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Physical Review

Investigation of Ground-State Correlations for a Model Hamiltonian of the Nucleus

Jitendra C. Parikh, DAVID J. ROWE

Published Nov 20, 1968• 65 citations
Publisher

Abstract

The importance of ground-state correlations in the spectroscopy of a finite many-particle system, such as the nucleus, is investigated for the model Hamiltonian of Lipkin, Meshkov, and Glick. In particular, we have investigated their effect on the solution of the equations of motion for excitation energies, transition matrix elements, and the correlation energy of the ground state. Methods are studied for calculating the correlations needed to make the necessary corrections to the equations of motion. Also studied is the behavior, with increase of interaction strength, of the phase transition from the spherical to the strongly deformed intrinsic states, in the projected Hartree-Fock scheme.

DOI
10.1103/physrev.175.1293

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The importance of ground-state correlations in the spectroscopy of a finite many-particle system, such as the nucleus, is investigated for the model Hamiltonian of Lipkin, Meshkov, and Glick. In particular, we have investigated their effect on the solution of the equations of motion for excitation energies, transition matrix elements, and the correlation energy of the ground state. Methods are studied for calculating the correlations needed to make the necessary corrections to the equations of motion. Also studied is the behavior, with increase of interaction strength, of the phase transition from the spherical to the strongly deformed intrinsic states, in the projected Hartree-Fock scheme.
This paper presents the equations-of-motion method as a useful and flexible tool in the study of nuclear spectroscopy. It is partly a review, but also it introduces a new and much more powerful equations-of-motion technique which supercedes the older linearization methods. The older methods worked with operator equations. To obtain closed expressions they had to be linearized in a rather arbitrary manner. The present approach works with the ground-state expectation of operator equations and thereby avoids all problems of linearization. Thus, like the Green's function method, the equations-of-motion method becomes potentially exact. It has many advantages over Green's function methods, however, among which are its greater compactness, simplicity, and the physical insight it yields.The method is first applied to rederive the random phase approximation (RPA) and the quasi-particle RPA (QRPA) and to show precisely what terms they neglect. It is demonstrated that some of these terms have coherent phases. A higher RPA and QRPA are then derived to include these terms. The corrections have some interesting effects: notably, there is a reduction of the effective interaction strength and a stabilization of the nucleus against sudden phase transitions. The equations-of-motion method is also used to generalize, in a very simple and natural way, the Hartree-Fock (HF) and Hartree-Bogolyubov (HB) concepts of independent particles and quasi-particles to nonsimple ground states.The equations-of-motion method is presented as a simple extension of the shell model to the treatment of excitationg of a correlated ground state. By concentrating on the quantities of direct physical interest, the complexity of workins with correlated wavefunctions is avoided.
The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.
A new theory of treating the ground-state correlations and the correspond-ing renormalization effects is proposed. In the theory, two-particle-two-hole and four-particle-four-hole excitations are taken up as the important parts in the ground-state correlations. The correlated true ground state is given through multiplication of the unitary operator to the Hartree-Fock ground state. The unitary operator is determined from the condition of minimizing the energy of the correlated true ground state. New single-particle operators are defined through the unitary transformation of the Hartree-Fock single-particle operators in the second-quantization formalism. The transformed Hamiltonian represented by the new single-particle operators is established to display the renormalization effects in the single-particle energies and the two-body matrix elements, and further it is proved that in the Hamiltonian the two-body interactions inducing the ground-state correlati6ns are reduced to small and negligible values. With this Hamiltonian, we are led to a simple
It is well known that the second-order ground-state correlation energy calculated in the usual random-phase approximation (RPA) is off by a factor of 2. It has also been noted that there is a factor-of-2 discrepancy in the correlation corrections to the Hartree-Fock occupation probabilities. The reason for these phenomena is traced to the treatment of the exchange terms in the interaction matrix element. An illustration based on a simple single multipole interaction shows that the factor of 2 arises from a coherent sum of small higher-order terms. Results based on the consideration of one or a few eigenmodes at a time are seen to be free of this difficulty. The correlation energy, occupation-probability corrections, and equations of motion are reexamined in the Beliaev-Zelevinsky boson expansion, as an illustration of clearly defined higher-order treatments.
A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.