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arXiv:1201.6297

Some exact results for the particle number projected BCS approach of the isovector proton-neutron pairing

A. A. Raduta, M. I. Krivoruchenko, Amand Faessler

Published Jan 30, 2012• 0 citations
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Abstract

The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, $N$. The function Q(N) is calculated using a recursion in $N$ in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.

DOI
10.48550/arxiv.1201.6297

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The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, NNN. The function Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function Q(N)Q(N)Q(N) depending on the number of particles, NNN. The function Q(N)Q(N)Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.
Background: Within the quartet condensation model (QCM), the isovector pairing correlations for N=ZN=ZN=Z nuclei are described with a very high accuracy by a ``condensate'' of $\ensuremath{\alpha}$-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes.Purpose: We present the final analytical expressions for the above-mentioned quantities, for all cases up to four quartets in the valence shell.Method: The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system cadabra2.Results: The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas.Conclusions: We introduce in this work a method of computer-aided analytical calculus for a many-body setting. In particular, we provide precise and easy-to-use tools for the description of isovector pairing correlations.
The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.
A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.
The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of the products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In the case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J = 0. Realistic applications of the quartet formalism are carried out for Sn isotopes with valence neutrons in the 50–82 neutron shell. The exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected BCS approach.