The Shell Model, the Renormalization Group and the Two-Body Interaction

The no-core shell model and the effective interaction $V_{{\rm low} k}$ can both be derived using the Lee-Suzuki projection operator formalism. The main difference between the two is the choice of basis states that define the model space. The effective interaction $V_{{\rm low} k}$ can also be derived using the renormalization group. That renormalization group derivation can be extended in a straight forward manner to also include the no-core shell model. In the nuclear matter limit the no-core shell model effective interaction in the two-body approximation reduces identically to $V_{{\rm low} k}$. The same considerations apply to the Bloch-Horowitz version of the shell model and the renormalization group treatment of two-body scattering by Birse, McGovern and Richardson

Projection Operator Formalisms and the Nuclear Shell Model

The shell model solve the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrodinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.Comment: 4 pages, no figure

In-Medium Similarity Renormalization Group for Nuclei

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-model Hamiltonians and operators. We present first results for the ground-state energies of $^4$He, $^{16}$O and $^{40}$Ca, which have accuracies comparable to coupled-cluster calculations.Comment: 4pages, 4 figures, to be published in PR

Density Matrix Expansion for Low-Momentum Interactions

A first step toward a universal nuclear energy density functional based on low-momentum interactions is taken using the density matrix expansion (DME) of Negele and Vautherin. The DME is adapted for non-local momentum-space potentials and generalized to include local three-body interactions. Different prescriptions for the three-body DME are compared. Exploratory results are given at the Hartree-Fock level, along with a roadmap for systematic improvements within an effective action framework for Kohn-Sham density functional theory.Comment: 50 pages, 10 figure

From low-momentum interactions to nuclear structure

We present an overview of low-momentum two-nucleon and many-body interactions and their use in calculations of nuclei and infinite matter. The softening of phenomenological and effective field theory (EFT) potentials by renormalization group (RG) transformations that decouple low and high momenta leads to greatly enhanced convergence in few- and many-body systems while maintaining a decreasing hierarchy of many-body forces. This review surveys the RG-based technology and results, discusses the connections to chiral EFT, and clarifies various misconceptions.Comment: 76 pages, 57 figures, two figures updated, published versio