Modified Non-Euclidian 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

TL;DR

This paper introduces a modified non-Euclidean transformation on the SO(2N+2)/U(N+1) Grassmannian, deriving a new form of the extended TD Hartree-Bogoliubov equation and an SO(2N+1) RPA for unified Bose and Fermi collective excitations.

Contribution

It proposes a novel transformation and derives a unified RPA framework for Bose and Fermi excitations using group theoretical methods.

Findings

Derived a modified non-Euclidean transformation on Grassmannian.

Established a new form of extended TD Hartree-Bogoliubov equation.

Developed an SO(2N+1) RPA with geometric and symplectic structures.

Abstract

In a slight different way from the previous one, we propose a modified non-Euclidian transformation on the SO(2N+2)/U(N+1) Grassmannian which give the projected SO(2N+1) Tamm-Dancoff equation. We derive a classical time dependent (TD) SO(2N+1) Lagrangian which, through the Euler-Lagrange equation of motion for SO(2N+2)/U(N+1) coset variables, brings another form of the previous extended-TD Hartree-Bogoliubov (HB) equation. The SO(2N+1) random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the SO(2N) 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 omega.