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arXiv:1010.1642 · International Journal of Modern Physics A

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

Published Apr 18, 2012• 2 citations
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Abstract

-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.

DOI
10.1142/s0217751x12500546

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-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.
An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.
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.
A new many-body theory for fermions is proposed wihich is based on the SO(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A New cannonical transformation, which is the extension of the Bogoliubov transformation to the SO(2N+1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the SO(2N+1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the SO(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the SO(2N+1) canonical transformation. The physical fermion space is shown to be the irreducible spinor representation of the SO(2N+1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N+1 dimensional rotator.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
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.
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].