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Physical Review C

Comparison between the temperature and the time-dependent Hartree-Fock-Bogoliubov descriptions of the pairing interaction

M. C. Cambiaggio, F. G. Carbone, G.G. Dussel

Published Jan 1, 1994• 1 citation
Publisher

Abstract

The thermal and the time-dependent versions of the Hartree-Fock-Bogoliubov approximation are used to describe the effects of the pairing interaction in a simple model. Similarities and differences between the two approaches are discussed. In the one-shell case analytic results are obtained which clearly show that as the degeneracy increases the exact energies tend to the limits provided by the approximate methods. A relation between the quasiparticle occupation probability and the seniority is established. In the two-shell case the time-dependent approach approximates all the families of exact states labeled by the shell seniority and gives information on the two different phase transitions present in the model. The finite-temperature method, instead, only approximates the exact energy for the lowest state of each family.

DOI
10.1103/physrevc.49.194

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The thermal and the time-dependent versions of the Hartree-Fock-Bogoliubov approximation are used to describe the effects of the pairing interaction in a simple model. Similarities and differences between the two approaches are discussed. In the one-shell case analytic results are obtained which clearly show that as the degeneracy increases the exact energies tend to the limits provided by the approximate methods. A relation between the quasiparticle occupation probability and the seniority is established. In the two-shell case the time-dependent approach approximates all the families of exact states labeled by the shell seniority and gives information on the two different phase transitions present in the model. The finite-temperature method, instead, only approximates the exact energy for the lowest state of each family.
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.
We present simple equations for a canonical-basis (Cb) formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. The equations are obtained from the TDHFB theory with an approximation that the pair potential is assumed to be diagonal in the Cb. The Cb formulation significantly reduces the computational cost. We apply the method to linear-response calculations for even-even light nuclei and demonstrate its capability and accuracy by comparing our results with recent calculations of the quasiparticle random-phase approximation with Skyrme functionals. We show systematic studies of E1E1E1 strength distributions for Ne and Mg isotopes. The evolution of the low-lying pygmy strength seems to be determined by the interplay of several factors, which include the neutron excess, the separation energy, the neutron-shell effects, the deformation, and the pairing.
We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the Hamiltonian. In contrast, the theories based on the BCS approximation are more problematic. In the usual formulation of time-dependent Hartree-Fock (TDHF)+BCS(\mathrm{TDHF})+\mathrm{BCS}(TDHF)+BCS, the equation of continuity is violated and one sees unphysical oscillations in particle densities. This can be ameliorated by freezing the occupation numbers during the evolution in TDHF+BCS\mathrm{TDHF}+\mathrm{BCS}TDHF+BCS, but there are other problems with the BCS approximation that make it doubtful for reaction dynamics. We also compare different numerical implementations of the time-dependent HFB equations. The equations of motion for the UUU and VVV Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient. Finally, we compare the time-dependent HFB solutions with numerically exact solutions of the two-particle Schr\"odinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schr\"odinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory.
The scaling properties of the exact wave functions in the two-level pairing model are studied and a well-defined limit, when the number of pairs goes to infinity, is found. An approximate method for obtaining the scaled wave functions is discussed. Well-known methods for relating finite-difference equations with differential ones are used, together with a semiclassical expansion. The approximate results obtained agree well with the exact ones. A comparison with the time-dependent Hartree-Fock approach is also done.
An extension of the time-dependent Hartree-Fock-Bogoliubov theory (ETDHFB) which includes higher-order effects such as screening of the pairing correlation is proposed. ETDHFB is applied to a pairing model and a fermion system trapped in a harmonic potential to test its feasibility by comparison with exact solutions. With the use of perturbative expressions for the pairing tensor and the two-body density matrix derived from ETDHFB, the screening effect is investigated for atomic fermion systems and isotopes of tin nuclei. It is found that the screening effect on the pairing correlation is not significant.