The numerical suite TAURUS performs symmetry-projected variational calculations for atomic nuclei. Its founding paper (Bally, Sánchez-Fernández and Rodríguez, 2021) solved the variation-after-particle-number-projection (VAPNP) equations for real general Bogoliubov quasiparticle states, and left open the extension of the projected variational principle beyond particle number: restoring angular momentum, parity, and ultimately isospin directly in the variational minimization, using fully general (complex, complex-conjugation-breaking) Bogoliubov matrices, treating the three-nucleon part of chiral Hamiltonians beyond the normal-ordered two-body approximation, and adapting the optimization to the projected energy landscape. The 2024 companion codes TAURUS_pav and TAURUS_mix implemented projection after variation plus configuration mixing — a different, variationally weaker strategy — and explicitly deferred complex reference states and isospin projection. One point of terminology matters for the formulation: real Bogoliubov states already break time reversal, while complex matrices are what is needed to break complex conjugation, e.g. to constrain the real part of the angular-momentum component along the y axis. The surviving open problem is therefore: extend the variation-after-projection variational principle to simultaneous restoration of particle number, angular momentum, and parity (with isospin as the envisioned completion) using general complex Bogoliubov trial states and chiral two- plus three-nucleon Hamiltonians, with optimization adapted to the projected energy landscape, validated against exact shell-model diagonalization. As of August 2026 this core remains open.

Background

Nuclear mean-field theory describes the atomic nucleus with a Hartree-Fock-Bogoliubov (HFB) product state: a quasiparticle vacuum built from two matrices (U,V)(U,V) that mix particle creation and annihilation operators and encode both mean field and pairing. To capture correlations, such trial states are allowed to break symmetries of the underlying Hamiltonian — rotational invariance (shape deformation), particle number (pairing), parity (octupole shapes), isospin (the proton-neutron symmetry; breaking it mixes protons and neutrons), and time reversal. The broken symmetries are then restored by projection: one integrates the intrinsic state over the symmetry group (a gauge angle for particle number, Euler angles for rotations, discrete operations for parity and isospin) to build a state with good quantum numbers. Three strategies are distinguished: projection after variation (PAV), where the unprojected energy is minimized first; variation after projection (VAP), where the symmetry-projected energy itself is minimized with respect to the intrinsic state — variationally superior but far more expensive because the projection integrals enter every iteration; and the projected generator coordinate method (PGCM), which diagonalizes the Hamiltonian within a basis of several symmetry-projected intrinsic states. In VAP/PAV the projected-energy gradient is evaluated through Bogoliubov overlap and transition-density kernels between rotated states, whose computation for general (triaxial, time-reversal-breaking) states relies on Pfaffian algebra to fix otherwise ambiguous phases. The founding paper, TAURUS I, presented the code TAURUS_vap, which solves the VAP equations with particle-number projection (VAPNP) for real general Bogoliubov states in a spherical harmonic-oscillator single-particle basis, usable both in shell-model valence spaces and no-core ab initio model spaces with Hamiltonians up to two-body terms; the minimization uses a Heavy-Ball (momentum) gradient method whose step parameters can be preset from empirical 'recipes' based on the HFB quasiparticle spectrum, recipes the authors show to perform poorly on the projected energy landscape. Chiral effective-field-theory Hamiltonians — the standard ab initio input — contain genuine three-nucleon (3N) forces, which the code includes only through normal-ordered two-body (NO2B) recasts; exact 3N matrix elements would be memory-prohibitive. Two documented limitations motivate the open problem: with real matrices (U,V)(U,V) the real part of the expectation value of the angular-momentum component JyJ_y cannot be constrained (its matrix elements are purely imaginary in the chosen basis), and in no-core chiral calculations the VAPNP minimum of 24Mg lies more than 40 MeV above the experimental binding energy, reflecting missing correlations that many-body renormalizations such as the in-medium similarity renormalization group (IMSRG) can only partly supply. Variational use of general symmetry-projected HFB configurations, including complex transformations that break complex conjugation (and thereby generically also time-reversal invariance), has a long history in small valence spaces with phenomenological two-body interactions (the VAMPIR approach reviewed by Schmid), but has never been combined with chiral three-nucleon Hamiltonians, multi-symmetry (angular momentum, parity, isospin) variational projection, and no-core model spaces.

