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arXiv:0802.4239 · Physical Review C

Unitary correlation operator method and similarity renormalization group: Connections and differences

Robert Roth, S. Reinhardt, H. Hergert

Published Jun 20, 2008• 76 citations
arXivHTMLPDF

Abstract

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 $^{208}\mathrm{Pb}$. Both interactions exhibit very similar convergence properties in light nuclei but show a different systematic behavior as function of particle number.

DOI
10.1103/physrevc.77.064003

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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.
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 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.
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) 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.