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

Abstract

It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many-particle-many-hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number N = 2 and, numerically, for N = 8. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation, which opens up new possibilities for realistic calculations in many-body problems.

Equation of Motion Method to strongly correlated Fermi systems and Extended RPA approaches

“…If there are N particles in the system and one pushes above expansion up to the N p − N h configuration, the exact result will be recovered. A demonstration of this is given in [40]. Of course, the more terms are kept in the expansion (2.8), the more difficult it will become to solve the ensuing equations (2.5), for instance, from the numerical point of view.…”

“…Next, we present the results for the Lipkin model [157] and the Hubbard model using the original form of ERPA with the three-body correlation matrix. We also review the main results for the quadrupole states in 16 O and 40 Ca calculated using STDDM.…”

“…Applications to the Isoscalar Giant Quadrupole Resonances (ISGQR) in 16 O [204] and 40 Ca [208] have been done using the simplified version of ERPA, that is, STDDM. A simplified Skyrme force has been adapted as the effective interaction for the mean-field potential and also as the residual interaction.…”

“…A simplified Skyrme force has been adapted as the effective interaction for the mean-field potential and also as the residual interaction. Assuming that single-particle states around the Fermi energy are most important for ground-state correlations, we use the 1p 3/2 , 1p 1/2 , 1d 5/2 and 2s 1/2 states to calculate n αα ′ , C αβα ′ β ′ and X µ αβα ′ β ′ for 16 O and the 1d 5/2 , 2s 1/2 , 1d 3/2 and 1f 1/2 states for 40 Ca, and only the 2p-2h and 2h-2p components of C αβα ′ β ′ and X µ αβα ′ β ′ are considered to facilitate the numerics. To satisfy the energy weighted sum rule, we use a large number of single-particle states to calculate the one-body transition amplitude: The continuum states are discretized by confining the single-particle wave functions in a sphere of radius 20 fm and all singleparticle states with ǫ ≤ 40 MeV and the orbital angular momentum ℓ ≤ 5 are taken.…”

A possible trial beyond the conventional random phase approximation for the su(2)-Lipkin model including the case of non-closed shell system

“…This is our motivation for the present work. Concerning this point, we are interested in the paper [7].…”