Progress in Many Body Theory with the Equation of Motion method. Time dependent Density Matrix meets Self-Consistent RPA. Applications to solvable Models

TL;DR

This paper advances many-body theory by developing a time-dependent density matrix approach with a three-body correlation approximation, demonstrating its importance and linking it to the Self-Consistent RPA, with applications to solvable models.

Contribution

It introduces a truncated TDDM hierarchy including three-body correlations and connects it to SCRPA, showing improved accuracy in solvable models.

Findings

Three-body correlations are crucial for precise results.

The nonlinear TDDM equations are equivalent to SCRPA in the small amplitude limit.

The methods are validated against exactly solvable models.

Abstract

The Bogoliubov-Born-Green-Kirkwood-Yvon or Time-Dependent Density Matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three body level in approximating the three body correlation function by a quadratic form of two body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in non-trivial model cases that the approximate inclusion of three body correlation functions is very important to obtain precise results. A small amplitude approximation of this time dependent nonlinear equation for the two body correlation function is performed (STDDM*-b) and it is shown that the one body sector of this generalised non-linear second RPA equation is equivalent to the Self-Consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.