SO(2N)/U(N) Riccati-Hartree-Bogoliubov Equation Based on the SO(2N) Lie Algebra of the Fermion Operators

TL;DR

This paper derives the time-dependent Hartree-Bogoliubov equation on the SO(2N)/U(N) coset space using Lie algebra techniques, providing a geometric and algebraic framework for fermion pair condensates.

Contribution

It introduces a novel derivation of the TDHB equation on the SO(2N)/U(N) coset space using Lie algebra and Riccati equations, linking geometric structures with fermionic systems.

Findings

Derived the TDHB equation on the coset space

Provided solutions for specific coset variables

Connected Riccati equations with fermion condensate dynamics

Abstract

In this paper we present the induced representation of SO(2N) canonical transformation group and introduce SO(2N)/U(N) coset variables. We give a derivation of the time dependent Hartree-Bogoliubov (TDHB) equation on the Kaehler coset space G/H=SO(2N)/U(N) 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 TDRHB equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Moebius transformation together with the geodesic flow on the manifold.