ADDITION AND REMOVAL ENERGIES VIA THE IN-MEDIUM SIMILARITY RENORMALIZATION GROUP METHOD By Fei Yuan A DISSERTATION Submitted to Michigan State University in partial fulllment of the requirements for the degree of Physics—Doctor of Philosophy 2018 ABSTRACT ADDITION AND REMOVAL ENERGIES VIA THE IN-MEDIUM SIMILARITY RENORMALIZATION GROUP METHOD By Fei Yuan The in-medium similarity renormalization group (IM-SRG) is an ab initio many-body method suitable for systems with moderate numbers of particles due to its polynomial scaling in computa- tional cost. The formalism is highly exible and admits a variety of modications that extend its utility beyond the original goal of computing ground state energies of closed-shell systems. In this work, we present an extension of IM-SRG through quasidegenerate perturbation theory (QDPT) to compute addition and removal energies (single particle energies) near the Fermi level at low computational cost. This expands the range of systems that can be studied from closed-shell ones to nearby systems that dier by one particle. The method is applied to circular quantum dot systems and nuclei, and compared against other methods including equations-of-motion (EOM) IM-SRG and EOM coupled-cluster (CC) theory. The results are in good agreement for most cases. As part of this work, we present an open-source implementation of our exible and easy-to-use J-scheme framework as well as the HF, IM-SRG, and QDPT codes built upon this framework. We include an overview of the overall structure, the implementation details, and strategies for maintaining high code quality and eciency. Lastly, we also present a graphical application for manipulation of angular momentum coupling coecients through a diagrammatic notation for angular momenta (Jucys diagrams). The tool enables rapid derivations of equations involving angular momentum coupling – such as in J-scheme – and signicantly reduces the risk of human errors. In memory of Shichao Yuan (1964-2016) and Erik “Kexie” Gustafsson (1992-2017) iii ACKNOWLEDGMENTS I am forever indebted to my parents, who have made this journey at all possible. My interest in software was strongly inspired by my father’s projects, whereas my foundations in mathematics would not have been as solid without the tutoring from my mother. Throughout my studies, they have gone above and beyond to support my education, in spite of the limited resources our family had. I also extend my thanks to my grandparents who cared for me during my youngest years. On the academic side, I would like to thank my thesis advisor Morten Hjorth-Jensen for his knowledge, wisdom, and, most importantly, support. He had helped reignite my interest in physics by oering a pathway from experimental to computational physics. He encouraged me to research in areas that are most relevant to my interests, allowing me to learn and explore far more than I would have otherwise. I also wish to thank my co-advisor Scott Bogner, who has been very patient and oered valuable assistance and technical discussions. I thank the rest of my committee – Alexandra Gade, Carlo Piermarrochi, and Scott Pratt – for their helpful inputs, as well as my former advisor Chong-Yu Ruan who guided me during both undergraduate research and my rst two years of graduate research. I thank my colleagues for our productive discussions, including Heiko Hergert, Titus Morris, Justin Lietz, as well as Nathan Parzuchowski and Sam Novario, who have contributed substantially to the results of this work. On a personal level, I wish to thank Nathan, my hard-working ocemate who is well-versed in the intricacies of IM-SRG and Fortran and has a great sense of humor, Titus, whose mastery in many-body theory, quantum chemistry, and social situations is unmatched, Vinzent Steinberg, with whom many fun discussions