NUCLEAR SPECTROSCOPY WITH THE IN-MEDIUM SIMILARITY RENORMALIZATION GROUP By Nathan Michael Parzuchowski A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics – Doctor of Philosophy 2017 ABSTRACT NUCLEAR SPECTROSCOPY WITH THE IN-MEDIUM SIMILARITY RENORMALIZATION GROUP By Nathan Michael Parzuchowski The in-medium similarity renormalization group (IMSRG) is an ab initio many-body method which features soft polynomial scaling with system size and a Hermitian framework to create Hamiltonians tailored for use with low-level approximations such as Hartree-Fock (HF) theory or the random phase approximation (RPA). The flexibility that comes with these characteristics has made the IMSRG a mainstay in contemporary nuclear structure theory. However, spectroscopy with IMSRG calculations has been limited to scalar observables in nuclei accessible with shell model machinery, where the IMSRG is used to construct effective valence-space interactions. In this thesis, we present two novel developments which have greatly extended the IMSRG’s capability to perform spectroscopic calculations. First is the introduction of the equations-of-motion IMSRG (EOM-IMSRG), which uses an approximate, but systematically improvable diagonalization scheme in conjunction with the IMSRG to produce spectra and wave functions. The method does not suffer the model-space limitations of the shell model, but sacrifices some accuracy due to the approximate diagonalization. We benchmark this new method with the well established equations-of-motion coupled cluster and full configuration interaction methods, where we demonstrate that the method is indeed viable for closed-shell systems, encouraging expansion to open shells using a multireference formalism. We also introduce a perturbative framework to add systematic corrections to the EOM-IMSRG, showing results for closed shell nuclei and quantum dots. The second development is a generalized effective operator formalism for the IMSRG, capable of consistently evolving non-scalar operators relevant for electroweak transitions and moments. This general framework is applicable to both the EOM-IMSRG and the valence-space IMSRG approaches. We benchmark electromagnetic transition strengths and moments using both of these methods, also comparing with the quasi-exact no-core shell model and experiment when available. We demonstrate that consistent renormalization of observables is critical for precise computations with the IMSRG. We find that our methods perform well for transitions that are strongly single particle in nature, but for collective transitions involving many particles, we note that work remains to properly incorporate these effects in the IMSRG. “...I must earnestly entreat both the patience and attention of the reader: his patience in order to examine a detail which may perhaps in some places appear unnecessarily tedious; and his attention in order to understand what may, perhaps, after the fullest explication which I am capable of giving of it, appear still in some degree obscure. I am always willing to run some hazard of being tedious in order to be sure that I am perspicuous; and after taking the utmost pains that I can to be perspicuous, some obscurity may still appear to remain upon a subject in its own nature extremely abstracted.” - Adam Smith iv ACKNOWLEDGMENTS First and foremost, I would like to say thanks to my family. I want to thank my parents Scott and Mary, for teaching me how to behave like a well adjusted adult with integrity. Thank you also to my siblings Kristen and Kevin for being supportive and inspirational throughout my studies. I would be remiss if I didn’t also mention the numerous other family members who have inspired me to care about stuff like physics. If you’re related to me and you have bothered to read this, you’ve probably been a positive influence at some point. Most of all, I want to say thank you to Erica Yurk for being by my side for the whole thing, and never once agreeing with my constant insistence that I wasn’t going to graduate. Thank you. I would also like to thank my academic family. Most importantly my adviser Scott Bogner, who is the finest scientist I know. It has been a pleasure learning how to study physics under his guidance. Thanks also to my committee, Filomena Nunes, Morten Hjorth-Jensen, Piotr Piecuch and Alexandra Gade, who have all provided useful inputs and criticism during my journey. I’m also indebted to Heiko Hergert, Richard Cyburt, Gaute Hagen, Gustav Jansen, Jason Holt, Wayne Repko, Thomas Papenbrock, Carlo