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Similarity Renormalization Group Methods in Ab-Initio Nuclear and Hadronic Physics Volume III -Independent Study

Syed Muntasir Mamun

Published Jan 1, 2026• 0 citations

Abstract

The Similarity Renormalization Group (SRG), introduced in its modern Hamiltonian form by G■azek & Wilson and by Wegner, and pedagogically clarified by Szpigel & Perry (2000) through the two-dimensional delta-function laboratory, has become one of the dominant tools of ab-initio nuclear theory. Continuous unitary transformations that suppress off-diagonal matrix elements with respect to an energy-difference cutoff systematically soften nuclear interactions, accelerate the convergence of many-body methods, and generate a controlled hierarchy of induced many-body forces. This independent study traces the trajectory from the original pedagogical prototype to the present state of the art. It examines the SRG evolution of two-, three- and four-body forces, the in-medium similarity renormalization group (IMSRG), applications to nuclear structure and reactions, and the adaptation of the error-scaling diagnostics pioneered by Szpigel and Perry to the quantification of residual cutoff dependence. Emphasis is placed on the systematic inclusion of higher-body forces and on the necessity of rigorous uncertainty quantification if SRG-based predictions are to meet modern standards of scientific reliability. The study concludes with an assessment of open frontiers in continuum observables, electroweak currents, and the interface with lattice QCD and chiral effective field theory. Also available on Academia.edu: https://www.academia.edu/attachments/133749625/download_file?s=portfolio

DOI: 10.5281/zenodo.22990620

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The Similarity Renormalization Group (SRG), introduced in its modern Hamiltonian form by G■azek & Wilson and by Wegner, and pedagogically clarified by Szpigel & Perry (2000) through the two-dimensional delta-function laboratory, has become one of the dominant tools of ab-initio nuclear theory. Continuous unitary transformations that suppress off-diagonal matrix elements with respect to an energy-difference cutoff systematically soften nuclear interactions, accelerate the convergence of many-body methods, and generate a controlled hierarchy of induced many-body forces. This independent study traces the trajectory from the original pedagogical prototype to the present state of the art. It examines the SRG evolution of two-, three- and four-body forces, the in-medium similarity renormalization group (IMSRG), applications to nuclear structure and reactions, and the adaptation of the error-scaling diagnostics pioneered by Szpigel and Perry to the quantification of residual cutoff dependence. Emphasis is placed on the systematic inclusion of higher-body forces and on the necessity of rigorous uncertainty quantification if SRG-based predictions are to meet modern standards of scientific reliability. The study concludes with an assessment of open frontiers in continuum observables, electroweak currents, and the interface with lattice QCD and chiral effective field theory. Also available on Academia.edu: https://www.academia.edu/attachments/133749625/download_file?s=portfolio

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 (

\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.

I examine the evolution of nuclear forces under the similarity renormalization group (SRG) using traces of the many-body configuration-space Hamiltonian. While SRG is often said to “soften” the nuclear interaction, I provide numerical examples which paint a complementary point of view: the primary effect of SRG, using the kinetic energy as the generator of the evolution, is to shift downward the diagonal matrix elements in the model space, while the off-diagonal elements undergo significantly smaller changes. By employing traces, I argue that this is a very natural outcome as one diagonalizes a matrix, and helps one to understand the success of SRG.

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.

Efforts to describe nuclear structure and dynamics from first principles have advanced significantly in recent years. Exact methods for light nuclei are now able to include continuum degrees of freedom and treat structure and reactions on the same footing, and multiple approximate, computationally efficient many-body methods have been developed that can be routinely applied for medium-mass nuclei. This has made it possible to confront modern nuclear interactions from Chiral Effective Field Theory, that are rooted in Quantum Chromodynamics with a wealth of experimental data. Here, we discuss one of these efficient new many-body methods, the In-Medium Similarity Renormalization Group (IMSRG), and its applications in modern nuclear structure theory. The IMSRG evolves the nuclear many-body Hamiltonian in second-quantized form through continuous unitary transformations that can be implemented with polynomial computational effort. Through suitably chosen generators, we drive the matrix representation of the Hamiltonian in configuration space to specific shapes, e.g., to implement a decoupling of low-and high-energy scales, or to extract energy eigenvalues for a given nucleus. We present selected results from Multireference IMSRG (MR-IMSRG) calculations of open-shell nuclei, as well as proof-of-principle applications for intrinsically deformed medium-mass nuclei. We discuss the successes and prospects of merging the (MR-)IMSRG with many-body methods ranging from Configuration Interaction to the Density Matrix Renormalization Group, with the goal of achieving an efficient simultaneous description of dynamic and static correlationsin atomic nuclei.