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

Abstract

Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical deformed nucleus ${}^{162}$Dy, a typical spherical nucleus ${}^{148}$Sm, and a transitional nucleus ${}^{150}$Sm in which the pairing condensate is deformed. We compare the results of this projection with results from the saddle-point approximation and exact shell model Monte Carlo calculations. We find that the approximate canonical HF entropy in the particle-number projection decreases monotonically to zero in the limit when the temperature goes to zero. However, in a nucleus with a strong pairing condensate, the approximate canonical HFB entropy in the particle-number projection decreases monotonically to a negative value, reflecting the violation of particle-number conservation. Computationally, the exact particle-number projection is more efficient than calculating the derivatives required in the saddle-point approximation.

11 pages, 5 figures

References in corpus (6)

• The sign of the overlap of HFB wave functions
• Variation after Particle-Number Projection for the HFB Method with the Skyrme Energy Density Functional
• Heavy deformed nuclei in the shell model Monte Carlo method
• Crossover from vibrational to rotational collectivity in heavy nuclei in the shell-model Monte Carlo approach
• Benchmarking mean-field approximations to level densities
• Auxiliary-field quantum Monte Carlo methods in nuclei

Cited by in corpus (10)

• Symmetry restoration in mean-field approaches
• Ab Initio Finite Temperature Auxiliary Field Quantum Monte Carlo
• Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes
• State densities of heavy nuclei in the static-path plus random-phase approximation
• Finite-temperature mean-field approximations for shell model Hamiltonians: the code HF-SHELL
• Projection after variation in the finite-temperature Hartree-Fock-Bogoliubov approximation
• Nuclear level densities: from empirical models to microscopic methods
• Quantum computing of the pairing Hamiltonian at finite temperatures
• In-Medium Similarity Renormalization Group at Finite Temperature
• Reply to the Comment of S. Ayik and D. Lacroix, posted as arXiv:1909.1361v1, on the recent article "Fission Dynamics of 240Pu from Saddle-to-Scission and Beyond" by Bulgac et al, published as Phys. Rev. C 100, 034615 (2019)