Barbieri, and numerous others who have willfully given their time so that I might not fail miserably. Titus Morris, my predecessor from our group, has been an inspiration to me throughout this process. He is a good friend, a great physicist, and was a worthy adversary in our beloved group meetings. Likewise, Justin Lietz, Sam Novario, Fei Yuan, and Sushant More have been excellent companions in the quests to solve the problems of nuclear physics, and also in the equally worthy quest to distract ourselves from those problems. I should not forget Ragnar Stroberg, who has taught me how to do physics correctly. Thank you. Finally I want to thank also the other friends that I’ve had along the way who have encouraged me and made life worth living. There are too many of them to name without forgetting a few and potentially offending them, but if you’re reading this, and haven’t seen your name yet, you should know that I added this paragraph just for you! v TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 12 14 20 22 24 26 CHAPTER 3 IN-MEDIUM SIMILARITY RENORMALIZATION GROUP 3.1 The Similarity Renormalization Group . . . . . . . . . . . . . . . . 3.1.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 A Practical Example . . . . . . . . . . . . . . . . . . . . . 3.1.3 SRG Softening of Interactions . . . . . . . . . . . . . . . . 3.2 The In-Medium Similarity Renormalization Group . . . . . . . . . 3.2.1 The In-Medium Similarity Renormalization Group Equations 3.2.2 Choice of Generator . . . . . . . . . . . . . . . . . . . . . 3.2.3 Solving the IMSRG Equations . . . . . . . . . . . . . . . . 3.2.4 Center-of-Mass Factorization in the IMSRG . . . . . . . . . 3.2.5 Results for Ground States of Closed Shell Nuclei . . . . . . 3.2.6 Perturbative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 30 30 32 34 35 36 37 40 48 51 56 CHAPTER 4 EXCITED STATE EXTENSIONS . . . . . . . . . . . . . . . . . . . . . 4.1 Sequential Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 TDA-IMSRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 vTDA-IMSRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Full Configuration Interaction Analysis of TDA-IMSRG(2) and vTDAIMSRG(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Equations-of-Motion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Improving the EOM-IMSRG(2,2) . . . . . . . . . . . . . . . . . . . . 4.2.2 Comparison of the EOM-IMSRG and vTDA-IMSRG via Quantum Dots 4.2.3 Results in Finite Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Perturbative Corrections for Nuclei . . . . . . . . . . . . . . . . . . . . 4.2.5 Extensions of EOM-IMSRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 65 65 66 . . . . . . . . . . . . . . 68 72 75 76 80 87 89 CHAPTER 2 THE NUCLEAR MANY-BODY PROBLEM 2.1 Nuclear Interactions . . . . . . . . . . . . . . . . . 2.2 Independent Particle Model . . . . . . . . . . . . . 2.3 The Many-Body Basis and Hamiltonian . . . . . . 2.4 Hartree-Fock Theory . . . . . . . . . . . . . . . . 2.5 The Configuration Interaction Method . . . . . . . 2.6 Many-Body Perturbation Theory . . . . . . . . . . 2.7 Coupled Cluster Theory . . . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 5 CONSISTENT COMPUTATION OF OBSERVABLES . . . . 5.1 Effective Operators in the IMSRG . . . . . . . . . . . . . . . . . . 5.1.1 Transition Operators . . . . . . . . . . . . . . . . . . . . . 5.1.2 General Observables in the IMSRG . . . . . . . . . . . . . 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Center-of-Mass Treatment . . . . . . . . . . . . . . . . . . 5.2.2 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 p shell Nuclei: Comparison to NCSM and Small-Space FCI 5.2.4 Electric Quadrupole Observables in sd and f p shell Systems 5.2.5 Magnetic Dipole Observables in sd and f p shell Systems . 5.2.6 Electric Octupole Transitions . . . . . . . . . . . . . . . . . 5.2.7 Comparing and Contrasting Methods . . . . . . . . . . . . . 5.2.8 The Effects of Consistent Operator Evolution . . . . . . . . CHAPTER 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 94 94 96 98 98 101 102 106 111 114 116 119 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 122 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 vii LIST OF TABLES Table 4.1 Values of h¯ ω˜ in MeV, used in center-of-mass Hamiltonian for Lawson calculations of 22 O energy spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Table 5.1 Ecm for intrinsic ground state and first 2+ state of 14 C, computed at emax =14 and h¯ ω = 20 MeV using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 with EOM-IMSRG(2,2). Values are given for calculations using Hin (β =0), and Hin + Hcm (β =1). . . . . . . . . . . . . . . 