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.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.

arXiv:1610.08954Physical Review CJul 7, 201718 citations

Similarities and differences between the Gaussian and random phase approximation for the partition function

G. Puddu

Analogies and differences between the random phase approximation (RPA) and the Gaussian approximation at finite temperature are discussed. A comparison is made for Hartree and Hartree-Fock temperature dependent mean fields. Some ambiguities, in the definition of the Hartree mean field and in the relative correction, are analyzed. In a solvable model with a repulsive potential, the Gaussian approximation and the RPA, both built from the Hartree mean field, are found more reliable than the RPA built from a Hartree-Fock mean field.Analogies and differences between the random phase approximation (RPA) and the Gaussian approximation at finite temperature are discussed. A comparison is made for Hartree and Hartree-Fock temperature dependent mean fields. Some ambiguities, in the definition of the Hartree mean field and in the relative correction, are analyzed. In a solvable model with a repulsive potential, the Gaussian approximation and the RPA, both built from the Hartree mean field, are found more reliable than the RPA built from a Hartree-Fock mean field.

Physical Review CJan 1, 19940 citations

The Hartree-Fock Approximation

W. Cassing

Springer eBooksJan 1, 20250 citations

Time-Dependent Hartree–Fock Approximation and Mean-Field Decoupling

Giancarlo Calvanese Strinati

Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.

Cambridge University Press eBooksFeb 6, 20260 citations

^{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.