Nuclear energy density functionals (EDFs) such as Skyrme and Gogny are not expectation values of a single Hamiltonian, because they contain density-dependent terms, often with fractional powers of the density. Dobaczewski, Stoitsov, Nazarewicz, and Reinhard (Phys. Rev. C 76, 054315 (2007)) showed that when such a functional is used beyond mean field — particle-number projection, variation after projection, or configuration mixing — the projected energy is an integral over the gauge angle whose integrand has poles on the imaginary axis and branch cuts from fractional density powers, so the resulting energies are multivalued and depend on the chosen integration contour. The open problem is to formulate a multi-reference EDF framework in which projected and configuration-mixed energies are uniquely defined (contour-independent) for functionals not derived from one Hamiltonian, including fractional density dependence, that supports stable variation-after-projection iterations and smooth generator-coordinate-method energy surfaces, and extends to nondiagonal kernels and to higher symmetry groups — angular-momentum projection (non-Abelian SU(2)) and simultaneous proton-neutron projection (two-parameter Abelian U(1)_p × U(1)_n) — or to prove such a framework cannot exist and characterize the admissible functional forms instead.

Background

Low-energy nuclear structure is dominated by the energy density functional (EDF) method: the exact ground-state energy of a nucleus is approximated by a functional of one-body densities, whose two standard phenomenological families — the Skyrme and Gogny parametrizations — contain explicit density-dependent terms (e.g. terms proportional to ρ1/3\rho^{1/3}) fitted phenomenologically across the nuclear chart. The standard mean-field description uses a Hartree-Fock-Bogoliubov (HFB) quasiparticle vacuum — a self-consistent product state mixing particle numbers — which therefore breaks the conservation of particle number associated with the global U(1)U(1) gauge symmetry. Exact particle number (and other broken symmetries) are restored by projection: the physical state is extracted by integrating the broken-symmetry product over the symmetry group, here the gauge angle φ\varphi conjugate to particle number.

For a genuine Hamiltonian expectation value this is unproblematic. Dobaczewski, Stoitsov, Nazarewicz, and Reinhard (arXiv:0708.0441; Phys. Rev. C 76, 054315 (2007)) analyzed what happens when the projected energy is instead built from an EDF kernel evaluated on transition density matrices — the densities connecting the product state to its gauge-rotated copy, which are the natural multi-reference extension of a density functional. The transition density matrix elements, as functions of the complexified gauge angle, have poles located on the imaginary axis of the complex gauge-angle plane. If the functional derives from a single Hamiltonian, the pole residues cancel and the projected energy is independent of the integration contour. If it does not — as for Skyrme and Gogny functionals with density-dependent terms, or the widely used Slater (local-density) approximation to Coulomb exchange, which behaves as ρ4/3\rho^{4/3} — the fractional powers of complex-valued transition densities make the integrand multivalued: branch cuts appear, and the projected energy depends on where the integration contour is drawn relative to poles and cuts. Self-interaction errors (a particle spuriously interacting with itself) and self-pairing errors in the functional add further spuriosities. The practical mitigations proposed in that paper — contours passing near or above the poles, and removing self-interaction and self-pairing contributions — were shown to be partial: they do not remove the multivaluedness induced by fractional powers.

This is not a corner case: poles close to the contour deform projected and generator-coordinate-method (GCM, a configuration-mixing framework in which the variational state is a superposition of constrained mean-field states, e.g. at different quadrupole deformations) energy surfaces, and variation after projection (VAP, minimizing the projected rather than the intrinsic energy) becomes unstable because the poles move as the intrinsic state is varied. The pathology is thus embedded in every beyond-mean-field observable — fission barriers, shape coexistence, projection-corrected masses — computed with standard EDFs.

Problem Statement

Formulate a particle-number-projected and configuration-mixed (multi-reference) nuclear energy density functional framework whose energies are uniquely defined — that is, independent of the gauge-angle integration contour and free of pole, branch-cut, and multivaluedness pathologies — even when the functional is not the expectation value of a single Hamiltonian and contains fractional powers of the density, as current Skyrme and Gogny parametrizations do. The framework should (i) come with a precise prescription for evaluating projected energies of such functionals, (ii) support stable and convergent variation-after-projection iterations robust against motion of the poles during variation and yield smooth generator-coordinate-method energy surfaces, (iii) extend the analytic singularity analysis and its regularization to the nondiagonal matrix elements needed for angular-momentum projection (non-Abelian SU(2)) and simultaneous proton-neutron projection (two-parameter Abelian U(1)_p × U(1)_n), and (iv) establish principled theoretical constraints on admissible functional forms (e.g. on admissible density dependence or Hamiltonian representability). Alternatively, prove that no such contour-independent extension exists for the class of functionals currently in use, thereby characterizing rigorously which functional forms admit well-defined symmetry restoration. A complete solution is either (A) a framework satisfying all of (i)-(iv), or (B) a proof that no such contour-independent extension exists together with the resulting characterization of admissible functional forms; a solution of (i) alone, or of (i)-(ii) only, is a partial result.

