually absorbed phenomenologically into energy density functionals (EDFs, e.g. Skyrme or Gogny functionals fitted to nuclear data), while static correlations are recovered by symmetry breaking. The broken symmetry can then be restored by projection: the projected state is a weighted sum (integral) over the group of the broken transformation, restoring good quantum numbers such as total angular momentum (angular-momentum projection) or particle number (particle-number projection). Restoring the global U(1) gauge symmetry associated with particle-number conservation is the standard remedy for the particle-number mixing introduced by Bardeen-Cooper-Schrieffer (BCS) and Hartree-Fock-Bogoliubov (HFB) pairing theories, which describe superfluidity in nuclei. Beyond-mean-field methods combining EDFs with projection are mature (reviewed by Sheikh, Dobaczewski, Ring, Robledo and Yannouleas, J. Phys. G 48, 123001 (2021)), and symmetry-broken and restored formulations also exist at the coupled-cluster level (Duguet, J. Phys. G 42, 025107 (2014); Duguet and Signoracci, J. Phys. G 44, 015103 (2016)). However, exactly (or quasi-exactly) solvable benchmark systems against which the effectiveness of breaking and restoration can be measured quantitatively are rare. Cesca and Simenel (Phys. Rev. C 108, 054307 (2023)) introduced such a benchmark: NN identical, fully spin-polarized fermions constrained to a circle of radius RR (a one-dimensional quantum ring), interacting through model two-body potentials depending on the through-the-ring distance — repulsive and attractive 'Coulomb' forms VC±(r)=±V0/rV_{C\pm}(r)=\pm V_0/r and two 'nuclear' forms VN1V_{N1}, VN2V_{N2} with hard-core repulsion and longer-range attraction. They compared (a) unbroken-symmetry HF (USHF), (b) broken-symmetry HF (BSHF, rotational invariance not enforced), and (c) restored-symmetry HF (RSHF, obtained by angular-momentum projection, i.e. integrating the rotated broken-symmetry determinant over the rotation angle, evaluated by Monte Carlo sampling) against a variational Monte Carlo (VMC) quasi-exact ground state with a Jastrow-correlated trial function (a trial wave function multiplying the Slater determinant by pairwise correlation factors). The figure of merit is the fraction of correlation energy (defined relative to USHF) recovered by each method. Their findings: symmetry breaking helps little for weak repulsion or attractive systems on small rings, helps substantially for strong repulsion and for self-bound attractive systems on larger rings, and projection recovers most of the remaining correlation energy in repulsive systems. The precursor work of Bray and Simenel (Phys. Rev. C 103, 014302 (2021)) established the VMC reference energies and the failure modes of USHF on the same ring model. Natural higher-dimensional analogues exist in quantum chemistry: spherium, fermions constrained to the surface of a two-sphere (D=2\mathcal{D}=2), and glomium, fermions on the surface of a three-sphere (D=3\mathcal{D}=3); the two-electron versions are quasi-exactly solvable, and the correlation energy of spherium is known to depend strongly on dimensionality (Loos and Gill, Phys. Rev. Lett. 103, 123008 (2009)). Angular-momentum-projected HF has also been benchmarked against shell-model calculations by Lauber, Frye and Johnson (J. Phys. G 48, 095107 (2021)). What does not yet exist is any measurement, against quasi-exact references, of how effective symmetry breaking and restoration is for the U(1) gauge symmetry, for higher-dimensional versions of the ring model, or when dynamical correlations are included through EDFs or renormalized shell-model interactions.

