Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1708.02601 · Physics Letters B

Tracing the evolution of nuclear forces under the similarity renormalization group

Calvin W. Johnson

Published Oct 12, 2017• 3 citations
arXivHTMLPDF

Abstract

I examine the evolution of nuclear forces under the similarity renormalization group (SRG) using traces of the many-body configuration-space Hamiltonian. While SRG is often said to “soften” the nuclear interaction, I provide numerical examples which paint a complementary point of view: the primary effect of SRG, using the kinetic energy as the generator of the evolution, is to shift downward the diagonal matrix elements in the model space, while the off-diagonal elements undergo significantly smaller changes. By employing traces, I argue that this is a very natural outcome as one diagonalizes a matrix, and helps one to understand the success of SRG.

DOI
10.1016/j.physletb.2017.10.011

Related papers

Applications of the Similarity Renormalization Group to the Nuclear Interaction

Eric Jurgenson

The Similarity Renormalization Group (SRG) is investigated as a powerful yet practical method to modify nuclear potentials so as to reduce computational requirements for calculations of observables. The key feature of SRG transformations that leads to computational benefits is the decoupling of low-energy nuclear physics from high-energy details of the inter-nucleon interaction. We examine decoupling quantitatively for two-body observables and few-body binding energies. The universal nature of this decoupling is illustrated and errors from suppressing high-momentum modes above the decoupling scale are shown to be perturbatively small. To explore the SRG evolution of many-body forces, we use as a laboratory a one-dimensional system of bosons with short-range repulsion and mid-range attraction, which emulates realistic nuclear forces. The free-space SRG is implemented for few-body systems in a symmetrized harmonic oscillator basis using a recursive construction analogous to no-core shell model implementations. Building on one-dimensional results we performed the first practical evolution of three-dimensional many-body forces within the No-Core Shell Model basis. Results for the 3H binding energy are consistent with previous calculations involving momentum-space evolution of only two-body forces, and validate expectations from calculations in the one-dimensional oscillator basis. When applied to 4He calculations, the two- and three-body oscillator matrix elements yield rapid convergence of the ground-state energy with a small net contribution of the induced four-body force. The radius of light nuclei is also explored in the three-dimensional basis.

arXiv:0912.2937•Dec 15, 2009•5 citations

Alternative similarity renormalization group generators in nuclear structure calculations

Nuiok Dicaire, Conor M. B. Omand, P. Navrátil

The similarity renormalization group (SRG) has been successfully applied to soften interactions for ab initio nuclear calculations. In almost all practical applications in nuclear physics, an SRG generator with the kinetic energy operator is used. With this choice, a fast convergence of many-body calculations can be achieved, but at the same time substantial three-body interactions are induced even if one starts from a purely two-nucleon ($\mathit{NN}$) Hamiltonian. Three-nucleon ($3N$) interactions can be handled by modern many-body methods. However, it has been observed that when including initial chiral $3N$ forces in the Hamiltonian, the SRG transformations induce a non-negligible four-nucleon interaction that cannot be currently included in the calculations for technical reasons. Consequently, it is essential to investigate alternative SRG generators that might suppress the induction of many-body forces while at the same time might preserve the good convergence. In this work we test two alternative generators with operators of block structure in the harmonic oscillator basis. In the no-core shell model calculations for ${}^{3}\mathrm{H}$, ${}^{4}\mathrm{He}$, and ${}^{6}\mathrm{Li}$ with chiral $\mathit{NN}$ force, we demonstrate that their performances appear quite promising.

arXiv:1406.1815•Physical Review C•Sep 4, 2014•8 citations

Similarity Renormalization Group Methods in Ab-Initio Nuclear and Hadronic Physics Volume III -Independent Study

Syed Muntasir Mamun

The Similarity Renormalization Group (SRG), introduced in its modern Hamiltonian form by G■azek & Wilson and by Wegner, and pedagogically clarified by Szpigel & Perry (2000) through the two-dimensional delta-function laboratory, has become one of the dominant tools of ab-initio nuclear theory. Continuous unitary transformations that suppress off-diagonal matrix elements with respect to an energy-difference cutoff systematically soften nuclear interactions, accelerate the convergence of many-body methods, and generate a controlled hierarchy of induced many-body forces. This independent study traces the trajectory from the original pedagogical prototype to the present state of the art. It examines the SRG evolution of two-, three- and four-body forces, the in-medium similarity renormalization group (IMSRG), applications to nuclear structure and reactions, and the adaptation of the error-scaling diagnostics pioneered by Szpigel and Perry to the quantification of residual cutoff dependence. Emphasis is placed on the systematic inclusion of higher-body forces and on the necessity of rigorous uncertainty quantification if SRG-based predictions are to meet modern standards of scientific reliability. The study concludes with an assessment of open frontiers in continuum observables, electroweak currents, and the interface with lattice QCD and chiral effective field theory. Also available on Academia.edu: https://www.academia.edu/attachments/133749625/download_file?s=portfolio

•Jan 1, 2026•0 citations

Similarity renormalization group for nucleon-nucleon interactions

S. K. Bogner, R. J. Furnstahl, Robert J. Perry

The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the Hamiltonian toward a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables and provides a method to consistently evolve many-body potentials and other operators.

arXiv:nucl-th/0611045•Physical Review C•Jun 13, 2007•469 citations

Group-theoretical approach to interaction renormalization forab initiosolutions of the nuclear problem

K. D. Sviratcheva, Jerry Paul Draayer, T. Dytrych

For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations ) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions.

