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Progress of Theoretical Physics

A Possible Canonical Theory for Description of Pairing Correlation: Comparison with the BCS Plus RPA Approach

M. Yamamura, A. Kuriyama

Published Apr 1, 1982• 0 citations
Publisher

Abstract

In order to go beyond the conventional BCS plus RPA approach for the description of pairing correlation, we have developed a possible canonical theory. In this theory, we can exactly get rid of the well-known difficulty of the particle-number non-conservation in the BCS approximation. We give, in this paper, its approximate form in order to clarify the relation with the conventional approach. We show that our theory in its approximate form can reproduce the same result as that obtained by the BCS plus RPA approach. The equivalence between the approach developed by Suzuki et al. and the present one is also shown, within the RPA order at least. We give some perspective for the description of higher-order non-linear effect, such as the coupling between the pairing rotation and pairing vibration, in conclusion.

DOI
10.1143/ptp.67.1135

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In order to go beyond the conventional BCS plus RPA approach for the description of pairing correlation, we have developed a possible canonical theory. In this theory, we can exactly get rid of the well-known difficulty of the particle-number non-conservation in the BCS approximation. We give, in this paper, its approximate form in order to clarify the relation with the conventional approach. We show that our theory in its approximate form can reproduce the same result as that obtained by the BCS plus RPA approach. The equivalence between the approach developed by Suzuki et al. and the present one is also shown, within the RPA order at least. We give some perspective for the description of higher-order non-linear effect, such as the coupling between the pairing rotation and pairing vibration, in conclusion.
Recent development in particle-particle random phase approximation (pp-RPA) broadens the perspective on ground state correlation energies [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013), Y. Yang, H. van Aggelen, S. N. Steinmann, D. Peng, and W. Yang, J. Chem. Phys. 139, 174110 (2013); D. Peng, S. N. Steinmann, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 104112 (2013)] and N ± 2 excitation energies [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. So far Hartree-Fock and approximated density-functional orbitals have been utilized to evaluate the pp-RPA equation. In this paper, to further explore the fundamentals and the potential use of pairing matrix dependent functionals, we present the linear-response time-dependent density-functional theory with pairing fields with both adiabatic and frequency-dependent kernels. This theory is related to the density-functional theory and time-dependent density-functional theory for superconductors, but is applied to normal non-superconducting systems for our purpose. Due to the lack of the proof of the one-to-one mapping between the pairing matrix and the pairing field for time-dependent systems, the linear-response theory is established based on the representability assumption of the pairing matrix. The linear response theory justifies the use of approximated density-functionals in the pp-RPA equation. This work sets the fundamentals for future density-functional development to enhance the description of ground state correlation energies and N ± 2 excitation energies.
The BCS and/or Hartree-Fock-Bogoliubov theories are extended by treating the effect of four quasiparticle states perturbatively. The approach is tested on the pairing Hamiltonian, showing that it combines the advantage of standard perturbation theory, valid at low pairing strength, and of nonperturbative approaches breaking the particle number, valid at higher pairing strength. Including the restoration of particle number further improves the description of pairing correlation. In the presented test, the agreement between the exact solution and the combined perturbative plus projection is almost perfect. The proposed method scales friendly when the number of particles increases and provides a simple alternative to other more complicated approaches.
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.