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

TL;DR

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, finding that the hunter-gatherer scheme differs by up to up to $7\,\mathrm{MeV}$ for ground-state energies and $0.5\,\mathrm{MeV}$ for excitation energies from standard IMSRG(2) approaches.

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.

Figures (3)

• Figure 1: Comparison of split Magnus and hunter-gatherer approaches for varying splitting thresholds $\varepsilon_\mathrm{split}$. The flow equation, Magnus (without splitting), split Magnus, and hunter-gatherer results are indicated by the black, gray, blue, and red lines, respectively. I consider the ground-state energy of $^{48}\mathrm{Ca}$ computed using the IMSRG(2) in a model space of 7 major oscillator shells with frequency $\hbar\omega=16\,\mathrm{MeV}$.
• Figure 2: Comparison of ground-state energies and charge radii of $^{48}$Ca predicted using the flow equation (black stars), split Magnus (blue circles), and hunter-gatherer (red squares) approaches for varying $\varepsilon_\mathrm{split}$. I consider single-reference IMSRG(2) calculations (left) and VS-IMSRG(2) calculations using neutron $pf$-shell and full $pf$-shell valence spaces (middle and right, respectively).
• Figure 3: Energies from the IMSRG(2) as a function of the flow parameter $s$. I compare direct integration of the flow equation (black line) with the split Magnus scheme (blue) and the hunter-gatherer scheme with splitting threshold $\varepsilon_\mathrm{split} = 2.0$, $1.0$, $0.25$ (orange, green, red, respectively). Small circles indicate the integration steps, and large open circles indicate integration steps where the Magnus operator is split.