100 Table 5.2 Computed E2 transition strengths from first excited 2+ state to 0+ ground state for even-even nuclei, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experiment [1] and Weisskopf [2] single particle estimates are compared with IMSRG calculations. . . . . . . . . . 110 Table 5.3 Operator matrix elements computed for select nuclei with the EOM-IMSRG(2,2), using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Calculations are performed for bare operators and operators dressed by consistent IMSRG evolution. Values are expressed in e f m2 and µN for E2 and M1 operators respectively. . . . . . . . . . . . . . . . . . . . 120 viii LIST OF FIGURES Figure 1.1 Figure 2.1 Taken from [3]. The progress of ab initio nuclear structure from 2005 to 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chiral EFT forces up to N3 LO (ν = 4). Solid and dashed lines represent nucleons and pions respectively. Vertex order ∆ is 0 for small dots, 1 for filled circles, 2 for filled squares, and 4 for filled diamonds. Figure taken from [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 2.2 Schematic diagram of the Woods-Saxon potential. Discrete levels exist in the potential well, and a continuum of scattering states exist above it. Orbitals are filled up to the Fermi level, εF . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.3 Schematic depiction of the reference state (0p0h), single (1p1h), double (2p2h) and triple (3p3h) excitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Figure 3.1 SRG evolution of the two-body Hamiltonian in momentum space. In the left panel, we show solution of eq. 3.8 for select values of s. In the right panel we show the non-unitary decay of matrix elements when given a half-life corresponding to eq. 3.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Figure 3.2 Schematic representation of the initial and ground-state-decoupled Hamilto¯ ¯ nians, H(0) and H(∞), in the many-body Hilbert space spanned by particlehole excitations of the reference state. . . . . . . . . . . . . . . . . . . . . . . 38 Figure 3.3 Schematic zero-body flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Figure 3.4 Schematic resummation of MBPT diagrams. . . . . . . . . . . . . . . . . . . . 41 Figure 3.5 IMSRG ground-state-decoupling for a 6-electron quantum dot, with ω = 1.0 and four major oscillator shells. Hartree-Fock energy, second-, and thirdorder MBPT energies are plotted as function of flow parameter s. Exact energies for the bare and dressed Hamiltonians are given for reference. . . . . 42 Figure 3.6 Flowing reference energy E0 (s) for 16 O at with the EM input NN potential, N3LO(500) [5] at λSRG = 2.0 fm−1 , emax = 8 and h¯ ω = 24 MeV. Computed with Magnus and flow equation methods using forward Euler step. Results are benchmarked against flow equation solved by high-order adaptive step solver [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ix Figure 3.7 The top panels show the relative magnitude of each term in the Magnus expansion, eq. 3.37, computed for 16 O with the EM input NN potential, N3LO(500) [5] at emax = 8 and h¯ ω = 24 MeV. The potential is SRG evolved to (a) λSRG = 2.0 fm−1 , and (b) λSRG = 3.0 fm−1 . The bottom panels show similar plots for the BCH expansion of eq. 3.39 [6]. . . . . . . . . . . . . . . . 47 Figure 3.8 Center-of-mass diagnostics for IMSRG(2) calculations of 14 C starting from the NN+3N(400) interaction, SRG evolved to λSRG = 2.0 fm−1 . See the text for details. The calculations were done in an emax =14 model space. . . . . . . . 50 Figure 3.9 Convergence of 4 He, 16 O, and 40 Ca IMSRG(2) ground-state energies w.r.t. singleparticle basis size emax , for a chiral N3 LO NN interaction with λ = ∞ (left panels) and λ = 2.0 fm−1 (right panels). Notice the significant differences in the energy scales between the left and right panels. Gray dashed lines indicate energies from extrapolation the emax ≥ 10 data sets to infinite basis size (see text and Refs. [7, 8]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 3.10 Ground-state energies of stable closed-shell nuclei as a function of the resolution scale λ for the SRG-evolved NN + 3N-induced (left column) and NN + 3N-full Hamiltonians (right column), using different many-body methods: IMSRG(2) (●), CCSD ( ), Λ-CCSD(T) ( ), and IT-NCSM ( ). IMSRG and CC results are obtained with emax = 14, and the IT-NCSM results have been extrapolated to infinite model space, with error bars indicating the uncertainties of the importance truncation and the model space extrapolation (for 4 He, the error bars are smaller than the symbols). The gray solid line for 4 He in the top left panel is the result of a converged NCSM calculation with the bare NN Hamiltonian, E = −25.39 MeV (see e.g. [9]). Dashed black lines are experimental values from [10]. . . . . . . . . . . . . . . . . . . . . . 