References

  1. J. Dobaczewski, M. V. Stoitsov, W. Nazarewicz, P.-G. Reinhard, 'Particle-number projection and the density functional theory', Physical Review C 76, 054315 (2007). DOI: 10.1103/physrevc.76.054315; arXiv:0708.0441. URL: https://doi.org/10.1103/physrevc.76.054315
  2. D. Lacroix, T. Duguet, M. Bender, 'Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels', Physical Review C 79, 044318 (2009). DOI: 10.1103/physrevc.79.044318. URL: https://doi.org/10.1103/physrevc.79.044318
  3. M. Bender, T. Duguet, D. Lacroix, 'Particle-number restoration within the energy density functional formalism', Physical Review C 79, 044319 (2009). DOI: 10.1103/physrevc.79.044319. URL: https://doi.org/10.1103/physrevc.79.044319
  4. L. M. Robledo, 'Remarks on the use of projected densities in the density-dependent part of Skyrme or Gogny functionals', Journal of Physics G: Nuclear and Particle Physics 37, 064020 (2010). DOI: 10.1088/0954-3899/37/6/064020. URL: https://doi.org/10.1088/0954-3899/37/6/064020
  5. T. Duguet, J. Sadoudi, 'On the breaking and restoration of symmetries within the nuclear energy density functional formalism', International Journal of Modern Physics E 20, 270 (2011). DOI: 10.1142/S0218301311017612. URL: https://doi.org/10.1142/S0218301311017612
  6. G. Hupin, D. Lacroix, M. Bender, 'Formulation of functional theory for pairing with particle number restoration', Physical Review C 84, 014309 (2011). DOI: 10.1103/physrevc.84.014309. URL: https://doi.org/10.1103/physrevc.84.014309
  7. J. Sadoudi, M. Bender, K. Bennaceur, D. Davesne, R. Jodon, T. Duguet, 'Skyrme pseudo-potential-based EDF parametrization for spuriousity-free MR EDF calculations', Physica Scripta T154, 014013 (2013). DOI: 10.1088/0031-8949/2013/T154/014013. URL: https://doi.org/10.1088/0031-8949/2013/T154/014013
  8. J. L. Egido, 'State-of-the-art of beyond mean field theories with nuclear density functionals', Physica Scripta 91, 073003 (2016). DOI: 10.1088/0031-8949/91/7/073003. URL: https://doi.org/10.1088/0031-8949/91/7/073003
  9. M. Grasso, 'Effective density functionals beyond mean field', Progress in Particle and Nuclear Physics 106, 256 (2019). DOI: 10.1016/j.ppnp.2019.02.002. URL: https://doi.org/10.1016/j.ppnp.2019.02.002
  10. K. Bennaceur, J. Dobaczewski, T. Haverinen, M. Kortelainen, 'Properties of spherical and deformed nuclei using regularized pseudopotentials in nuclear DFT', Journal of Physics G: Nuclear and Particle Physics 47, 105101 (2020). DOI: 10.1088/1361-6471/ab9493. URL: https://doi.org/10.1088/1361-6471/ab9493
  11. G. Salvioni, J. Dobaczewski, C. Barbieri, G. Carlsson, A. Idini, A. Pastore, 'Model nuclear energy density functionals derived from ab initio calculations', Journal of Physics G: Nuclear and Particle Physics 47, 085107 (2020). DOI: 10.1088/1361-6471/ab8d8e. URL: https://doi.org/10.1088/1361-6471/ab8d8e
  12. J. A. Sheikh, R. N. Ali, 'Symmetry projection in atomic nuclei', European Physical Journal Special Topics 229, 2555 (2020). DOI: 10.1140/epjst/e2020-000111-3. URL: https://doi.org/10.1140/epjst/e2020-000111-3
  13. B. Bally, A. Sánchez-Fernández, T. R. Rodríguez, 'Symmetry-projected variational calculations with the numerical suite TAURUS', European Physical Journal A 57, 69 (2021). DOI: 10.1140/epja/s10050-021-00369-z. URL: https://doi.org/10.1140/epja/s10050-021-00369-z
  14. 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, 123001 (2021). DOI: 10.1088/1361-6471/ac288a. URL: https://doi.org/10.1088/1361-6471/ac288a
  15. B. Bally, G. Giacalone, M. Bender, 'Structure of 128,129,130^{128,129,130}Xe through multi-reference energy density functional calculations', European Physical Journal A 58, 187 (2022). DOI: 10.1140/epja/s10050-022-00833-4. URL: https://doi.org/10.1140/epja/s10050-022-00833-4