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arXiv:1507.06725 · Physical Review C

Magnus expansion and in-medium similarity renormalization group

Titus Morris, Nathan Parzuchowski, S. K. Bogner

Published Sep 30, 2015• 143 citations
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Abstract

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 $^{16}\mathrm{O}$ nucleus are used as test beds to illustrate the efficacy of the Magnus expansion.

DOI
10.1103/physrevc.92.034331

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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.
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.
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.
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.
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.