55 Figure 3.11 Connected Hugenholtz diagrams for the fourth-order energy correction E (4) (cf. [11]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 3.12 Induced two-body force with intermediate three-body force. Above the dashed blue line is the intermediate three-body force. The portion to the left of the dashed red line is an effective one-body insertion term. . . . . . . . . . . . . . 59 Figure 3.13 Convergence of IMSRG(2*) ground state energy for 4 He with the input EM N3 LO NN potential, SRG evolved to λSRG = 2.0 fm−1 . IMSRG ground state energies are computed to emax =9 for IMSRG(2), IMSRG(2*), and IMSRG({3}*). Results from CCSD and Λ-CCSD(T) are also shown. . . . . . . . 60 Figure 3.14 Convergence of IMSRG(2*) ground state energy for 16 O with the input EM N3 LO NN potential, SRG evolved to λSRG = 2.0 (3.0) fm−1 in the left (right) panel. IMSRG ground state energies are computed to emax =9 for IMSRG(2), IMSRG(2*), and IMSRG({3}*). . . . . . . . . . . . . . . . . . . . . . . . . . 61 x Figure 4.1 Schematic representation of the sequentially-decoupled Hamiltonians in the many-body Hilbert space spanned by particle-hole excitations of the reference state. The left panel corresponds to the use of Eq. 4.4 where the entire 1p1h sector is decoupled, and the right panel corresponds to Eq. 4.10 where just the valence 1v1h excitations are decoupled. The latter corresponds to the small block in the upper left corner of the full 1p1h-block. . . . . . . . . . . . 67 Figure 4.2 (ML , MS ) = (1, 0) excited states of a 6-electron quantum dot in an ω = 1.0 trap model space composed of four major oscillator shells. Three FCI calculations are shown; two using the flowing TDA-IMSRG(2) and vTDAIMSRG(2) Hamiltonians corresponding to each choice of the secondary decoupling Ux (s) (where x = 2 or 3), and one using the bare Hamiltonian. . . . . 70 Figure 4.3 FCI and TDA calculations of the first (ML , MS ) = (1, 0) excited state using the flowing vTDA-IMSRG(2) Hamiltonian for a 6-electron quantum dot with N = 3 and ω = 1.0. For reference, the flowing 0-body part of the Hamiltonian Ere f (s) and the FCI results for the bare Hamiltonian are also shown. . . . . . . 71 Figure 4.4 Selected excitation spectra of 6-electron quantum dots for ω = 1.0 (left) and ω = 0.5 (right) performed in an N = 3 single-particle basis. The quantum numbers of the various states are color-coded as (ML ,MS ) = (0,0), (1,0), (2,1), (3,0). The calculated spectra are displayed for five different many-body approaches, where the subscript indicates which Hamiltonian the respective method is applied to. For example FCI0 and FCI1 denote FCI calculations on the bare and ground-state-decoupled Hamiltonians respectively, TDA31 denotes a TDA calculation on the vTDA-decoupled Hamiltonian, etc. The lengths of the plotted energy levels indicate the 1p1h content of the state as defined in Eqs. 4.26-4.29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Figure 4.5 Absolute difference between quantum dot excitation energies calculated via IMSRG methods and those calculated with FCI on the bare Hamiltonian. Each point corresponds to an EOM or TDA energy level in Figure 4.4. The left panel is plotted against FCI 1p1h content, while the right is plotted against the EOM 1p1h amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . 79 Figure 4.6 Lowest 16 O excitation energies plotted for various quantum numbers, calculated with EOM-IMSRG(2,2) and vTDA-IMSRG(2) starting from the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 2.0 fm−1 . The single-particle basis is given by h¯ ω = 24.0 MeV and emax = 8. Above each plotted energy level from the EOM-IMSRG(2,2) calculation is the 1p1h partial norm of Eq. 4.28. . . . . . . . . . . . . . . . . . 