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The Similarity Renormalization Group - for three bosons in one-dimensional momentum space

Oscar Åkerlund, Jonathan Lindgren, Pablo Lerner, Ross Linscott, Björn Grevholm, Johan Bergsten

Published Jan 1, 2011• 1 citation
OpenAlex

Abstract

The Similarity Renormalization Group (SRG) has been studied in a one-dimensional system. It has been implemented for two- and three-body systems in continuous momentum space using Jacobi coordinates. Various potentials have been used, describing different physical attributes such as hard-core repulsion and long-range attraction. The unitarity of the transformation has been verified by examining the conservation of observables, for example of the binding energy. The importance of the induced three-body force has been addressed. We show that the decoupling of high and low energies occurs and that because of this computational requirements are lowered.

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The similarity renormalization group for three-body interactions in one dimension

Oscar Åkerlund, Lindgren, E. J., Johan Bergsten +5 more

We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane-wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low- from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of observables.

•Jan 1, 2011•6 citations

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.

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The Similarity Renormalization Group for three bosons in a momentum-space partial-wave basis

B. D. Carlsson

The Similarity Renormalization Group (SRG) flow equation is explored for systems of two and three spinless bosons in a momentum-space partial-wave basis. The two- and three-body binding energies as well as the phaseshifts are used to gauge that the transformation is unitary and to study how well the SRG decouples high- and low-energy physics. I consider four different potentials with different characteristics: Two simplied nucleon potentials and two inter-atomic helium potentials (a soft-core potential and the state-of-the-art LM2M2 potential that is fitted to a wealth of experimental data). An initial three-body force is included for two of these potentials. Even with only two-body terms in the initial hamiltonian, SRG induced many-body forces are shown to arise during the transformation. These induced forces are computed for the three-body system and their evolution is studied as a function of the flow parameter. In all cases the SRG transformed potentials display greatly improved decoupling. This is achieved with a three-body binding-energy deviation of less than 0.1% in all cases except for the soft-core helium potential.\n

•Jan 1, 2012•0 citations

Toward Universality in Similarity Renormalization Group Evolved Few-body Potential Matrix Elements

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We first examine how T-matrix equivalence drives the flow of similarity renormalization group (SRG) evolved potential matrix elements to a universal form, with the ultimate goal of gaining insight into universality for three-nucleon forces. In agreement with observations made previously for Lee-Suzuki transformations, regions of universal potential matrix elements are restricted to where half-on-shell T-matrix equivalence holds, but the potentials must also reproduce binding energies. We find universality in local energy regions, reflecting a local decoupling by the SRG. To continue the study in the 3-body sector, we create a simple 1-D spinless boson "theoretical laboratory" for a dramatic improvement in computational efficiency. We introduce a basis-transformation, harmonic oscillator (HO) basis, which is used for current many-body calculations and discuss the imposed truncations. When SRG evolving in a HO-basis, we show that the evolved matrix elements, once transformed back into momentum-representation, differ from those when evolving with momentum representation. This is because the generator in each basis is not exactly the same due to the truncation, which creates evolution artifacts in the 3-body potential matrix elements. In the 2- body sector, this can be avoided by increasing the basis size, but it remains unclear whether this is possible in the 3-body sector, as truncation errors in the 3-body sector are more difficult to avoid, and the computational power required is greatly increased for three-body evolution.

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The similarity renormalization group (SRG), based on the simple one-body free harmonic oscillator Hamiltonian, is applied to various nucleon-nucleon realistic interactions to investigate the unitarity of the SRG transformations. Two-body and three-body contributions to the SRG-evolved Hamiltonian are studied in the framework of spectral distribution theory for reasonable SRG cutoffs and in multiparticle systems, with up through 28 particles considered. The outcome points to the first evidence for the overall importance of three-body SRG-induced interactions and especially, of its two-body effective content in multinucleon systems, without the need for large-scale shell model calculations for many light to heavier nuclei.

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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
The Similarity Renormalization Group (SRG) has been studied in a one-dimensional system. It has been implemented for two- and three-body systems in continuous momentum space using Jacobi coordinates. Various potentials have been used, describing different physical attributes such as hard-core repulsion and long-range attraction. The unitarity of the transformation has been verified by examining the conservation of observables, for example of the binding energy. The importance of the induced three-body force has been addressed. We show that the decoupling of high and low energies occurs and that because of this computational requirements are lowered.
We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane-wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low- from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of 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.
The Similarity Renormalization Group (SRG) flow equation is explored for systems of two and three spinless bosons in a momentum-space partial-wave basis. The two- and three-body binding energies as well as the phaseshifts are used to gauge that the transformation is unitary and to study how well the SRG decouples high- and low-energy physics. I consider four different potentials with different characteristics: Two simplied nucleon potentials and two inter-atomic helium potentials (a soft-core potential and the state-of-the-art LM2M2 potential that is fitted to a wealth of experimental data). An initial three-body force is included for two of these potentials. Even with only two-body terms in the initial hamiltonian, SRG induced many-body forces are shown to arise during the transformation. These induced forces are computed for the three-body system and their evolution is studied as a function of the flow parameter. In all cases the SRG transformed potentials display greatly improved decoupling. This is achieved with a three-body binding-energy deviation of less than 0.1% in all cases except for the soft-core helium potential.\n
We first examine how T-matrix equivalence drives the flow of similarity renormalization group (SRG) evolved potential matrix elements to a universal form, with the ultimate goal of gaining insight into universality for three-nucleon forces. In agreement with observations made previously for Lee-Suzuki transformations, regions of universal potential matrix elements are restricted to where half-on-shell T-matrix equivalence holds, but the potentials must also reproduce binding energies. We find universality in local energy regions, reflecting a local decoupling by the SRG. To continue the study in the 3-body sector, we create a simple 1-D spinless boson "theoretical laboratory" for a dramatic improvement in computational efficiency. We introduce a basis-transformation, harmonic oscillator (HO) basis, which is used for current many-body calculations and discuss the imposed truncations. When SRG evolving in a HO-basis, we show that the evolved matrix elements, once transformed back into momentum-representation, differ from those when evolving with momentum representation. This is because the generator in each basis is not exactly the same due to the truncation, which creates evolution artifacts in the 3-body potential matrix elements. In the 2- body sector, this can be avoided by increasing the basis size, but it remains unclear whether this is possible in the 3-body sector, as truncation errors in the 3-body sector are more difficult to avoid, and the computational power required is greatly increased for three-body evolution.
The similarity renormalization group (SRG), based on the simple one-body free harmonic oscillator Hamiltonian, is applied to various nucleon-nucleon realistic interactions to investigate the unitarity of the SRG transformations. Two-body and three-body contributions to the SRG-evolved Hamiltonian are studied in the framework of spectral distribution theory for reasonable SRG cutoffs and in multiparticle systems, with up through 28 particles considered. The outcome points to the first evidence for the overall importance of three-body SRG-induced interactions and especially, of its two-body effective content in multinucleon systems, without the need for large-scale shell model calculations for many light to heavier nuclei.
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.