The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei

Abstract

We present a comprehensive review of the In-Medium Similarity Renormalization Group (IM-SRG), a novel ab inito method for nuclei. The IM-SRG employs a continuous unitary transformation of the many-body Hamiltonian to decouple the ground state from all excitations, thereby solving the many-body problem. Starting from a pedagogical introduction of the underlying concepts, the IM-SRG flow equations are developed for systems with and without explicit spherical symmetry. We study different IM-SRG generators that achieve the desired decoupling, and how they affect the details of the IM-SRG flow. Based on calculations of closed-shell nuclei, we assess possible truncations for closing the system of flow equations in practical applications, as well as choices of the reference state. We discuss the issue of center-of-mass factorization and demonstrate that the IM-SRG ground-state wave function exhibits an approximate decoupling of intrinsic and center-of-mass degrees of freedom, similar to Coupled Cluster (CC) wave functions. To put the IM-SRG in context with other many-body methods, in particular many-body perturbation theory and non-perturbative approaches like CC, a detailed perturbative analysis of the IM-SRG flow equations is carried out. We conclude with a discussion of ongoing developments, including IM-SRG calculations with three-nucleon forces, the multi-reference IM-SRG for open-shell nuclei, first non-perturbative derivations of shellmodel interactions, and the consistent evolution of operators in the IM-SRG. We dedicate this review to the memory of Gerry Brown, one of the pioneers of many-body calculations of nuclei.3 made a significant impact on nuclear structure theory since the pioneering applications in the early 2000's. Gerry would have been quite pleased with the IM-SRG, as he long advocated for the increased use of RG and Effective Field Theory (EFT) methods in nuclear physics, dating back to when two of us (SKB and AS) were beginning Ph.D. students at Stony Brook in the late 1990's. It was then that Gerry provided our first exposure to these powerful techniques, challenging us to recast in RG language the low-momentum NN interaction V low k and to revisit the calculations of Fermi liquid parameters and shell model Hamiltonians from a modern RG perspective. This was vintage Gerry, in that his intuitive style of doing physics told him that these problems were intimately related to Wilsonian RG ideas, even if he didn't know yet the details. Indeed, if pressed on any of the formalism or technical details, he would give a wry smile and say that such things were the responsibilities of young people to work through.While Gerry's research interests shifted towards astrophysics, heavy-ion and hadronic physics in his later years, the nuclear many-body problem always held a privileged place in his heart. As students, Gerry told us on more than one occasion that his work with Tom Kuo in the 1960's deriving shell model Hamiltonians from the NN interaction [1, 2] was his proudest achievement. Gerry was similarly fond...

In-medium similarity renormalization group for closed and open-shell nuclei

“…Slater determinants that are variationally optimized through a Hartree-Fock (HF) procedure have been shown to be reasonable reference states for interactions with low resolution scales around λ = 2.0 fm −1 (see, e.g., Refs. [33,51,53,58,62,115] and references therein), allowing post-HF methods like MBPT, CC, or the IMSRG discussed below to converge rapidly to the exact FCI result. Starting from such a HF reference state |Φ HF , we can obtain a basis consisting of the state itself and up to A-particle, A-hole (ApAh) excitations:…”

“…They generate ladder and ring summations in the limit s → ∞, but also ring-ladder interference diagrams with rich topologies that go far beyond traditional re-summation methods [41,124,125]. A detailed perturbative analysis is presented in [62]. For general reference states, the MR-IMSRG flow equations also include couplings to correlated pairs and triples of nucleons through the irreducible density matrices λ (2) and λ (3) .…”

“…However, we still have enormous freedom in choosing generators that implement this decoupling, especially if we are only interested in the limit s → ∞ [62]. Here, we restrict ourselves to those generators that we found to be most useful in practical applications.…”

“…[62] for a detailed discussion). We assume a Hartree-Fock reference state, and partition the Hamiltonian as…”

“…This leads us to the socalled In-Medium SRG (IMSRG), which is the main focus of the present work [60][61][62]. In a nutshell, we want to use SRG-like flow equations to decouple physics at different excitation energy scales of the nucleus, and render the Hamiltonian matrix in configuration space block or band diagonal in the process.…”