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Reviews of Modern Physics

Linked-Cluster Expansions for the Nuclear Many-Body Problem

B. H. Brandow

Published Oct 1, 1967• 1,275 citations
Publisher

Abstract

The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms $\ensuremath{\Delta}E$ are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.

DOI
10.1103/revmodphys.39.771

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Compact-Cluster Expansion for the Nuclear Many-Body Problem

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Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\ensuremath{\kappa}}^{{h}^{\ensuremath{'}}\ensuremath{-}1}$, where ${h}^{\ensuremath{'}}$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\ensuremath{\kappa}=\ensuremath{\rho}\ensuremath{\int}{|\ensuremath{\zeta}|}^{2}d\ensuremath{\tau}$, where $\ensuremath{\rho}$ is the ordinary nuclear density and $\ensuremath{\zeta}=\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\ensuremath{\kappa}$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" ${M}^{\mathrm{on}}$. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.

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The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms \ensuremathΔE\ensuremath{\Delta}E\ensuremathΔE are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.
Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\ensuremath{\kappa}}^{{h}^{\ensuremath{'}}\ensuremath{-}1}$, where ${h}^{\ensuremath{'}}$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\ensuremath{\kappa}=\ensuremath{\rho}\ensuremath{\int}{|\ensuremath{\zeta}|}^{2}d\ensuremath{\tau}$, where $\ensuremath{\rho}$ is the ordinary nuclear density and $\ensuremath{\zeta}=\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\ensuremath{\kappa}$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" Mon{M}^{\mathrm{on}}Mon. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.
An approximation method developed previously to deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem where it is shown that the correction arising from the first cluster term involving three particles is less than the leading term by a factor of about ${10}^{\ensuremath{-}4}$. The smallness of the correction is largely a result of the action of the exclusion principle.
The hypotheticai system of nucleons known as nuclear matter provides a very interesting field of study with relevance to both nuclear physics and astrophysics. The task of finding nucleon-nucleon potentials that explain the empirical properties of nuclear matter, as well as the two-body data, has been hampered by the lack of an adequate many-body theory for treating the complicated forces involved. Realistic nucleon-nucleon potentials have strong spin, isospin, tensor and spin-orbit components which induce correspondingly strong correlations in nuclear matter. In this thesis we present a variational theory designed to cope with these complicated correlations. To place the theory in perspective we give an overview of several different many-body theories and compare their predictions in the somewhat simpler systems of the 4He and 3He liquids, and for a simple central potential model of nuclear matter. The variational theory is then developed, starting with the choice of wave function. A generalized Jastrow product of two-body correlation functions containing central, spin, isospin, tensor, and spin-orbit operators is used. The correlation function is generated by a series of two-body Schrodinger equations with boundary conditions which require the wave function to heal at a distance d. The wave function is parameterized by d and by the magnitudes of the non-central correlations. Expectation values with variational wave functions of the Jastrow type may be studied conveniently with generalized Mayer diagrams. Diagram rules are given and a general diagrammatic cluster expansion is derived. It is an expansion in powers of commutators of the non-central operators, and in their absence reduces to the usual Mayer cluster expansion. Evaluation of diagrams in the cluster expansion is complicated by the presence of the non-commuting operators. Some simple rules and useful methods for calculating the contribution due to the central, spin, isospin, and tensor operators are given. Spin-orbit operators are neglected beyond this point, but the approach used is extendible to them. Large classes of diagrams contributing to an expectation value can be summed by means of integral equations. The development of single chain, hypernetted chain, and Fermi hypernetted chain (FHNC) equations for central correlations is reviewed. The chain summation methods are extended to sum single operator chains (SOC) for the non-central operators. Methods are also given for treating the leading commutator corrections in the expansi0n by simple vertex factors. The energy expectation value for potentials with six operators are evaluated using the FHNC/SOC functions, with an exact treatment of the commutators involved. The energy is found to have a minimum with respect to variations in all parameters. Results of calculations with model potentials based on the Reid, Bethe-Johnson, Hamada-Johnston, and Gammel-Thaler potentials are reported. A crude estimate of the effect of the neglected spin-orbit potentials on the nuclear matter binding energy and saturation density indicates that it could be significant. A correct treatment of the spin-orbit potentials is the next major step that should be taken.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
The generalization of coupled cluster (CC) theory is far from being a standard method to account for correlation effects of ubiquitous open-shell systems. This dilemma is largely due to three problems. The first concerns the incorporation of multiple reference “guess” wave functions into nonvariational theories. Next is the size-extensivity issue. Finally, and perhaps most importantly is the notorious intruder state problem. Brillouin–Wigner (BW) and generalized Brillouin–Wigner (gBW) perturbation theories are used to aid in the development of new Fock-space coupled cluster theories in an attempt to alleviate some of these problems. Bloch equations are derived which can be used to formulate BWCC and gBWCC theories of arbitrary dimension in all sectors of Fock space. Since this is our first study of Brillouin–Wigner coupled cluster theories in Fock space we have chosen to keep our approach very simple. Explicit effective Hamiltonian and amplitude equations for up to two-body S-amplitudes in the (0,1) and (1,0) sectors are given for the single reference case. Aspects concerning the connectivity of the amplitude equations are addressed.