81 xi Figure 4.7 Low lying states of 22 O at h¯ ω = 28.0 MeV and emax = 11 for several values of the Lawson parameter β , using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 3.0 fm−1 . The COM frequency h¯ ω˜ = 17.28 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Figure 4.8 Selected excitation spectra of 22 O at h¯ ω = 20.0 MeV and emax = 11 using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 2.0 fm−1 . The left frame shows the excitation energies calculated with the intrinsic Hamiltonian, and the right frame shows the result ˜ with β = 5.0. of adding a Lawson center-of-mass term H = Hint +β HCM (ω), Π Different colors indicate different J . . . . . . . . . . . . . . . . . . . . . . . 83 Figure 4.9 Selected excitation spectra of 22 O at h¯ ω = 28.0 MeV and emax =11 using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 3.0 fm−1 . The left frame shows the excitation energies calculated with the intrinsic Hamiltonian, and the right frame shows the result of adding a Lawson center-of-mass term H = Hint + β HCM , with β = 5.0. Different colors indicate different JΠ . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 4.10 Low lying spectra of 14 C and 22 O at h¯ ω = 20.0 MeV and emax =14 using the standard input NN+3N(400) interaction, softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. . . . . . . . 86 Figure 4.11 Triples correction for low lying spectra of 14 C and 22 O at h¯ ω = 20.0 MeV and emax =8 using the N3 LO (500 MeV) NN interaction of EM [5] with N2 LO 3N force (400 MeV) [13], softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. . . . . . . . . . . . . . . . . 87 Figure 4.12 Calculated (ML , MS ) = (0, 1) particle addition and removal energies for a 6electron quantum dot with ω = 1.0 state with EOM-IMSRG, EOM-CCSD and Hartree-Fock theory. Results are plotted as a function of oscillator shells used for the single-particle basis. . . . . . . . . . . . . . . . . . . . . . . . . . 90 Figure 4.13 Low lying spectra of 14 N at h¯ ω = 20.0 MeV and emax =14 using the standard NN+3N(400) softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. Experimental spectra beyond the 1− 2 state are omitted for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Figure 5.1 Observables calculated at several values of the Lawson scaling parameter β , for 14 C, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 , at emax = 14. . . . . . . . . . . . . . . . . . . . . . 99 xii Figure 5.2 Ground state properties of the deuteron calculated with a full diagonalization (labeled FCI), compared to the same properties calculated in the 0s valence space using operators transformed with the IMSRG. The line labeled “exact” is obtained by diagonalizing in the Jacobi basis with emax = 200. Here we use the EM input N3 LO nucleon-nucleon interaction SRG evolved to λSRG = 2.0 fm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure 5.3 14 Convergence of the 2+ 1 excitation energy and B(E2) to ground state of C with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Figure 5.4 14 Convergence of the 1+ 1 excitation energy and B(M1) to ground state of C with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Figure 5.5 Convergence of 0+ 1 excitation energy, B(M1) to ground state, and magnetic dipole moment of 14 N with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. . . . . . . . . . . . . . . . . . 105 Figure 5.6 22 Convergence of the 2+ 1 excitation energy and B(E2) to ground state of O with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 5.7 Convergence of the 2+ 1 excitation energy, corresponding B(E2) to ground state and quadrupole moment of 32 S with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. . . . . . . . . . . 108 Figure 5.8 First excited 2+ states and corresponding B(E2) transition strengths for 48 Ca and 56,60 Ni, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Convergence patterns are shown, computed with EOM-IMSRG(2,2). Results are compared with experimental points in the left panel [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure 5.9 22 Convergence of the 1+ 1 excitation energy, B(M1) to ground state of O with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . . . . . . . . . . . . . . . 111 xiii + Figure 5.10 Convergence of 1+ 1 excitation energy, B(M1) to ground state and 21 magnetic dipole moment of 32 S with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. . . . . . . . . . . . . . . . 112 Figure 5.11 Computed ground state energy and magnetic dipole moment of 32 Cl with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Figure 5.12 Excitation energies of the first 3− state of 16 O and 40 Ca, along with corresponding B(E3) strength for transition to ground state. These values are computed using the EOM-IMSRG(2,2) with N2 LOsat and the standard NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 .. Results are compared with experiment [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 32 Figure 5.13 Computed ground-state and 2+ 1 energies for S, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Here we use VS-IMSRG with a variety of truncations on particles allowed into the 0d3/2 shell. Results are compared with EOM-IMSRG values. . . . . . . . . . . 