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Progress of Theoretical Physics

Note on the New Type of the SO(2N+1) Time-Dependent Hartree-Bogoliubov Equation

S. Nishiyama

Published Aug 1, 1982• 8 citations
Publisher

Abstract

Embedding an SO(2N+1) group into an SO(2N+2) group and using the variables of the coset space SO(2N+2)/ U(N+1), we obtain a time-dependent Hartree-Bogoliubov (TDHB) theory which includes unpaired mode effects. The new type of the SO(2N+1) TDHB equation applicable to both even and odd fermion systems is given through the variables as the representatives of the paired and unpaired modes.

DOI
10.1143/ptp.68.680

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Progress of Theoretical Physics•Feb 1, 1980•2 citations
Embedding an SO(2N+1) group into an SO(2N+2) group and using the variables of the coset space SO(2N+2)/ U(N+1), we obtain a time-dependent Hartree-Bogoliubov (TDHB) theory which includes unpaired mode effects. The new type of the SO(2N+1) TDHB equation applicable to both even and odd fermion systems is given through the variables as the representatives of the paired and unpaired modes.
A time dependent theory is developed for the SO(2N+1) wave function which yields both paired and unpaired modes of Fermion. The time dependent SO(2N+1) equation is derived and shown to have two kinds of solution, the Hartree-Bogoliubov solution which describes even Fermion systems and the improper Hartree-Bogoliubov solution which describes odd Fermion systems. The time dependent improper Hargree-Bogoliubov equation leads to a new random phase approximation which gives a self-consistent unified description of Bose and Fermi type collective excitations in an odd Fermion system.
SO(2N)U(N)\frac{{\rm SO}(2N)}{U(N)}U(N)SO(2N)​ Riccati–Hartree–Bogoliubov equation based on the SO (2N) Lie algebra of the fermion operators
In this paper we present the induced representation of SO (2N) canonical transformation group and introduce [Formula: see text] coset variables. We give a derivation of the time-dependent Hartree–Bogoliubov (TDHB) equation on the Kähler coset space [Formula: see text] from the Euler–Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati–Hartree–Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO (2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.
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 [Formula: see text] 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 [Formula: see text] 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 coset space [Formula: see text] on the Kähler 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 [Formula: see text] 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.
Properties and implications of the SO(2N+1) regular representation of a fermion system, which is a representation on the SO(2N+1) group proposed recently by us, are studied. By using the independent variables of the SO(2N+1) group, a representation of fermion annihilation-creation and pair operators is obtained which is an extension of the Dyson representation to cover both paired and unpaired modes. The structures of the SO(2N+1) spinor space and the SO(2N+1) wave function are determined. An integral and a differential Schrödinger equation on the SO(2N+1) spinor space are derived which are applicable to both even and odd fermion systems. The structures of the symmetry projected SO(2N+1) wave function and Schrödinger operator are determined. It is shown that odd fermion systems can be treated with an analogous way as even fermion systems by using a special SO(2N+1) canonical transformation which is very similar to the Bogoliubov transformation but corresponds to an improper rotation in the O(2N) group.
A possible quantization is proposed with respect to time-dependent Hartree-Bogoliubov theory based on the SO(2N+1) algebra which was presented by Fukutome, Nishiyama and the present author. The adopted quantization procedure is an extension of Marshalek and Holzwarth's idea used in the case of the conventional time-dependent Hartree-Bogoliubov theory based on the SO(2N) algebra. As a natural consequent of the quantization, Holstein-Primakoff type boson expansion theory can be derived in a unified form of even and odd particle system.