Progress in many-body theory with the equation of motion method: Time-dependent density matrix meets self-consistent RPA and 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.

Reproduction of exact solutions of Lipkin model by nonlinear higher random-phase approximation

“…The analytical solutions for N = 2 are summarized in Table I If the equations of the self-consistent second RPA [19,20,26] with c k are solved exactly for N = 2, our result should be obtained. The vacuum condition for the Lipkin model is solved in Ref.…”

“…The extension of the RPA to include also the 2p-2h excitations, so called the second RPA, has been investigated and used by many authors [10][11][12][13][14][15][16][17][18][19][20]. Currently the frontier of this approach is the self-consistent second RPA [20], which is still used only for schematic models (their equation is solved only approximately).…”

“…The extension of the RPA to include also the 2p-2h excitations, so called the second RPA, has been investigated and used by many authors [10][11][12][13][14][15][16][17][18][19][20]. Currently the frontier of this approach is the self-consistent second RPA [20], which is still used only for schematic models (their equation is solved only approximately). The aim of this work is the inclusion in the RPA framework of states with arbitrary order of particle-hole excitations from the ground state and the nonlinearity of the eigenequation [14,[19][20][21][22], and our equations are solved exactly.…”

“…Currently the frontier of this approach is the self-consistent second RPA [20], which is still used only for schematic models (their equation is solved only approximately). The aim of this work is the inclusion in the RPA framework of states with arbitrary order of particle-hole excitations from the ground state and the nonlinearity of the eigenequation [14,[19][20][21][22], and our equations are solved exactly. It turns out that such extension reproduces the exact solutions of the Lipkin model [23].…”

Isospin dynamics in nuclear structure

“…As in the present formulation we consider only Hamiltonians with two-body interactions, the major quantity of interest to describe nuclear spectra is the in-medium two-fermion propagator in various channels. Respectively, the technique of choice is the Green function formalism and the equation of motion method which was extensively discussed in the past [5][6][7]. As follows from its formulation, the approach can be, if needed, straightforwardly generalized to three-body forces.…”