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

TL;DR

This paper derives exact analytical expressions for the expectation values of a Hamiltonian with proton-neutron pairing within a particle number projected BCS framework, using a recursive function Q(N).

Contribution

It introduces a recursive method to compute key quantities in the particle number projected BCS approach for proton-neutron pairing, providing exact results and insights.

Findings

Analytical expressions for Hamiltonian expectation values in projected BCS states.

Recursive calculation of the function Q(N) for particle number dependence.

Comparison of standard BCS, projection after variation, and variation after projection methods.

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.