Fanto, Alhassid, and Bertsch (Phys. Rev. C 96, 014305 (2017)) derived exact particle-number projection after variation (PAV) for the finite-temperature Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) approximations and identified open directions. Later literature resolved several of them: the sign ambiguity of the general projection formula was eliminated by a pfaffian method valid for time-reversal-breaking condensates (Fanto, 2017); static fluctuations beyond the mean field and the missing rotational enhancement of state densities were captured by the static-path plus random-phase approximation benchmarked against exact shell-model Monte Carlo (Fanto and Alhassid, 2021); and exact finite-temperature PAV has since been implemented in shell-model codes (HF-SHELL) and in self-consistent density-functional calculations (Chen, Qiang, and Pei, 2026). The surviving open core is the central item of that outlook: a practical finite-temperature variation-after-projection (VAP) method for HF/HFB, i.e., a computationally tractable and reliable evaluation of the particle-number-projected entropy and free energy in the paired (superfluid) phase, enabling particle-number-conserving variation for mid-mass and heavy nuclei and removing the PAV artifacts (negative thermodynamic canonical entropy below the pairing transition, sharp phase-transition kinks, and violation of the variational principle). Reviews (2021) and the most recent primary literature (2026) explicitly attest that this remains unresolved.

Background

Hot nuclei are described statistically: the state and level densities and thermodynamic functions of an excited compound nucleus are basic inputs to the Hauser-Feshbach theory of statistical nuclear reactions and to fission modeling. Because a nucleus is a finite, isolated system with fixed proton and neutron numbers, the physically relevant ensemble is the canonical one (fixed particle number), whereas mean-field approximations are formulated in the grand canonical ensemble (variable particle number, chemical potential). In the Hartree-Fock (HF) approximation - the self-consistent one-body approximation to the interacting many-body problem - particle number is conserved. When pairing correlations are strong, the appropriate mean field is the Hartree-Fock-Bogoliubov (HFB) approximation, a generalization of the Bardeen-Cooper-Schrieffer (BCS) pairing theory whose finite-temperature density operator is proportional to the exponential of a quadratic quasi-particle Hamiltonian; the pairing condensate spontaneously breaks the U(1) gauge symmetry associated with particle-number conservation, so fixed-N quantities extracted from it are intrinsically approximate.

The exact canonical partition function is obtained by particle-number projection (PNP): inserting the projector P^N\hat P_N onto fixed N into the statistical trace, ZN=Tr(P^ND^)Z_N=\mathrm{Tr}(\hat P_N\hat D). Projection combines with the mean field in two ways. In projection after variation (PAV), the grand-canonical mean field is determined first and the projection applied afterwards; this is tractable even for heavy nuclei. In variation after projection (VAP), the mean field is determined by extremizing the projected quantity - the energy at zero temperature, where VAP for HFB is a standard tool of nuclear energy-density-functional theory (the density-functional framework of nuclear structure, in which the total energy is expressed as a functional of the one-body density; Stoitsov et al., 2007), or the free energy at finite temperature. Finite-temperature VAP requires minimizing FN=EN−TSNF_N=E_N-TS_N with the projected entropy SN=−Tr(D^Nln⁡D^N)S_N=-\mathrm{Tr}(\hat D_N\ln\hat D_N), where D^N∝P^ND^P^N\hat D_N\propto\hat P_N\hat D\hat P_N, and this is the obstruction: the projected density operator is not the exponential of a one-body operator, so Gaudin's theorem - the finite-temperature extension of Wick's theorem that reduces all grand-canonical traces of quadratic Hamiltonians to determinants and pfaffians (a pfaffian being the polynomial of a skew-symmetric matrix that generalizes the determinant) - cannot be applied to its logarithm. Exact evaluation of SNS_N requires explicit traces over the fixed-N many-particle Hilbert space, whose dimension grows combinatorially with the number of single-particle orbitals; this has been done only in small BCS-type model spaces (Esebbag and Egido, 1993; Gambacurta and Lacroix, 2012). A formal alternative based on thermo-field dynamics exists (Tanabe and Nakada, 2005) but was never reduced to practical large-scale calculations. Finite-temperature projection itself has a long history: the general PAV formalism, including its sign ambiguity for general condensates, goes back to Rossignoli and Ring (1994); Robledo's pfaffian formula (2009) resolves the sign of the HFB overlap; and Egido, Robledo, and Martin (2000) applied finite-temperature restoration to shell effects in nuclei.

