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arXiv:2508.08595 · Physical Review C

In-medium similarity renormalization group for a pairing-plus-particle-hole model

L. H. Chen, Y. G. Yao, B. C. He

Published Nov 19, 2025• 0 citations
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Abstract

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.

DOI
10.1103/2d2w-2j5t

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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.
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).
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.
Abstract We present a pedagogical introduction to the in-medium similarity renormalization group (IMSRG) framework for ab initio calculations of nuclei. The IMSRG performs continuous unitary transformations of the nuclear many-body Hamiltonian in second-quantized form, which can be implemented with polynomial computational effort. Through suitably chosen generators, it is possible to extract eigenvalues of the Hamiltonian in a given nucleus, or drive the Hamiltonian matrix in configuration space to specific structures, e.g., band- or block-diagonal form. Exploiting this flexibility, we describe two complementary approaches for the description of closed- and open-shell nuclei: the first is the multireference IMSRG (MR-IMSRG), which is designed for the efficient calculation of nuclear ground-state properties. The second is the derivation of non-empirical valence-space interactions that can be used as input for nuclear shell model (i.e., configuration interaction (CI)) calculations. This IMSRG+shell model approach provides immediate access to excitation spectra, transitions, etc, but is limited in applicability by the factorial cost of the CI calculations. We review applications of the MR-IMSRG and IMSRG+shell model approaches to the calculation of ground-state properties for the oxygen, calcium, and nickel isotopic chains or the spectroscopy of nuclei in the lower sd shell, respectively, and present selected new results, e.g., for the ground- and excited state properties of neon isotopes.
We present two new methods for performing ab initio calculations of excited states for closed-shell systems within the in-medium similarity renormalization group (IMSRG) framework. Both are based on combining the IMSRG with simple many-body methods commonly used to target excited states, such as the Tamm-Dancoff approximation (TDA) and equations-of-motion (EOM) techniques. In the first approach, a two-step sequential IMSRG transformation is used to drive the Hamiltonian to a form where a simple TDA calculation (i.e., diagonalization in the space of 1p1h1\mathrm{p}1\mathrm{h}1p1h excitations) becomes exact for a subset of eigenvalues. In the second approach, EOM techniques are applied to the IMSRG ground-state-decoupled Hamiltonian to access excited states. We perform proof-of-principle calculations for parabolic quantum dots in two dimensions and the closed-shell nuclei 16O^{16}\mathrm{O}16O and 22O^{22}\mathrm{O}22O. We find that the TDA-IMSRG approach gives better accuracy than the EOM-IMSRG when calculations converge, but it is otherwise lacking the versatility and numerical stability of the latter. Our calculated spectra are in reasonable agreement with analogous EOM-coupled-cluster calculations. This work paves the way for more interesting applications of the EOM-IMSRG approach to calculations of consistently evolved observables such as electromagnetic strength functions and nuclear matrix elements, and extensions to nuclei within one or two nucleons of a closed shell by generalizing the EOM ladder operator to include particle-number nonconserving terms.
By integrating the quantum Monte Carlo technique into the similarity renormalization group (SRG), we have developed a stochastic SRG framework (SRGQMC) capable of both free-space two-body and in-medium many-body evolutions. This approach circumvents the combinatorial tensor-space explosion of many-body flow equations by mapping continuous unitary transformations onto an ensemble of signed random walkers. We benchmark the SRGQMC against deterministic free-space SRG evolutions of realistic nucleon-nucleon (NN) interactions, as well as against in-medium SRG (IMSRG) many-body calculations with the Richardson pairing model at two- and three-body levels [IMSRG(2)/(3)]. While a deterministic extension to the four-body level [IMSRG(4)] remains unfeasible due to prohibitive computational costs, we have achieved the first IMSRG(4) calculation by using the stochastic technique, demonstrating a substantial improvement toward the full configuration-interaction limit. This stochastic framework provides a practical pathway to higher-order IMSRG calculations.
We explore the impact of retaining three-body operators within the in-medium similarity renormalization group (IMSRG), as well as various approximations schemes. After studying two toy problems, identical fermions with a contact interaction and the Lipkin-Meshkov-Glick model, we employ the valence-space formulation of the IMSRG to investigate the even-AAA carbon isotopes with a chiral two-body potential. We find that retaining only those commutators expressions that scale as N7{N}^{7}N7 provides an excellent approximation of the full three-body treatment.