Remarks on the mean-field theory based on the SO(2N+1) Lie algebra of the fermion operators

Toward a unified algebraic theory for mean-field Hamiltonian describing paired- and unpaired-mode effects, in this paper, we propose a generalized Hartree-Bogoliubov mean-field Hamiltonian in terms of fermion pair and creation-annihilation operators of the SO(2N + 1) Lie algebra. We diagonalize the generalized Hartree-Bogoliubov mean-field Hamiltonian and throughout its diagonalization we can first obtain the unpaired mode amplitudes which are given by the self-consistent field parameters appeared in the Hartree Bogoliubov theory together with the additional self-consistent field parameter in the generalized Hartree-Bogoliubov mean-field Hamiltonian and by the parameter specifying the property of the SO(2N + 1) group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such results. We construct the Killing potential in the (2N) coset space SO(2N)/U(N) on the Kahler symmetric space which is equivalent to the generalized density matrix. We show another approach to the fermion mean-field Hamiltonian based on such a generalized density matrix. We derive an SO(2N + 1) generalized Hartree-Bogoliubov mean-field Hamiltonian operator and a modified Hartree-Bogoliubov eigenvalue equation. We discuss on the mean-field theory related to the algebraic mean-field theory based on the generalized density matrix and the coadjoint orbit leading to the nondegenerate symplectic form.