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arXiv:1512.00944 · International Journal of Geometric Methods in Modern Physics

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

Seiya Nishiyama, João da Providência

Published Jan 14, 2016• 0 citations
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Abstract

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].

DOI
10.1142/s0219887816500432

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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].
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 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.
First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group 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.
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.
We study the variational properties of the action functional appeared in the time-dependent Hartree-Bogoliubov (TDHB) theory, Through the calculus of the second variation on the coset manifold SO(2N)/U(N), the Jacobi equation is obtained. Assuming the periodic Jacobi field, we derive in a natural way the equation for the quasi-particle random phase approximation (RPA) describing the collective excitation around certain static HB fields. The present method of obtaining the SO(2N) RPA may be useful to get the SO(2N+1) RPA.