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

27 pages, 14 figures

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• The density-matrix renormalization group in the age of matrix product states
• Quasiparticle Coupled Cluster Theory for Pairing Interactions
• Subtraction method in the second random--phase approximation: first applications with a Skyrme energy functional
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• Reproduction of exact solutions of Lipkin model by nonlinear random-phase approximation
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• Examining the justification for the introduction of a fermion localization function
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