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Physical Review C

α -cluster matter reexamined

J. W. Clark, E. Krotscheck

Published Mar 21, 2024• 4 citations
Publisher

Abstract

We examine in detail two alternative descriptions of a system of $\ensuremath{\alpha}$ particles interacting via local interactions of different character, highlighting the fact that a faithful microscopic description of such systems demands a consistent treatment of both short- and long-range correlations. In preparation, we examine four different versions of modern microscopic many-body theory and conclude by emphasizing that these approaches, although a priori very different, actually lead to the same equations for their efficient application. The only quantity that depends on the formulation of many-body theory chosen is an irreducible interaction correction. In the language of Green's functions and Feynman diagrams, it is the set of both particle-particle and particle-hole irreducible diagrams, and in variational Jastrow-Feenberg theory it is determined by multipartite correlations and elementary diagrams. We apply these theoretical methods to the calculation of the energetics, structure, thermodynamics, and dynamics of $\ensuremath{\alpha}$ matter, as well as its condensate fraction. In dimensionless units, $\ensuremath{\alpha}$ matter appears to be remarkably similar to the much-studied $^{4}\mathrm{He}$ quantum fluid, its low-temperature properties now basically solved in the Jastrow-Feenberg framework. Accordingly, one can have confidence in the results of application of the same procedure to $\ensuremath{\alpha}$ matter. Even so, closer examination reveals significant differences between the physics of the two systems. Within an infinite nuclear medium, $\ensuremath{\alpha}$ matter is subject to a spinoidal instability. Extended mixtures of nucleons and $\ensuremath{\alpha}$ particles are yet to be given rigorous consideration in a corresponding theoretical framework.

DOI
10.1103/physrevc.109.034315

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We examine in detail two alternative descriptions of a system of \ensuremathα\ensuremath{\alpha}\ensuremathα particles interacting via local interactions of different character, highlighting the fact that a faithful microscopic description of such systems demands a consistent treatment of both short- and long-range correlations. In preparation, we examine four different versions of modern microscopic many-body theory and conclude by emphasizing that these approaches, although a priori very different, actually lead to the same equations for their efficient application. The only quantity that depends on the formulation of many-body theory chosen is an irreducible interaction correction. In the language of Green's functions and Feynman diagrams, it is the set of both particle-particle and particle-hole irreducible diagrams, and in variational Jastrow-Feenberg theory it is determined by multipartite correlations and elementary diagrams. We apply these theoretical methods to the calculation of the energetics, structure, thermodynamics, and dynamics of \ensuremathα\ensuremath{\alpha}\ensuremathα matter, as well as its condensate fraction. In dimensionless units, \ensuremathα\ensuremath{\alpha}\ensuremathα matter appears to be remarkably similar to the much-studied 4He^{4}\mathrm{He}4He quantum fluid, its low-temperature properties now basically solved in the Jastrow-Feenberg framework. Accordingly, one can have confidence in the results of application of the same procedure to \ensuremathα\ensuremath{\alpha}\ensuremathα matter. Even so, closer examination reveals significant differences between the physics of the two systems. Within an infinite nuclear medium, \ensuremathα\ensuremath{\alpha}\ensuremathα matter is subject to a spinoidal instability. Extended mixtures of nucleons and \ensuremathα\ensuremath{\alpha}\ensuremathα particles are yet to be given rigorous consideration in a corresponding theoretical framework.
We examine in detail two alternative descriptions of a system of ααα particles interacting via local interactions of different character, highlighting the fact that a faithful microscopic description of such systems demands a consistent treatment of both short- and long-range correlations. In preparation, we examine four different versions of modern microscopic many-body theory and conclude by emphasizing that these approaches, although {\it a priori} very different, actually lead to the same equations for their efficient application. The only quantity that depends on the formulation of many-body theory chosen is an {\it irreducible} interaction correction. In the language of Green's functions and Feynman diagrams, it is the set of both particle-particle and particle-hole irreducible diagrams, and in variational Jastrow-Feenberg theory it is determined by {\it multipartite correlations} and {\it elementary diagrams}. We apply these theoretical methods to the calculation of the energetics, structure, thermodynamics, and dynamics of ααα matter, as well as its condensate fraction. In dimensionless units, ααα matter appears to be remarkably similar to the much-studied 4^44He quantum fluid, its low-temperature properties now basically solved in the Jastrow-Feenberg framework. Accordingly, one can have confidence in the results of application of the same procedure to ααα matter. Even so, closer examination reveals significant differences between the physics of the two systems. Within an infinite nuclear medium, alpha matter is subject to a spinoidal instability. Extended mixtures of nucleons and alpha particles are yet to be given rigorous consideration in a corresponding theoretical framework.
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called v8v_8v8​ v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the 3P2^3\textrm{P}_23P2​ 3 P 2 and 3P2^3\textrm{P}_23P2​ 3 P 2 - 3F2^3\textrm{F}_23F2​ 3 F 2 gaps. We also find pairing in 3P0^3\textrm{P}_03P0​ 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
We develop the variational/parquet diagram approach to the structure of nuclear systems with strongly state-dependent interactions. For that purpose, we combine ideas of the general Jastrow-Feenberg variational method and the local parquet-diagram theory for bosons with state-dependent interactions [R. A. Smith and A. D. Jackson, Nucl. Phys. A 476, 448 (1988)]. The most tedious aspect of variational approaches, namely the symmetrization of an operator dependent variational wave function, is thereby avoided. We carry out calculations for neutron matter interacting via the Reid and Argonne v6{v}_{6}v6​ models of the nucleon-nucleon interaction. While the equation of state is a rather robust quantity that comes out reasonably well even in very simplistic approaches, we show that effective interactions, which are the essential input for calculating dynamic properties, depend sensitively on the quality of the treatment of the many-body problem.
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.
Recent developments in the many-body theory of interacting fermions are discussed employing a self-consistent Green function approach. This scheme is outlined and its application to nuclear systems is presented. Special attention is paid to the consistent inclusion of short-range and long-range correlations induced by realistic nucleon-nucleon interactions. Such correlations lead to occupation probabilities which deviate from the simple mean-field or shell-model description. The scheme is extended to incorporate relativistic effects. Also applications of field theoretical models for hadrons in a nuclear medium and their relation to QCD are discussed.
We investigate the quartet correlations in four-component fermionic systems at the thermodynamic limit within a variational many-body theory. The Bardeen-Cooper-Schrieffer (BCS)--type variational wave function is extended to the systems with the coexistence of pair and quartet correlations at zero temperature. Special attention is paid to the application of the present framework to an $\ensuremath{\alpha}$-particle condensation in symmetric nuclear matter, where the coexistence of deuteron and $\ensuremath{\alpha}$ condensations is anticipated. We also discuss how physical properties, such as quasiparticle dispersion, can be modified by the pair and quartet correlations and show a hierarchical structure of in-medium cluster formations in infinite nuclear matter. The present results may also contribute to the interdisciplinary understanding of fermionic condensations beyond the BCS paradigm in many-body systems.