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:2106.01302 · Physical Review C

Singular value decomposition and similarity renormalization group evolution of nuclear interactions

Bin Zhu, R. Wirth, H. Hergert

Published Oct 11, 2021• 9 citations
arXivHTMLPDF

Abstract

One of the main challenges for ab initio nuclear many-body theory in the coming decade is the growth of computational and storage costs as calculations are extended to increasingly heavy, exotic, and structurally complex nuclei. Here, we investigate the factorization of nuclear interactions as a means to address this issue. We perform singular value decompositions (SVD) of current nucleon-nucleon interactions in partial wave representation and study the dependence of the singular value spectrum on interaction characteristics like regularization schemes and resolution scales. Next, we develop and implement the similarity renormalization group (SRG) evolution of the interaction in terms of the relevant singular vectors, and demonstrate that this SVD-SRG approach accurately preserves two-nucleon observables. We find that low-resolution interactions naturally allow the truncation of the SVD at low rank, and that a small number of relevant components is sufficient to capture the nuclear interaction and perform an accurate SRG evolution, while the Coulomb interaction requires special consideration. The rank is uniform across all partial waves and almost independent of the basis choice in the tested cases. This suggests an interpretation of the relevant singular components as mere representations of a small set of abstract operators that can describe the interaction and its SRG flow. Following the traditional workflow for nuclear interactions, we discuss how the transformation between the center-of-mass and laboratory frames creates redundant copies of the partial wave components when implemented in matrix representation, and we discuss strategies for mitigation. Finally, we test the low-rank approximation to the SRG-evolved interactions in many-body calculations using the in-medium SRG. By including nuclear radii in our analysis, we verify that the implementation of the SRG using the singular vectors of the interaction does not spoil the evolution of other observables.

DOI
10.1103/physrevc.104.044002

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

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

In-Medium Similarity Renormalization Group For Nuclei

Koshiroh Tsukiyama, S. K. Bogner, A. Schwenk

We present a new ab initio method that uses similarity renormalization group (SRG) techniques to continuously diagonalize nuclear many-body Hamiltonians. In contrast with applications of the SRG to two- and three-nucleon interactions in free space, we perform the SRG evolution "in medium" directly in the A-body system of interest. The in-medium approach has the advantage that one can approximately evolve 3,…,A-body operators using only two-body machinery based on normal-ordering techniques. The method is nonperturbative and can be tailored to problems ranging from the diagonalization of closed-shell nuclei to the construction of effective valence-shell Hamiltonians and operators. We present first results for the energies of 4He, 16O, and 40Ca, which have accuracies comparable to coupled-cluster calculations.

arXiv:1006.3639•Physical Review Letters•Jun 1, 2011•273 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

Dynamics and applicability of the similarity renormalization group

Kristina D. Launey, T. Dytrych, Jerry Paul Draayer +1 more

The similarity renormalization group (SRG) concept (or flow equations methodology) is studied with a view toward the renormalization of nucleon–nucleon interactions for ab initio shell-model calculations. For a general flow, we give quantitative measures, in the framework of spectral distribution theory, for the strength of the SRG-induced higher order (many-body) terms of an evolved interaction. Specifically, we show that there is a hierarchy among the terms, with those of the lowest particle rank being the most important. This feature is crucial for maintaining the unitarity of SRG transformations and key to the method's applicability.

Journal of Physics A Mathematical and Theoretical•Dec 6, 2011•1 citation

Low-rank matrix decompositions for ab initio nuclear structure

A. Tichai, P. Arthuis, K. Hebeler +3 more

The extension of ab initio quantum many-body theory to higher accuracy and larger systems is intrinsically limited by the handling of large data objects in form of wave-function expansions and/or many-body operators. In this work we present matrix factorization techniques as a systematically improvable and robust tool to significantly reduce the computational cost in many-body applications at the price of introducing a moderate decomposition error. We demonstrate the power of this approach for the nuclear two-body systems, for many-body perturbation theory calculations of symmetric nuclear matter, and for non-perturbative in-medium similarity renormalization group simulations of finite nuclei. Establishing low-rank expansions of chiral nuclear interactions offers possibilities to reformulate many-body methods in ways that take advantage of tensor factorization strategies.

arXiv:2105.03935•Physics Letters B•Sep 6, 2021•11 citations
One of the main challenges for ab initio nuclear many-body theory in the coming decade is the growth of computational and storage costs as calculations are extended to increasingly heavy, exotic, and structurally complex nuclei. Here, we investigate the factorization of nuclear interactions as a means to address this issue. We perform singular value decompositions (SVD) of current nucleon-nucleon interactions in partial wave representation and study the dependence of the singular value spectrum on interaction characteristics like regularization schemes and resolution scales. Next, we develop and implement the similarity renormalization group (SRG) evolution of the interaction in terms of the relevant singular vectors, and demonstrate that this SVD-SRG approach accurately preserves two-nucleon observables. We find that low-resolution interactions naturally allow the truncation of the SVD at low rank, and that a small number of relevant components is sufficient to capture the nuclear interaction and perform an accurate SRG evolution, while the Coulomb interaction requires special consideration. The rank is uniform across all partial waves and almost independent of the basis choice in the tested cases. This suggests an interpretation of the relevant singular components as mere representations of a small set of abstract operators that can describe the interaction and its SRG flow. Following the traditional workflow for nuclear interactions, we discuss how the transformation between the center-of-mass and laboratory frames creates redundant copies of the partial wave components when implemented in matrix representation, and we discuss strategies for mitigation. Finally, we test the low-rank approximation to the SRG-evolved interactions in many-body calculations using the in-medium SRG. By including nuclear radii in our analysis, we verify that the implementation of the SRG using the singular vectors of the interaction does not spoil the evolution of other observables.
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.
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.
We present a new ab initio method that uses similarity renormalization group (SRG) techniques to continuously diagonalize nuclear many-body Hamiltonians. In contrast with applications of the SRG to two- and three-nucleon interactions in free space, we perform the SRG evolution "in medium" directly in the A-body system of interest. The in-medium approach has the advantage that one can approximately evolve 3,…,A-body operators using only two-body machinery based on normal-ordering techniques. The method is nonperturbative and can be tailored to problems ranging from the diagonalization of closed-shell nuclei to the construction of effective valence-shell Hamiltonians and operators. We present first results for the energies of 4He, 16O, and 40Ca, which have accuracies comparable to coupled-cluster calculations.
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) concept (or flow equations methodology) is studied with a view toward the renormalization of nucleon–nucleon interactions for ab initio shell-model calculations. For a general flow, we give quantitative measures, in the framework of spectral distribution theory, for the strength of the SRG-induced higher order (many-body) terms of an evolved interaction. Specifically, we show that there is a hierarchy among the terms, with those of the lowest particle rank being the most important. This feature is crucial for maintaining the unitarity of SRG transformations and key to the method's applicability.
The extension of ab initio quantum many-body theory to higher accuracy and larger systems is intrinsically limited by the handling of large data objects in form of wave-function expansions and/or many-body operators. In this work we present matrix factorization techniques as a systematically improvable and robust tool to significantly reduce the computational cost in many-body applications at the price of introducing a moderate decomposition error. We demonstrate the power of this approach for the nuclear two-body systems, for many-body perturbation theory calculations of symmetric nuclear matter, and for non-perturbative in-medium similarity renormalization group simulations of finite nuclei. Establishing low-rank expansions of chiral nuclear interactions offers possibilities to reformulate many-body methods in ways that take advantage of tensor factorization strategies.