Particle-number projection in the finite-temperature mean-field approximation

Finite-temperature mean-field theories, such as the Hartree–Fock (HF) and Hartree–Fock–Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen–Cooper–Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei ${}^{162}\mathrm{Dy}, {}^{148}\mathrm{Sm}$, and ${}^{150}\mathrm{Sm}$. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.

Figure 1

Canonical entropy of ${}^{162}\mathrm{Dy}$ vs $\beta$ in the HF approximation. The approximate PNP canonical entropy (31) (solid black line) is compared with the approximate canonical entropy from DG1 (dashed red line) and from DG2 (dashed-dotted blue line). The open circles represent the SMMC entropy. The inset shows the various entropies at higher $\beta$ values.

Figure 2

Canonical entropies of ${}^{148}\mathrm{Sm}$ vs $\beta$ in the BCS limit of the HFB approximation. Lines and symbols are as in Fig. 1. The inset shows the entropies for higher $\beta$ values.

Figure 3

Excitation energy of ${}^{150}\mathrm{Sm}$ vs $\beta$ in the HFB approximation. The approximate canonical energy calculated from the PNP (solid black line) is compared with the approximate canonical energy from DG2 (dashed-dotted blue line). DG1 becomes unstable for this nucleus and is not shown. The open circles are the SMMC excitation energies. The inset shows higher $\beta$ values.

Figure 4

Canonical entropy of ${}^{150}\mathrm{Sm}$ vs $\beta$ in the HFB approximation. Lines and symbols for the PNP, DG2, and SMMC are as in Fig. 3. The inset shows an expanded scale at large values of $\beta$.

Figure 5

State density of ${}^{150}\mathrm{Sm}$ vs excitation energy $E_x$ in the HFB approximation. Lines and symbols are as in Fig. 3. The inset shows the low-excitation-energy results.