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AIP conference proceedings

The Magnus expansion and the in-medium similarity renormalization group

Titus Morris, S. K. Bogner

Published Jan 1, 2014• 3 citations
Publisher

Abstract

We present a variant of the in-medium similarity renormalization group(IMSRG) based on the Magnus expansion. In this new variant, the unitary transformation of the IMSRG is constructed explicitly, which allows for the transformation of observables quickly and easily. Additionally, the stiffness of equations encountered by the traditional solution of the IMSRG can be alleviated greatly. We present results and comparisons for the 3d electron gas.

DOI
10.1063/1.4899225

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We present a variant of the in-medium similarity renormalization group(IMSRG) based on the Magnus expansion. In this new variant, the unitary transformation of the IMSRG is constructed explicitly, which allows for the transformation of observables quickly and easily. Additionally, the stiffness of equations encountered by the traditional solution of the IMSRG can be alleviated greatly. We present results and comparisons for the 3d electron gas.
We present an improved variant of the in-medium similarity renormalization group (IM-SRG) based on the Magnus expansion. In the new formulation, one solves flow equations for the anti-Hermitian operator that, upon exponentiation, yields the unitary transformation of the IM-SRG. The resulting flow equations can be solved using a first-order Euler method without any loss of accuracy, resulting in substantial memory savings and modest computational speedups. Since one obtains the unitary transformation directly, the transformation of additional operators beyond the Hamiltonian can be accomplished with little additional cost, in sharp contrast to the standard formulation of the IM-SRG. Ground state calculations of the homogeneous electron gas (HEG) and 16O^{16}\mathrm{O}16O nucleus are used as test beds to illustrate the efficacy of the Magnus expansion.
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 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.
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.
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.