Zero-pairing and zero-temperature limits of finite-temperature Hartree-Fock-Bogoliubov theory

Abstract

Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed-or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases. Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits. Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction. Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases. Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.

Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

“…V. As a consequence, however, the first point of the discretization is at π M γ instead of zero. Therefore, the operator (114) does not reduce to the identity for M γ = 1, and the case of "no projection" has thus to be treated separately. (ii) Note that because of the identical choice of quadrature for the integrals over α and γ , the action of the discretized projection operator is trivially symmetric under the exchange of the bra and ket as long as the same number of discretization points is used for both of these angles.…”

Finite-temperature mean-field approximations for shell model Hamiltonians: the code HF-SHELL

“…The T → 0 limit is less straightforward when pairing correlations collapse at low temperatures or in the HF approximation; see Ref. [36] and Appendix B of Ref. [37].…”

Clustering in nuclei at finite temperature

“…Up to now, the temperature effects have been studied for the clustering of nucleons only in nuclear matter [1,3,5,7,[31][32][33][34]. Concomitantly, comprehensive studies have also been conducted using the relativistic and nonrelativistic NEDFs to understand the properties and behavior of nuclei with increasing temperature [35][36][37][38][39][40][41][42][43]. However, there is no study on the temperature dependence of the localization and clustering features in finite nuclei, which is the aim of the present work.…”