Graduate Lunch Talk: Paul Fanto, Yale University, “Particle-number projection in the finite-temperature mean-field approximation”

Finite temperature mean-field approximations are computationally efficient methods for predicting statistical properties of nuclei, particularly the nuclear level density, which is a crucial input to models of nuclear reactions. These theories, namely the Hartree-Fock and Hartree-Fock-Bogoliubov approximations, are formulated in the grand-canonical ensemble. To predict statistical properties in a finite nucleus, it is necessary to reduce these mean-field theories to the canonical ensemble. This reduction is usually carried out by a saddle-point approximation. An alternate and more conceptually satisfying approach is the particle-number projection, which exactly cancels the contributions of any states without the correct particle number. In this talk, I will discuss our application of the exact particle-number projection to these grand-canonical finite-temperature mean-field approximations. I will present a straightforward projection formula for the Hartree-Fock theory, which is appropriate for nuclei without strong pairing condensates, and more complicated formulas for the Hartree-Fock-Bogoliubov theory, which is applied to nuclei in which the pairing condensate is strong. For the latter theory, I will discuss both a general projection expression applicable to condensates with time-reversal invariance and a simpler expression for the BCS limit, which is realized in nuclei with spherically symmetric condensates. Finally, I will discuss our results for the canonical energy, canonical entropy, and state density in specific nuclei. We benchmark our results against the shell model Monte Carlo, and we compare our results with the results from the more commonly used saddle-point approximation.