Journal of Physics Conference Series•Mar 1, 2011•0 citations

Operator evolution via the similarity renormalization group: The deuteron

E. R. Anderson, S. K. Bogner, R. J. Furnstahl +1 more

Similarity renormalization group (SRG) flow equations can be used to unitarily soften nuclear Hamiltonians by decoupling high-energy intermediate-state contributions to low-energy observables while maintaining the natural hierarchy of many-body forces. Analogous flow equations can be used to consistently evolve operators so that observables are unchanged if no approximations are made. The question in practice is whether the advantages of a softer Hamiltonian and less-correlated wave functions might be offset by complications in approximating and applying other operators. Here we examine the properties of SRG-evolved operators, focusing in this article on applications to the deuteron but leading toward methods for few-body systems. We find the advantageous features generally carry over to other operators with additional simplifications in some cases from factorization of the unitary transformation operator.

arXiv:1008.1569•Physical Review C•Nov 3, 2010•67 citations
I examine the evolution of nuclear forces under the similarity renormalization group (SRG) using traces of the many-body configuration-space Hamiltonian. While SRG is often said to “soften” the nuclear interaction, I provide numerical examples which paint a complementary point of view: the primary effect of SRG, using the kinetic energy as the generator of the evolution, is to shift downward the diagonal matrix elements in the model space, while the off-diagonal elements undergo significantly smaller changes. By employing traces, I argue that this is a very natural outcome as one diagonalizes a matrix, and helps one to understand the success of SRG.
The Similarity Renormalization Group (SRG) is investigated as a powerful yet practical method to modify nuclear potentials so as to reduce computational requirements for calculations of observables. The key feature of SRG transformations that leads to computational benefits is the decoupling of low-energy nuclear physics from high-energy details of the inter-nucleon interaction. We examine decoupling quantitatively for two-body observables and few-body binding energies. The universal nature of this decoupling is illustrated and errors from suppressing high-momentum modes above the decoupling scale are shown to be perturbatively small. To explore the SRG evolution of many-body forces, we use as a laboratory a one-dimensional system of bosons with short-range repulsion and mid-range attraction, which emulates realistic nuclear forces. The free-space SRG is implemented for few-body systems in a symmetrized harmonic oscillator basis using a recursive construction analogous to no-core shell model implementations. Building on one-dimensional results we performed the first practical evolution of three-dimensional many-body forces within the No-Core Shell Model basis. Results for the 3H binding energy are consistent with previous calculations involving momentum-space evolution of only two-body forces, and validate expectations from calculations in the one-dimensional oscillator basis. When applied to 4He calculations, the two- and three-body oscillator matrix elements yield rapid convergence of the ground-state energy with a small net contribution of the induced four-body force. The radius of light nuclei is also explored in the three-dimensional basis.
The similarity renormalization group (SRG) has been successfully applied to soften interactions for ab initio nuclear calculations. In almost all practical applications in nuclear physics, an SRG generator with the kinetic energy operator is used. With this choice, a fast convergence of many-body calculations can be achieved, but at the same time substantial three-body interactions are induced even if one starts from a purely two-nucleon (NN\mathit{NN}NN) Hamiltonian. Three-nucleon (3N3N3N) interactions can be handled by modern many-body methods. However, it has been observed that when including initial chiral 3N3N3N forces in the Hamiltonian, the SRG transformations induce a non-negligible four-nucleon interaction that cannot be currently included in the calculations for technical reasons. Consequently, it is essential to investigate alternative SRG generators that might suppress the induction of many-body forces while at the same time might preserve the good convergence. In this work we test two alternative generators with operators of block structure in the harmonic oscillator basis. In the no-core shell model calculations for 3H{}^{3}\mathrm{H}3H, 4He{}^{4}\mathrm{He}4He, and 6Li{}^{6}\mathrm{Li}6Li with chiral NN\mathit{NN}NN force, we demonstrate that their performances appear quite promising.
The Similarity Renormalization Group (SRG), introduced in its modern Hamiltonian form by G■azek & Wilson and by Wegner, and pedagogically clarified by Szpigel & Perry (2000) through the two-dimensional delta-function laboratory, has become one of the dominant tools of ab-initio nuclear theory. Continuous unitary transformations that suppress off-diagonal matrix elements with respect to an energy-difference cutoff systematically soften nuclear interactions, accelerate the convergence of many-body methods, and generate a controlled hierarchy of induced many-body forces. This independent study traces the trajectory from the original pedagogical prototype to the present state of the art. It examines the SRG evolution of two-, three- and four-body forces, the in-medium similarity renormalization group (IMSRG), applications to nuclear structure and reactions, and the adaptation of the error-scaling diagnostics pioneered by Szpigel and Perry to the quantification of residual cutoff dependence. Emphasis is placed on the systematic inclusion of higher-body forces and on the necessity of rigorous uncertainty quantification if SRG-based predictions are to meet modern standards of scientific reliability. The study concludes with an assessment of open frontiers in continuum observables, electroweak currents, and the interface with lattice QCD and chiral effective field theory. Also available on Academia.edu: https://www.academia.edu/attachments/133749625/download_file?s=portfolio
For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations ) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions.
Similarity renormalization group (SRG) flow equations can be used to unitarily soften nuclear Hamiltonians by decoupling high-energy intermediate-state contributions to low-energy observables while maintaining the natural hierarchy of many-body forces. Analogous flow equations can be used to consistently evolve operators so that observables are unchanged if no approximations are made. The question in practice is whether the advantages of a softer Hamiltonian and less-correlated wave functions might be offset by complications in approximating and applying other operators. Here we examine the properties of SRG-evolved operators, focusing in this article on applications to the deuteron but leading toward methods for few-body systems. We find the advantageous features generally carry over to other operators with additional simplifications in some cases from factorization of the unitary transformation operator.