Problem Statement

The unified objective, carried over from Cesca and Simenel's benchmark, is to quantify — against quasi-exact (variational Monte Carlo or exact) ground-state references — the fraction of correlation energy recovered by breaking and restoring continuous symmetries at the mean-field level in controlled finite-fermion model systems, extended along the three axes the source itself identifies: (1) Symmetry axis: construct a version of the quantum-ring benchmark (or an analogous controlled finite-fermion system) in which the broken continuous symmetry is the global U(1) gauge symmetry associated with particle-number conservation, and measure how much pairing (and, where coexisting, particle-hole) correlation energy the broken and restored solutions recover relative to the quasi-exact reference, as a function of interaction strength, confinement size and particle number, for the same families of model interactions used in the source (VC±V_{C\pm}, VN1V_{N1}, VN2V_{N2}) supplemented by a stated pairing channel. (2) Dimensionality axis: extend the rotational-symmetry benchmark from the one-dimensional ring (a single rotation angle) to two-dimensional spherium and three-dimensional glomium, where rotational restoration involves higher-dimensional rotation groups, and quantify how the effectiveness of symmetry breaking and angular-momentum projection depends on the spatial dimension D\mathcal{D}, again relative to quasi-exact references. (3) Dynamical-correlation axis: evaluate, with interactions or functionals that include dynamical correlations — an EDF treatment or shell-model effective interactions in which the hard-core repulsion is renormalized — the quantitative benefit of symmetry breaking and restoration within the same controlled finite-fermion model systems and against the same quasi-exact reference, so as to assess the practical value of projection in realistic nuclear-structure calculations. The objective on each axis is the recovered correlation-energy fractions and their dependence on the physical parameters, together with the model definitions needed to make the comparison to the exact reference well defined; it is precisely the generalization program stated as future work in Section V of the source paper, no more and no less.

References

  1. J. Cesca and C. Simenel, 'Symmetry breaking and restoration on a fermionic quantum ring', Phys. Rev. C 108, 054307 (2023). DOI: 10.1103/PhysRevC.108.054307; arXiv:2304.05557; https://arxiv.org/abs/2304.05557
  2. A. W. Bray and C. Simenel, 'Fermions with long and finite-range interactions on a quantum ring', Phys. Rev. C 103, 014302 (2021). DOI: 10.1103/PhysRevC.103.014302; arXiv:2012.15451; https://arxiv.org/abs/2012.15451
  3. P.-F. Loos and P. M. W. Gill, 'Two Electrons on a Hypersphere: A Quasiexactly Solvable Model', Phys. Rev. Lett. 103, 123008 (2009). DOI: 10.1103/PhysRevLett.103.123008; https://doi.org/10.1103/PhysRevLett.103.123008
  4. J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, and C. Yannouleas, 'Symmetry restoration in mean-field approaches', J. Phys. G: Nucl. Part. Phys. 48, 123001 (2021). DOI: 10.1088/1361-6471/ac288a; https://doi.org/10.1088/1361-6471/ac288a
  5. T. Duguet and A. Signoracci, 'Symmetry broken and restored coupled-cluster theory: II. Global gauge symmetry and particle number', J. Phys. G: Nucl. Part. Phys. 44, 015103 (2016). DOI: 10.1088/0954-3899/44/1/015103; https://doi.org/10.1088/0954-3899/44/1/015103
  6. S. M. Lauber, H. C. Frye, and C. W. Johnson, 'Benchmarking angular-momentum projected Hartree-Fock as an approximation', J. Phys. G: Nucl. Part. Phys. 48, 095107 (2021). DOI: 10.1088/1361-6471/ac1390; https://doi.org/10.1088/1361-6471/ac1390
  7. N. Krishnan, R. Abdel Khaleq, and C. Simenel, 'Application of the Skyrme Hartree-Fock-Bogoliubov Theory to WIMP-Nucleus Interactions in 40Ar' (2026). arXiv:2606.11668; https://arxiv.org/abs/2606.11668
  8. P.-F. Loos, F. Romaniello, and J. A. Berger, 'Green Functions and Self-Consistency: Insights From the Spherium Model', J. Chem. Theory Comput. 14, 3071 (2018). DOI: 10.1021/acs.jctc.8b00260; https://doi.org/10.1021/acs.jctc.8b00260