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Effective interactions and the nuclear shell-model

D. J. Dean, T. Engel, Morten Hjorth-Jensen, M. P. Kartamyshev, E. Osnes

Published Jan 1, 2004• 44 citations
OpenAlex

Abstract

This review aims at a critical discussion of the interplay between effective interactions derived from various many-body approaches and spectroscopic data extracted from large scale shell-model studies. To achieve this, our many-body scheme starts with the free nucleon-nucleon (NN) interaction, typically modelled on various meson exchanges. The NN interaction is in turn renormalized in order to derive an effective medium dependent interaction. The latter is in turn used in shell-model calculations of selected nuclei. We also describe how to sum up the parquet class of diagrams and present initial uses of the effective interactions in coupled cluster many-body theory.

Related papers

Effective interactions for the nuclear shell model

Morten Hjorth-Jensen

Various perturbative and non-perturbative many-body techniques are discussed in this work. Especially, we will focus on the summation of so-called Parquet diagrams with emphasis on applications to finite nuclei. Here, the subset of two-body Parquet equations will be discussed. A practical implementation of the corresponding equations for studies of effective interactions for finite nuclei is outlined.

arXiv:nucl-th/9811101•Nov 30, 1998•0 citations

Nonobservable nature of the nuclear shell structure: Meaning, illustrations, and consequences

T. Duguet, H. Hergert, J. D. Holt +1 more

Background: The concept of single-nucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective single-particle energies (ESPEs) introduced by French [Proceedings of the International School of Physics ``Enrico Fermi,'' Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970)] represent the most appropriate tool to relate many-body observables to a single-nucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012)], the dependence of ESPEs on one-nucleon transfer probability matrices makes them purely theoretical quantities that ``run'' with the nonobservable resolution scale $\ensuremath{\lambda}$ employed in the calculation.Purpose: Given that ESPEs provide a way to interpret the many-body problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. State-of-the-art multireference in-medium similarity renormalization group and self-consistent Gorkov Green's function many-body calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale $\ensuremath{\lambda}$.Results: The formal proofs are confirmed by the results of ab initio many-body calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced many-body interactions beyond three-body operators and to the nonexact treatment of the many-body Schr\"odinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of $\ensuremath{\lambda}$ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasi-observable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.

arXiv:1411.1237•Physical Review C•Sep 14, 2015•76 citations

Realistic and effective interactions in the study of nuclear matter

V. Somà

The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced

•Jan 1, 2006•0 citations

The Canonical Nuclear Many-Body Problem as a Rigorous Effective Theory

W. C. Haxton, C.-L. Song

The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.

arXiv:nucl-th/9906082•Jun 27, 1999•0 citations

1Computational aspects of nuclear coupled-cluster theory

G. Hagen, Hai Ah Nam

We discuss computational aspects of the spherical coupled-cluster method specific to the nuclear many-body problem. Using chiral nucleon-nucleon interaction at next-to-next-to-next-to leading order (N3LO) with cutoff Λ = 500MeV, we present coupled-cluster results for the ground state of 40Ca. Scaling and performance studies are presented together with challenges we meet with when extending the coupled-cluster effort to nuclei mass hundred and beyond. The low-energy nuclear many-body problem is a challenging undertaking, how-ever, in the last decade there has been significant progress in first principle cal-culations of nuclei.1), 2), 3), 4), 5), 6), 7), 8), 9) With the advance of chiral effective field theory10), 11), 12), 13), 14), 15) which allows for a systematic and consistent derivation of

•Jan 1, 2012•3 citations

Renormalization Group Equation for Low Momentum Effective Nuclear Interactions

S. K. Bogner, A. Schwenk, T.T.S. Kuo +1 more

We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.

arXiv:nucl-th/0111042•Nov 13, 2001•11 citations
This review aims at a critical discussion of the interplay between effective interactions derived from various many-body approaches and spectroscopic data extracted from large scale shell-model studies. To achieve this, our many-body scheme starts with the free nucleon-nucleon (NN) interaction, typically modelled on various meson exchanges. The NN interaction is in turn renormalized in order to derive an effective medium dependent interaction. The latter is in turn used in shell-model calculations of selected nuclei. We also describe how to sum up the parquet class of diagrams and present initial uses of the effective interactions in coupled cluster many-body theory.
Background: The concept of single-nucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective single-particle energies (ESPEs) introduced by French [Proceedings of the International School of Physics ``Enrico Fermi,'' Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970)] represent the most appropriate tool to relate many-body observables to a single-nucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012)], the dependence of ESPEs on one-nucleon transfer probability matrices makes them purely theoretical quantities that ``run'' with the nonobservable resolution scale $\ensuremath{\lambda}$ employed in the calculation.Purpose: Given that ESPEs provide a way to interpret the many-body problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. State-of-the-art multireference in-medium similarity renormalization group and self-consistent Gorkov Green's function many-body calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale $\ensuremath{\lambda}$.Results: The formal proofs are confirmed by the results of ab initio many-body calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced many-body interactions beyond three-body operators and to the nonexact treatment of the many-body Schr\"odinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of $\ensuremath{\lambda}$ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasi-observable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.
The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced
The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.
We discuss computational aspects of the spherical coupled-cluster method specific to the nuclear many-body problem. Using chiral nucleon-nucleon interaction at next-to-next-to-next-to leading order (N3LO) with cutoff Λ = 500MeV, we present coupled-cluster results for the ground state of 40Ca. Scaling and performance studies are presented together with challenges we meet with when extending the coupled-cluster effort to nuclei mass hundred and beyond. The low-energy nuclear many-body problem is a challenging undertaking, how-ever, in the last decade there has been significant progress in first principle cal-culations of nuclei.1), 2), 3), 4), 5), 6), 7), 8), 9) With the advance of chiral effective field theory10), 11), 12), 13), 14), 15) which allows for a systematic and consistent derivation of
We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.