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arXiv:nucl-th/0603069 · Physical Review C

Beyond mean field study of excited states: Analysis within the Lipkin model

A. P. Severyukhin, M. Bender, P.-H. Heenen

Published Aug 18, 2006• 16 citations
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Abstract

We compare the generator coordinate method (GCM) and the random-phase approximation (RPA) in the framework of the exactly solvable Lipkin-Meshkov-Glick model. We show that the discretized GCM works quite well and permits to obtain results close to the exact results with a small number of discretization points.
DOI
10.1103/physrevc.74.024311

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Concerning the [Formula: see text]-Lipkin model, the calculation of the excitation energy to the first excited state gives rise to the following fact: The two results based on the exact treatment and the conventional random phase approximation (RPA) are in unexpected disagreement. In order to remove this discrepancy, the conventional RPA is renewed. With the aim of this renewal, the terms, which cannot be dealt with in the conventional one, are estimated. The basic framework of this new form is not reorganized from that of the conventional one. In addition, the [Formula: see text]-Lipkin model itself is also modified so as to be applicable to the case with nonclosed shell system. To this modification, one more [Formula: see text]-algebra, which has been already proposed by the present authors, is applied. Through the use of this algebra, the [Formula: see text]-Lipkin model is applicable to the case with any total fermion number permitted in the model.
The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
We develop and test efficient approximations to estimate ground state correlations associated with low- and zero-energy modes. The scheme is an extension of the generator coordinate method (GCM) within Gaussian overlap approximation (GOA). We show that the GOA fails in non-Cartesian topologies and presents a topologically correct generalization of the GOA (topGOA). A random-phase-approximation like (RPA-like) correction is derived as the small amplitude limit of topGOA, called topRPA. Using exactly solvable models, the topGOA and topRPA schemes are compared with conventional approaches (GCM-GOA, RPA, Lipkin-Nogami projection) for rotational-vibrational motion and for particle-number projection. The results shows that the new schemes perform very well in all regimes of coupling.
We present a benchmark study of the generator coordinate method (GCM) combined with eigenvector continuation (EC) in two different schemes for the low-lying states of the Lipkin-Meshkov-Glick (LMG) model, where the interaction strength is treated as a controlling parameter, simulating quantum many-body systems with the phase transition from noncollective to collective states. We demonstrate that the ECkmax{\mathrm{EC}}_{\mathrm{kmax}}ECkmax​ scheme accurately reproduces the low-lying states of the LMG model. In this scheme, the EC basis consists of the wave functions of low-lying states up to the kmax{k}_{\mathrm{max}}kmax​-th state of sampling Hamiltonians. Compared to EC1{\mathrm{EC}}_{1}EC1​, which only includes the wave functions of the kthk\mathrm{th}kth state of sampling Hamiltonians for the kthk\mathrm{th}kth state of a target Hamiltonian, the ECkmax{\mathrm{EC}}_{\mathrm{kmax}}ECkmax​ scheme exhibits significantly improved efficiency and accuracy. This study suggests the potential utilization of the extended EC scheme as an efficient emulator for GCM calculations.