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Bulletin of the American Physical Society

Operator Evolution in the Similarity Renormalization Group

E. R. Anderson

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Short-range interactions between the constituent nucleons of atomic nuclei usually generate strong coupling between high-and low-momentum degrees of freedom in a nuclear Hamiltonian.This coupling leads to significant difficulty in computing the properties of nuclei using traditional quantum many-body techniques.The Similarity Renormalization Group (SRG) flow equations can unitarily soften these nuclear Hamiltonians by decoupling high-energy intermediate state contributions to low-energy observables.Analogous flow equations can be used to consistently evolve operators so that observables other than the binding energy can be calculated.The question in practice is whether the advantages of a softer Hamiltonian and less correlated wave functions are offset by complications in approximating and applying other operators.We begin by examining the properties of SRG-evolved operators in the deuteron, and lead into methods for few-body systems.The nuclear Hamiltonian contains a hierarchy of many-body forces.The most significant I would like to express my sincerest thanks to Richard Furnstahl, my thesis advisor.He has provided me a thorough grounding in the study of nuclear physics.Thanks to him, I have been involved not only in research, but also the larger nuclear physics community since my first year of study at The Ohio State University (OSU).However, more than that, he has set an exceptional personal and professional example.He has cultivated an overwhelmingly positive atmosphere in our research group.He has an infectious energy for learning and research.He clearly loves what he does and sharing his knowledge with others.The level of care and personal attention he exhibits has been exceptional, and I am especially thankful for this.I would also like to thank the many other outstanding teachers I have had throughout my education here at OSU, the University of Kentucky, and the school system of Campbell County, Kentucky.With their help and dedication I have gained the tools and motivation needed to continue on this path.There are too many, unfortunately, to list here all by name.However, I must thankfully acknowledge the exceptional time, support, and positive encouragement provided by my undergraduate research

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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

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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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Nuclear interactions evolved via renormalization group methods to lower resolution become increasingly nonlocal (off-diagonal in coordinate space) as they are softened. This inhibits both the development of intuition about the interactions and their use with some methods for solving the quantum many-body problem. By applying ``local projections'', a softened interaction can be reduced to a local effective interaction plus a nonlocal residual interaction. At the two-body level, a local projection after similarity renormalization group (SRG) evolution manifests the elimination of short-range repulsive cores and the flow toward universal low-momentum interactions. The SRG residual interaction is found to be relatively weak at low energy, which motivates a perturbative treatment.

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Short-range interactions between the constituent nucleons of atomic nuclei usually generate strong coupling between high-and low-momentum degrees of freedom in a nuclear Hamiltonian.This coupling leads to significant difficulty in computing the properties of nuclei using traditional quantum many-body techniques.The Similarity Renormalization Group (SRG) flow equations can unitarily soften these nuclear Hamiltonians by decoupling high-energy intermediate state contributions to low-energy observables.Analogous flow equations can be used to consistently evolve operators so that observables other than the binding energy can be calculated.The question in practice is whether the advantages of a softer Hamiltonian and less correlated wave functions are offset by complications in approximating and applying other operators.We begin by examining the properties of SRG-evolved operators in the deuteron, and lead into methods for few-body systems.The nuclear Hamiltonian contains a hierarchy of many-body forces.The most significant I would like to express my sincerest thanks to Richard Furnstahl, my thesis advisor.He has provided me a thorough grounding in the study of nuclear physics.Thanks to him, I have been involved not only in research, but also the larger nuclear physics community since my first year of study at The Ohio State University (OSU).However, more than that, he has set an exceptional personal and professional example.He has cultivated an overwhelmingly positive atmosphere in our research group.He has an infectious energy for learning and research.He clearly loves what he does and sharing his knowledge with others.The level of care and personal attention he exhibits has been exceptional, and I am especially thankful for this.I would also like to thank the many other outstanding teachers I have had throughout my education here at OSU, the University of Kentucky, and the school system of Campbell County, Kentucky.With their help and dedication I have gained the tools and motivation needed to continue on this path.There are too many, unfortunately, to list here all by name.However, I must thankfully acknowledge the exceptional time, support, and positive encouragement provided by my undergraduate research
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.
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.
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
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.
Nuclear interactions evolved via renormalization group methods to lower resolution become increasingly nonlocal (off-diagonal in coordinate space) as they are softened. This inhibits both the development of intuition about the interactions and their use with some methods for solving the quantum many-body problem. By applying ``local projections'', a softened interaction can be reduced to a local effective interaction plus a nonlocal residual interaction. At the two-body level, a local projection after similarity renormalization group (SRG) evolution manifests the elimination of short-range repulsive cores and the flow toward universal low-momentum interactions. The SRG residual interaction is found to be relatively weak at low energy, which motivates a perturbative treatment.