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arXiv:2601.16133 · Physical Review C

Computational schemes for the Magnus expansion of the in-medium similarity renormalization group

M. Heinz

Published Apr 3, 2026• 0 citations
arXivHTMLPDF

Abstract

The in-medium similarity renormalization group (IMSRG) is a popular many-body method used for computations of nuclei. It solves the many-body Schrödinger equation through a continuous unitary transformation of the many-body Hamiltonian. The IMSRG transformation is typically truncated at the normal-ordered two-body level, the IMSRG(2), but recently several approaches have been developed to capture the effects of normal-ordered three-body operators, the IMSRG(3). In particular, a factorized approximation to the IMSRG(3) proposes to capture the leading effects of three-body operators at the same computational cost as the IMSRG(2) approximation. This approach often employs an approximate scheme for solving the IMSRG equations, the so-called hunter-gatherer scheme. In this work, I study the uncertainty associated with this scheme. I find that the hunter-gatherer scheme differs by up to $7\,\mathrm{MeV}$ for ground-state energies and $0.5\,\mathrm{MeV}$ for excitation energies from standard IMSRG(2) approaches. These differences are in some cases comparable to the expected size of IMSRG(3) corrections.

DOI
10.1103/n348-lvll

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•Apr 3, 2026•0 citations
The in-medium similarity renormalization group (IMSRG) is a popular many-body method used for computations of nuclei. It solves the many-body Schrödinger equation through a continuous unitary transformation of the many-body Hamiltonian. The IMSRG transformation is typically truncated at the normal-ordered two-body level, the IMSRG(2), but recently several approaches have been developed to capture the effects of normal-ordered three-body operators, the IMSRG(3). In particular, a factorized approximation to the IMSRG(3) proposes to capture the leading effects of three-body operators at the same computational cost as the IMSRG(2) approximation. This approach often employs an approximate scheme for solving the IMSRG equations, the so-called hunter-gatherer scheme. In this work, I study the uncertainty associated with this scheme. I find that the hunter-gatherer scheme differs by up to 7 MeV7\,\mathrm{MeV}7MeV for ground-state energies and 0.5 MeV0.5\,\mathrm{MeV}0.5MeV for excitation energies from standard IMSRG(2) approaches. These differences are in some cases comparable to the expected size of IMSRG(3) corrections.
Over the past decade the in-medium similarity renormalization group (IMSRG) approach has proven to be a powerful and versatile ab initio many-body method for studying medium-mass nuclei. So far, the IMSRG was limited to the approximation in which only up to two-body operators are incorporated in the renormalization group flow, referred to as the IMSRG(2). In this work, we extend the IMSRG(2) approach to fully include three-body operators yielding the IMSRG(3) approximation. We use a perturbative scaling analysis to estimate the importance of individual terms in this approximation and introduce truncations that aim to approximate the IMSRG(3) at a lower computational cost. The IMSRG(3) is systematically benchmarked for different nuclear Hamiltonians for 4He^{4}\mathrm{He}4He and 16O^{16}\mathrm{O}16O in small model spaces. The IMSRG(3) systematically improves over the IMSRG(2) relative to exact results. Approximate IMSRG(3) truncations constructed based on computational cost are able to reproduce much of the systematic improvement offered by the full IMSRG(3). We also find that the approximate IMSRG(3) truncations behave consistently with expectations from our perturbative analysis, indicating that this strategy may also be used to systematically approximate the IMSRG(3).
I present a method, IMSRG-Net, which utilizes machine-learning techniques as a solver for the in-medium similarity renormalization group (IMSRG). The primary objective of IMSRG-Net is to approximate the Magnus operators $\mathrm{\ensuremath{\Omega}}(s)$ in the IMSRG flow equation, thereby offering an alternative to the computationally intensive part of IMSRG calculations. The key idea of IMSRG-Net is its design of the loss function inspired by physics-informed neural networks to encode the underlying physics, i.e., the IMSRG flow equation, into the model. Through training on a dataset comprising ten data points with flow parameters up to s=20s=20s=20, capturing approximately one-eighth to one-quarter of the entire flow, IMSRG-Net exhibits remarkable accuracy in extrapolating the ground-state energies and charge radii of 16O^{16}\mathrm{O}16O and 40Ca^{40}\mathrm{Ca}40Ca. Furthermore, this model demonstrates effectiveness in deriving effective interactions for a valence space.
We benchmark two implementations of the in-medium similarity renormalization group (IMSRG) method, IMSRG(2) and IMSRG(2*), for the low-lying states of a pairing-plus-particle-hole model with varying numbers of fermions. In IMSRG(2), all operators are truncated up to the normal-ordered two-body terms, whereas IMSRG(2*) includes an additional term to partially account for higher-body contributions. The results are compared against exact solutions. We find that IMSRG(2*) consistently outperforms IMSRG(2) for both ground and excited states, although achieving convergence for excited states remains more challenging in strongly correlated systems than for the ground state.
I propose a data-driven surrogate model for the In-Medium Similarity Renormalization Group (IMSRG) method using Dynamic Mode Decomposition (DMD). First, the Magnus formulation of the IMSRG is leveraged to represent the unitary transformation of many-body operators of interest. Then, snapshots of these operators at different flow parameters are decomposed by DMD to approximate the IMSRG flow in a latent space. The resulting emulator accurately reproduces the asymptotic flow behavior while lowering computational costs. I demonstrate that the DMD-based emulator results in a three to five times speedup compared to the full IMSRG calculation in a few test cases based on the ground state properties of 56Ni, 16O, and 40Ca in realistic nuclear interactions. While this is still not an acceleration that is significant enough to enable us to fully quantify, e.g., statistical uncertainties using Bayesian methods, this work offers a starting point for constructing efficient surrogate models for the IMSRG.
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.
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.