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Moscow University Physics Bulletin

Nuclear Ground State Correlations

S. P. Kamerdzhiev, Шитов Михаил Игоревич

Published Jun 1, 2023• 2 citations
Publisher

Abstract

Abstract The article discusses nuclear ground state correlations (GSCs), which are particularly clearly manifested when using the formalism of quantum Green’s functions (GFs) and Feynman diagrams. In addition to the one-particle-one-hole ground state correlations in the well-known random phase approximation (RPA) method, there also exist and make a noticeable quantitative contribution GSCs that arise from using more complex configurations than in RPA, specifically, configurations containing phonons and those containing only three quasiparticles, phonon–GSCs and GSC3s, respectively. The latter case of GSC3 is discussed in detail in this paper. It has been shown that GSC3s make a significant quantitative contribution to problems related to explaining certain characteristics of low-lying excited states (phonons) and transitions between them in even–even nuclei.

DOI
10.3103/s0027134923030104

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Abstract The article discusses nuclear ground state correlations (GSCs), which are particularly clearly manifested when using the formalism of quantum Green’s functions (GFs) and Feynman diagrams. In addition to the one-particle-one-hole ground state correlations in the well-known random phase approximation (RPA) method, there also exist and make a noticeable quantitative contribution GSCs that arise from using more complex configurations than in RPA, specifically, configurations containing phonons and those containing only three quasiparticles, phonon–GSCs and GSC3s, respectively. The latter case of GSC3 is discussed in detail in this paper. It has been shown that GSC3s make a significant quantitative contribution to problems related to explaining certain characteristics of low-lying excited states (phonons) and transitions between them in even–even nuclei.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.
The problem of the microscopic description of excited states of the even-even open-shell atomic nuclei is considered. A model is formulated which allows one to go beyond the quasiparticle random phase approximation. The physical content of the model is determined by the quasiparticle time blocking approximation (QTBA) which enables one to include contributions of the two-quasiparticle and the two-phonon configurations, while excluding (blocking) more complicated intermediate states. In addition, the QTBA ensures consistent treatment of ground state correlations in the Fermi systems with pairing. The model is based on the generalized Green function formalism (GGFF) in which the normal and the anomalous Green functions are treated in a unified way in terms of the components of generalized Green functions in a space that is double the size of the usual single-particle space. Modification of the GGFF is considered in the case when the many-body nuclear Hamiltonian contains two-, three-, and other many-particle effective forces.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
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.
Working within a schematic model and in a boson formalism, we provide a descrip tion of the ground and first-excited states of a nuclear system in terms of two independent phonon operators. The description reveals itself quite effective in reproducing the energies of the states. A comparison between these two phonon operators allows us to localize a region where a correct description of these states is compatible with the Random Phase Approximation (RPA) requirement of a single phonon. This region is found to coincide approximately with that spanned by the standard RPA. Outside this region, the two phonons start differing rapidly therefore making the application of the RPA scheme impossible. This allows a better understanding of recently performed higher-order RPA calculations.