Problem Statement

Extend the variation-after-projection variational principle, currently realized in the TAURUS suite only for particle number (VAPNP), to simultaneous restoration of the symmetries broken by the intrinsic state — at minimum angular momentum and parity, with full isospin projection as the envisioned completion — directly inside the variational minimization, using fully general (complex, complex-conjugation-breaking) Bogoliubov trial states and chiral two- plus three-nucleon Hamiltonians. The problem, as left open by TAURUS I and explicitly deferred by TAURUS II (2024), involves interlinked aspects: (i) derive and implement the multi-symmetry-projected energy gradient for complex general Bogoliubov matrices, which requires stable Pfaffian-based overlap and transition-density kernels and makes constraints such as the real part of ⟨J_y⟩ — inaccessible with real matrices — available in the minimization; (ii) treat the three-nucleon part of chiral Hamiltonians beyond the normal-ordered two-body approximation — the source envisages the inclusion of the full three-body interaction in model spaces small enough for those matrix elements to remain tractable — addressing the memory-scaling barrier; (iii) achieve robust convergence of the minimization on the multi-symmetry projected energy landscape, for which the preset Heavy-Ball step recipes calibrated on unprojected HFB quasiparticle spectra are demonstrably miscalibrated; and (iv) validate the resulting method in valence spaces against exact shell-model diagonalization (for example in the sd shell), quantifying the energy changes of variation after multi-symmetry projection relative to VAPNP and to projection-after-variation/PGCM references on the same Hamiltonian. The source frames these developments disjunctively — 'many developments are possible in the future such as the inclusion of additionial symmetry projections in the variational scheme or the use of the full three-body interaction' (TAURUS I, Conclusions). The target is a variation-after-projection scheme with general complex Bogoliubov states that restores particle number, angular momentum, and parity inside the variational minimization, developed for a chiral two- plus three-nucleon Hamiltonian with the three-nucleon part at least through the normal-ordered two-body approximation, and validated in a valence space against exact shell-model diagonalization; projection onto a good isospin is an envisioned extension. The exact treatment of the three-nucleon interaction beyond the normal-ordered two-body approximation, further optimization suited to the projected landscape, and eventual no-core chiral applications are source-discussed accompanying developments of the same research program.

References

  1. B. Bally, A. Sánchez-Fernández, T. R. Rodríguez, 'Symmetry-projected variational calculations with the numerical suite TAURUS I. Variation after particle-number projection', Eur. Phys. J. A 57, 69 (2021). DOI: 10.1140/epja/s10050-021-00369-z, https://doi.org/10.1140/epja/s10050-021-00369-z (arXiv:2010.14169, https://arxiv.org/abs/2010.14169)
  2. B. Bally, T. R. Rodríguez, 'Symmetry-projected variational calculations with the numerical suite TAURUS II. Configuration mixing of symmetry-projected reference states', Eur. Phys. J. A 60, 62 (2024). DOI: 10.1140/epja/s10050-024-01271-0, https://doi.org/10.1140/epja/s10050-024-01271-0 (arXiv:2401.00941, https://arxiv.org/abs/2401.00941)
  3. B. Bally, M. Bender, 'Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states', Phys. Rev. C 103, 024315 (2021). DOI: 10.1103/PhysRevC.103.024315, https://doi.org/10.1103/PhysRevC.103.024315 (arXiv:2010.15224, https://arxiv.org/abs/2010.15224)
  4. W. Lin, E. F. Zhou, J. M. Yao, H. Hergert, 'Quantum-Number Projected Generator Coordinate Method for 21Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction', Symmetry 16(4), 409 (2024). DOI: 10.3390/sym16040409, https://doi.org/10.3390/sym16040409 (arXiv:2403.01177, https://arxiv.org/abs/2403.01177)
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