The problem was sharpened by Fanto, Alhassid, and Bertsch (2017), who derived exact PAV projection formulas for the finite-temperature HF and HFB approximations, benchmarked them against the shell-model Monte Carlo (SMMC) method - a Monte Carlo evaluation of the canonical shell-model partition function with controlled statistical errors - for 162^{162}Dy and 148,150^{148,150}Sm, and quantified the intrinsic PAV failures: below the pairing transition temperature TcT_c the approximate canonical HFB entropy decreases to an unphysical negative value (a direct consequence of number non-conservation) - this canonical entropy is the thermodynamic projected entropy Sc=ln⁡ZN+EN/TS_c=\ln Z_N+E_N/T (with ZN=Tr(P^ND^)Z_N=\mathrm{Tr}(\hat P_N\hat D)), a different object from the projected entropy SN=−Tr(D^Nln⁡D^N)S_N=-\mathrm{Tr}(\hat D_N\ln\hat D_N) that enters the free energy and is non-negative for a normalized D^N\hat D_N - and in deformed nuclei PAV misses the rotational enhancement of the state density generated by rotational bands built on intrinsic states. Their outlook called for (i) a practical finite-temperature VAP method, (ii) a particle-number-projected static-path approximation (SPA, which integrates over static fluctuations of the mean field beyond the self-consistent solution; particle-number conservation within the SPA was treated early by Kaneko and Schiller, 2007) with a systematic accuracy and cost assessment, and (iii) a general, numerically stable PNP for time-reversal-breaking condensates such as those of odd-mass and odd-odd nuclei. Post-2017 work resolved (iii) (Fanto, 2017, via the pfaffian method) and largely addressed (ii) together with the rotational-correlation failure, through the static-path plus random-phase approximation (SPA+RPA) benchmark of Fanto and Alhassid (2021); reviews of microscopic level-density methods (Alhassid, 2020) present exact PAV as the current state of the art, with collective enhancement factors still taken from phenomenology. What remains open, and is explicitly attested as impractical in the 2021 review literature and in the most recent primary work of 2026, is item (i) together with its computational-scaling aspect for heavy nuclei.

Problem Statement

Develop a practical finite-temperature variation-after-projection (VAP) method for the HF and HFB mean-field approximations: a computationally tractable procedure - exact, or approximate with controlled and quantifiable accuracy - to evaluate the particle-number-projected entropy SN=−Tr(D^Nln⁡D^N)S_N=-\mathrm{Tr}(\hat D_N\ln\hat D_N) (with D^N\hat D_N the particle-number-projected mean-field density operator) and the projected free energy FN=EN−TSNF_N=E_N-TS_N of a finite-temperature pairing condensate, and to determine the mean field by minimizing FNF_N, i.e., by carrying out the variation with particle number conserved, in the single-particle model spaces of mid-mass and heavy nuclei where explicit traces over the fixed-N many-particle Hilbert space are combinatorially intractable.

These aspects jointly constitute the single objective of particle-number-conserving finite-temperature variation. The method must be computationally tractable - its cost must scale at most polynomially in the number of single-particle orbitals, i.e., it must not grow combinatorially, and must be demonstrated on a model space of the realistic size used for mid-mass and heavy nuclei (of order 10210^2 single-particle orbitals) - and its accuracy must be established against exact references:

  1. Method: analytic expressions or a certified algorithm for SNS_N and for the projected free energy and its variational derivatives (or an equivalent practical minimization scheme), with the correct zero-pairing limit, the correct low-temperature limit, and agreement with explicit fixed-N traces to within about one percent in SNS_N wherever the latter are computable (small model spaces).
  2. Reliability in the paired phase: a non-negative canonical entropy below the pairing transition temperature TcT_c (removing the negative-entropy artifact that projection after variation produces in the thermodynamic projected entropy Sc=ln⁡ZN+EN/TS_c=\ln Z_N+E_N/T, as distinct from the projected entropy SN=−Tr(D^Nln⁡D^N)S_N=-\mathrm{Tr}(\hat D_N\ln\hat D_N), which is non-negative by construction), smooth behavior across mean-field phase transitions, and compliance with the variational principle.
  3. Validation and feasibility: quantitative benchmarking against exact references in regimes with a strong pairing condensate, accepting as a reference either shell-model Monte Carlo results or explicit projection and diagonalization in a space small enough for exact fixed-N traces to be computable, together with a demonstrated computational scaling with the number of single-particle orbitals that meets the tractability standard above.

This is the surviving open core of the outlook of Fanto, Alhassid, and Bertsch (2017): its sibling directions - a sign-ambiguity-free particle-number projection for time-reversal-breaking condensates, and the incorporation of static and collective fluctuations beyond the mean field - have since been resolved by Fanto (2017) and by Fanto and Alhassid (2021), respectively, while finite-temperature VAP remains unattained as of 2026.

References

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