and debates in programming have been made, and Tzong-Ru Terry Han, who has been very kind and supportive during my years in the laboratory. I would also like to thank Debbie Barratt and Kim Crosslan for their support iv throughout my long stay at Michigan State University. Lastly, I wish to thank all the friends I have made along the way for their support and encouragement, including my feline companion TipToe who has always had to put up with my chaotic schedule. v TABLE OF CONTENTS LIST OF TABLES ......................................... x LIST OF FIGURES ........................................ xi KEY TO SYMBOLS ........................................ xiv Chapter 1 Introduction ................................... 1 1.1 Contributions . 4 1.2 Outline . 5 Chapter 2 Many-body formalism ............................. 7 2.1 Many-particle states . 7 2.1.1 Product states . 8 2.1.2 Symmetrization and antisymmetrization . 11 2.2 Second quantization . 15 2.3 Many-body operators . 17 2.3.1 Zero-body operators . 17 2.3.2 One-body operators . 18 2.3.3 Two-body operators . 19 2.3.4 Three-body operators and beyond . 21 2.4 Particle-hole formalism . 22 2.5 Normal ordering . 24 2.5.1 Matrix elements relative to the Fermi vacuum . 27 2.5.2 Ambiguity of normal ordering on non-monomials . 29 2.6 Wick’s theorem . 30 2.6.1 Adjacent Wick contractions . 30 2.6.2 Normal-ordered Wick contractions . 32 2.6.3 Multiple Wick contractions . 33 2.6.4 Statement of Wick’s theorem . 34 2.6.5 Proof of Wick’s theorem . 35 2.7 Many-body diagrams . 38 2.7.1 Perturbative diagrams . 42 Chapter 3 Angular momentum coupling ......................... 44 3.1 Angular momentum and isospin . 44 3.2 Clebsch–Gordan coecients . 50 3.3 Wigner 3-jm symbol . 55 3.4 Angular momentum diagrams . 59 3.4.1 Nodes . 59 3.4.2 Lines . 60 3.4.3 Herring–Wigner 1-jm symbol . 61 3.4.4 Terminals . 62 vi 3.4.5 Closed diagrams . 63 3.4.6 Summed lines . 65 3.5 Phase rules . 65 3.6 Wigner–Eckart theorem . 69 3.7 Separation rules . 71 3.8 Recoupling coecients and 3n-j symbols . 73 3.8.1 Triangular delta . 73 3.8.2 6-j symbol . 75 3.8.3 9-j symbol . 78 3.9 Calculation of angular momentum coecients . 79 3.10 Graphical tool for angular momentum diagrams . 80 3.11 Fermionic states in J-scheme . 82 3.11.1 Two-particle states . 83 3.11.2 Three-particle states . 85 3.12 Matrix elements in J-scheme . 86 3.12.1 Standard-coupled matrix elements . 87 3.12.2 Pandya-coupled matrix elements . 88 3.12.3 Implicit-J convention . 90 Chapter 4 Many-body methods .............................. 93 4.1 Hartree-Fock method . 95 4.1.1 Hartree–Fock equations . 95 4.1.2 HF equations in J-scheme . 98 4.1.3 Solving HF equations . 98 4.1.4 Post-HF methods . 99 4.2 Similarity renormalization group methods . 101 4.2.1 Free space SRG . 101 4.2.2 In-medium SRG . 103 4.2.3 IM-SRG generators . 106 4.2.4 IM-SRG(2) equations . 108 4.2.5 IM-SRG(2) equations in J-scheme . 111 4.3 Quasidegenerate perturbation theory . 113 4.3.1 QDPT equations . 118 Chapter 5 Application to quantum systems ....................... 122 5.1 Quantum dots . 122 5.1.1 Quantum dot Hamiltonian . 122 5.1.2 Fock–Darwin basis . 124 5.1.3 Coulomb interaction in the Fock–Darwin basis . 127 5.2 Nuclei . 129 5.2.1 The nuclear Hamiltonian . 129 5.2.2 The nuclear interaction . 130 5.2.3 Spherical harmonic oscillator basis . 132 5.2.4 Matrix elements of kinetic energy . 135 vii Chapter 6 Implementation ................................. 137 6.1 Programming language . 137 6.1.1 Undened behavior . 140 6.1.2 Uniqueness and borrowing . 142 6.2 Structure of the program . 143 6.3 External libraries . 144 6.4 Basis and data layout . 144 6.4.1 Matrix types . 146 6.4.2 Basis charts . 150 6.4.3 Access of matrix elements . 154 6.4.4 Initialization of the basis . 157 6.5 Input matrix elements . 162 6.5.1 Inputs.