The Similarity Renormalization Group (SRG) flow equation is explored for systems of two and three spinless bosons in a momentum-space partial-wave basis. The two- and three-body binding energies as well as the phaseshifts are used to gauge that the transformation is unitary and to study how well the SRG decouples high- and low-energy physics. I consider four different potentials with different characteristics: Two simplied nucleon potentials and two inter-atomic helium potentials (a soft-core potential and the state-of-the-art LM2M2 potential that is fitted to a wealth of experimental data). An initial three-body force is included for two of these potentials. Even with only two-body terms in the initial hamiltonian, SRG induced many-body forces are shown to arise during the transformation. These induced forces are computed for the three-body system and their evolution is studied as a function of the flow parameter. In all cases the SRG transformed potentials display greatly improved decoupling. This is achieved with a three-body binding-energy deviation of less than 0.1% in all cases except for the soft-core helium potential.\n
The Similarity Renormalization Group (SRG) flow equation is explored for systems of two and three spinless bosons in a momentum-space partial-wave basis. The two- and three-body binding energies as well as the phaseshifts are used to gauge that the transformation is unitary and to study how well the SRG decouples high- and low-energy physics. I consider four different potentials with different characteristics: Two simplied nucleon potentials and two inter-atomic helium potentials (a soft-core potential and the state-of-the-art LM2M2 potential that is fitted to a wealth of experimental data). An initial three-body force is included for two of these potentials. Even with only two-body terms in the initial hamiltonian, SRG induced many-body forces are shown to arise during the transformation. These induced forces are computed for the three-body system and their evolution is studied as a function of the flow parameter. In all cases the SRG transformed potentials display greatly improved decoupling. This is achieved with a three-body binding-energy deviation of less than 0.1% in all cases except for the soft-core helium potential.\n
The Similarity Renormalization Group (SRG) has been studied in a one-dimensional system. It has been implemented for two- and three-body systems in continuous momentum space using Jacobi coordinates. Various potentials have been used, describing different physical attributes such as hard-core repulsion and long-range attraction. The unitarity of the transformation has been verified by examining the conservation of observables, for example of the binding energy. The importance of the induced three-body force has been addressed. We show that the decoupling of high and low energies occurs and that because of this computational requirements are lowered.
The Similarity Renormalization Group (SRG) is investigated as a powerful yet practical method to modify nuclear potentials so as to reduce computational requirements for calculations of observables. The key feature of SRG transformations that leads to computational benefits is the decoupling of low-energy nuclear physics from high-energy details of the inter-nucleon interaction. We examine decoupling quantitatively for two-body observables and few-body binding energies. The universal nature of this decoupling is illustrated and errors from suppressing high-momentum modes above the decoupling scale are shown to be perturbatively small. To explore the SRG evolution of many-body forces, we use as a laboratory a one-dimensional system of bosons with short-range repulsion and mid-range attraction, which emulates realistic nuclear forces. The free-space SRG is implemented for few-body systems in a symmetrized harmonic oscillator basis using a recursive construction analogous to no-core shell model implementations. Building on one-dimensional results we performed the first practical evolution of three-dimensional many-body forces within the No-Core Shell Model basis. Results for the 3H binding energy are consistent with previous calculations involving momentum-space evolution of only two-body forces, and validate expectations from calculations in the one-dimensional oscillator basis. When applied to 4He calculations, the two- and three-body oscillator matrix elements yield rapid convergence of the ground-state energy with a small net contribution of the induced four-body force. The radius of light nuclei is also explored in the three-dimensional basis.
We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane-wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low- from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of observables.
Similarity renormalization group (SRG) flow equations can be used to unitarily soften nuclear Hamiltonians by decoupling high-energy intermediate-state contributions to low-energy observables while maintaining the natural hierarchy of many-body forces. Analogous flow equations can be used to consistently evolve operators so that observables are unchanged if no approximations are made. The question in practice is whether the advantages of a softer Hamiltonian and less-correlated wave functions might be offset by complications in approximating and applying other operators. Here we examine the properties of SRG-evolved operators, focusing in this article on applications to the deuteron but leading toward methods for few-body systems. We find the advantageous features generally carry over to other operators with additional simplifications in some cases from factorization of the unitary transformation operator.
Short-range interactions between the constituent nucleons of atomic nuclei usually generate strong coupling between high-and low-momentum degrees of freedom in a nuclear Hamiltonian.This coupling leads to significant difficulty in computing the properties of nuclei using traditional quantum many-body techniques.The Similarity Renormalization Group (SRG) flow equations can unitarily soften these nuclear Hamiltonians by decoupling high-energy intermediate state contributions to low-energy observables.Analogous flow equations can be used to consistently evolve operators so that observables other than the binding energy can be calculated.The question in practice is whether the advantages of a softer Hamiltonian and less correlated wave functions are offset by complications in approximating and applying other operators.We begin by examining the properties of SRG-evolved operators in the deuteron, and lead into methods for few-body systems.The nuclear Hamiltonian contains a hierarchy of many-body forces.The most significant I would like to express my sincerest thanks to Richard Furnstahl, my thesis advisor.He has provided me a thorough grounding in the study of nuclear physics.Thanks to him, I have been involved not only in research, but also the larger nuclear physics community since my first year of study at The Ohio State University (OSU).However, more than that, he has set an exceptional personal and professional example.He has cultivated an overwhelmingly positive atmosphere in our research group.He has an infectious energy for learning and research.He clearly loves what he does and sharing his knowledge with others.The level of care and personal attention he exhibits has been exceptional, and I am especially thankful for this.I would also like to thank the many other outstanding teachers I have had throughout my education here at OSU, the University of Kentucky, and the school system of Campbell County, Kentucky.With their help and dedication I have gained the tools and motivation needed to continue on this path.There are too many, unfortunately, to list here all by name.However, I must thankfully acknowledge the exceptional time, support, and positive encouragement provided by my undergraduate research
We first examine how T-matrix equivalence drives the flow of similarity renormalization group (SRG) evolved potential matrix elements to a universal form, with the ultimate goal of gaining insight into universality for three-nucleon forces. In agreement with observations made previously for Lee-Suzuki transformations, regions of universal potential matrix elements are restricted to where half-on-shell T-matrix equivalence holds, but the potentials must also reproduce binding energies. We find universality in local energy regions, reflecting a local decoupling by the SRG. To continue the study in the 3-body sector, we create a simple 1-D spinless boson "theoretical laboratory" for a dramatic improvement in computational efficiency. We introduce a basis-transformation, harmonic oscillator (HO) basis, which is used for current many-body calculations and discuss the imposed truncations. When SRG evolving in a HO-basis, we show that the evolved matrix elements, once transformed back into momentum-representation, differ from those when evolving with momentum representation. This is because the generator in each basis is not exactly the same due to the truncation, which creates evolution artifacts in the 3-body potential matrix elements. In the 2- body sector, this can be avoided by increasing the basis size, but it remains unclear whether this is possible in the 3-body sector, as truncation errors in the 3-body sector are more difficult to avoid, and the computational power required is greatly increased for three-body evolution.