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arXiv:nucl-th/0703006 · Physical Review C

Unitary correlation operator method from a similarity renormalization group perspective

H. Hergert, Robert Roth

Published May 29, 2007• 50 citations
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Abstract

We investigate how the unitary correlation operator method (UCOM), developed to explicitly describe the strong short-range central and tensor correlations present in the nuclear many-body system, relates to the similarity renormalization group (SRG), a method to band-diagonalize Hamiltonians by continuous unitary transformations. We demonstrate that the structure of the UCOM transformation, originally motivated from the physically intuitive picture of correlations in coordinate space, arises naturally from the SRG flow equation. Apart from formal considerations we show that the momentum space matrix elements of the effective interactions obtained in both schemes agree extremely well.

DOI
10.1103/physrevc.75.051001

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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

Bulletin of the American Physical Society•Jan 1, 2012•0 citations
We investigate how the unitary correlation operator method (UCOM), developed to explicitly describe the strong short-range central and tensor correlations present in the nuclear many-body system, relates to the similarity renormalization group (SRG), a method to band-diagonalize Hamiltonians by continuous unitary transformations. We demonstrate that the structure of the UCOM transformation, originally motivated from the physically intuitive picture of correlations in coordinate space, arises naturally from the SRG flow equation. Apart from formal considerations we show that the momentum space matrix elements of the effective interactions obtained in both schemes agree extremely well.
Correlations play a crucial role in the nuclear many-body problem. We give an overview of recent developments in nuclear structure theory aiming at the description of these interaction-induced correlations by unitary transformations. We focus on the Unitary Correlation Operator Method (UCOM), which offers a very intuitive, universal and robust approach for the treatment of short-range correlations. We discuss the UCOM formalism in detail and highlight the connections to other methods for the description of short-range correlations and the construction of effective interactions. In particular, we juxtapose UCOM with the Similarity Renormalization Group (SRG) approach, which implements the unitary transformation of the Hamiltonian through a very flexi-ble flow-equation formulation. The UCOM- and SRG-transformed interactions are compared on the level of matrix elements and in many-body calculations within the no-core shell model and with Hartree-Fock plus perturbation theory for a variety of nuclei and observables. These calcu-lations provide a detailed picture of the similarities and differences as well as the advantages and limitations of unitary transformation methods.
We discuss relations and differences between two methods for the construction of unitarily transformed effective interactions, the Similarity Renormalization Group (SRG) and Unitary Correlation Operator Method (UCOM). The aim of both methods is to construct a soft phase-shift equivalent effective interaction which is well suited for many-body calculations in limited model spaces. After contrasting the two conceptual frameworks, we establish a formal connection between the initial SRG-generator and the static generators of the UCOM transformation. Furthermore we propose a mapping procedure to extract UCOM correlation functions from the SRG evolution. We compare the effective interactions resulting from the UCOM-transformation and the SRG-evolution on the level of matrix elements, in no-core shell model calculations of light nuclei, and in Hartree-Fock calculations up to 208Pb^{208}\mathrm{Pb}208Pb. Both interactions exhibit very similar convergence properties in light nuclei but show a different systematic behavior as function of particle number.
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.
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