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...

We use the recently proposed In-Medium Similarity Renormalization Group (IM-SRG) to carry out a systematic study of closed-shell nuclei up to 56 Ni, based on chiral two-plus three-nucleon interactions. We analyze the capabilities of the IM-SRG by comparing our results for the groundstate energy to Coupled Cluster calculations, as well as to quasi-exact results from the ImportanceTruncated No-Core Shell Model. Using chiral two-plus three-nucleon Hamiltonians whose resolution scales are lowered by free-space SRG evolution, we obtain good agreement with experimental binding energies in 4 He and the closed-shell oxygen isotopes, while the calcium and nickel isotopes are somewhat overbound.

We formulate the in-medium similarity renormalization group (IM-SRG) for open-shell nuclei using a multireference formalism based on a generalized Wick theorem introduced in quantum chemistry. The resulting multireference IM-SRG (MR-IM-SRG) is used to perform the first ab initio study of all even oxygen isotopes with chiral nucleon-nucleon and three-nucleon interactions, from the proton to the neutron drip lines. We obtain an excellent reproduction of experimental ground-state energies with quantified uncertainties, which is validated by results from the importance-truncated no-core shell model and the coupled cluster method. The agreement between conceptually different many-body approaches and experiment highlights the predictive power of current chiral two- and three-nucleon interactions, and establishes the MR-IM-SRG as a promising new tool for ab initio calculations of medium-mass nuclei far from shell closures.