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Progress of Theoretical Physics

Time Dependent SO(2N+1) Theory for Unified Description of Bose and Fermi Type Collective Excitations

H. Fukutome, Seiya Nishiyama

Published Aug 1, 1984• 21 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.72.239

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

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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.

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Modified non-Euclidean transformation on the SO(2N+2) U(N+1) Grassmannian and SO(2N + 1) random phase approximation for unified description of Bose and Fermi type collective excitations

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In a slight different way from the previous one, we propose a modified non-Euclidean transformation on the [Formula: see text] Grassmannian which gives the projected [Formula: see text] Tamm–Dancoff equation. We derive a classical time-dependent (TD) [Formula: see text] Lagrangian which, through the Euler–Lagrange equation of motion for [Formula: see text] coset variables, brings another form of the previous extended-TD Hartree–Bogoliubov (HB) equation. The [Formula: see text] random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the [Formula: see text] HB case, one boson and two boson excited states are realized. We, however, stress non-existence of a higher RPA vacuum. An integrable system is given by a geometrical concept of zero-curvature, i.e. integrability condition of connection on the corresponding Lie group. From the group theoretical viewpoint, we show the existence of a symplectic two-form [Formula: see text].

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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.

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A NEW DESCRIPTION OF MOTION OF THE FERMIONIC SO(2N+2) TOP IN THE CLASSICAL LIMIT UNDER THE QUASI-ANTICOMMUTATION RELATION APPROXIMATION

SEIYA NISHIYAMA, JOÃO DA PROVIDÊNCIA, CONSTANÇA PROVIDÊNCIA

-1 The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidean space (N: number of single-particle states of the fermions). The images of fermion annihilation–creation operators must satisfy the canonical anticommutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N+1) self-consistent field (SCF) Hartree–Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle–hole plus BCS-type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation due to the unpaired-made effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi-anticommutation relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters K and l in the Lagrange multipliers can be determined without any inconsistency.

arXiv:1010.1642•International Journal of Modern Physics A•Apr 18, 2012•2 citations
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.
In a slight different way from the previous one, we propose a modified non-Euclidean transformation on the [Formula: see text] Grassmannian which gives the projected [Formula: see text] Tamm–Dancoff equation. We derive a classical time-dependent (TD) [Formula: see text] Lagrangian which, through the Euler–Lagrange equation of motion for [Formula: see text] coset variables, brings another form of the previous extended-TD Hartree–Bogoliubov (HB) equation. The [Formula: see text] random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the [Formula: see text] HB case, one boson and two boson excited states are realized. We, however, stress non-existence of a higher RPA vacuum. An integrable system is given by a geometrical concept of zero-curvature, i.e. integrability condition of connection on the corresponding Lie group. From the group theoretical viewpoint, we show the existence of a symplectic two-form [Formula: see text].
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.
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.
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.
-1 The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidean space (N: number of single-particle states of the fermions). The images of fermion annihilation–creation operators must satisfy the canonical anticommutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N+1) self-consistent field (SCF) Hartree–Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle–hole plus BCS-type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation due to the unpaired-made effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi-anticommutation relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters K and l in the Lagrange multipliers can be determined without any inconsistency.