117 Figure A.1 Diagrammatic representation of Clebsch-Gordan coefficients . . . . . . . . . . 132 Figure A.2 Symmetries of Jucys diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Figure A.3 Jucys contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Figure A.4 Jucys factorization rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Figure A.5 Jucys diagrams for 6 j- and 9 j-symbols. . . . . . . . . . . . . . . . . . . . . . 134 Figure A.6 Jucys diagram for a two-body operator. . . . . . . . . . . . . . . . . . . . . . . 136 Figure A.7 Diagrammatic derivation of the Pandya transform. . . . . . . . . . . . . . . . . 136 Figure A.8 Expression of the first term of eq. 81 with Jucys Diagrams. . . . . . . . . . . . 141 Figure A.9 Contraction of all Jucys diagrams in eq. 82. . . . . . . . . . . . . . . . . . . . 141 Figure A.10 Derivation of Z λpq (22 → 1) with Jucys diagrams. . . . . . . . . . . . . . . . . . 142 (J J )λ 1 2 (11 → 2) using Jucys diagrams. . . . . . . . . . . . . . . 144 Figure A.11 Derivation of Z pqrs (J J )λ 1 2 (21 → 1) using Jucys diagrams. . . . . . . . . . . . . . . 145 Figure A.12 Derivation of Z pqrs xiv (J J )λ 1 2 (22 → 2(pp, hh)) using Jucys diagrams. . . . . . . . . . 147 Figure A.13 Derivation of Z pqrs Figure A.14 Derivation of induced three-body force with Jucys diagrams. . . . . . . . . . . 155 xv KEY TO ABBREVIATIONS CC – coupled cluster CCSD – coupled cluster with singles and doubles CI – configuration interaction COM – center-of-mass EFT – effective field theory EOM – equations-of-motion EOM-CC – equations-of-motion coupled cluster EOM-IMSRG – equations-of-motion in-medium similarity renormalization group EM – Entem and Machleidt EN – Epstein-Nesbet FCI – full configuration interaction HF – Hartree-Fock HO – harmonic oscillator IMSRG – in-medium similarity renormalization group IPM – independent particle model IT-NCSM – importance truncated no-core shell model LECs – low-energy constants MBPT – many-body perturbation theory MP – Møller-Plesset MR – multireference NCSM – no-core shell model NN – nucleon-nucleon NO2B – normal-ordered two-body N2 LO – next-to-next-to leading order N3 LO – next-to-next-to-next-to leading order xvi PA – particle attached PR – particle removed QMC – quantum Monte Carlo QCD – quantum chromodynamics RG – renormalization group RPA – random phase approximation SCGF – self-consistent Green’s function SRG – similarity renormalization group TDA – Tamm-Dancoff approximation VS – valence-space VS – valence-space in-medium similarity renormalization group WIMPs – weakly interacting massive particles WS – Woods-Saxon 3N – three-nucleon xvii CHAPTER 1 INTRODUCTION It has been apparent for some time that an important forefront of nuclear theory lies in ab initio methods. These are approaches which seek to describe nuclear properties starting from first principles. Over the past fifteen years, we have seen an explosion of applicability for this brand of theory, growing from the lightest nuclei well into the medium-mass region, with contemporary investigations reaching as far as the tin isotopes. It was not too long ago that the only reliable ab initio approaches for nuclear physics were quasi-exact methods such as the no-core shell model (NCSM) [16–18] or quantum Monte Carlo (QMC) [19–21]. At the time, approximate methods were not effectual because nucleon-nucleon potentials designed to reproduce elastic scattering phase shifts often exhibited strong short-range repulsion and tensor forces, which result in significant coupling to high-momentum modes, producing strongly correlated wave functions incapable of a perturbative description. While quasi-exact methods exhibited extraordinary success for nuclei existing in the lightest regions of the p-shell, the idea of one day computing a nucleus such as 100 Sn, or even a more modest system such as 56 Ni, was a veritable pipe dream. The brute-force nature of these quasi-exact methods results in exponential or factorial scaling with system size, and hence even with an optimistic view of Moore’s law, the medium-mass region seemed completely inaccessible using only these approaches. Meanwhile, the field of quantum chemistry enjoyed a much more bountiful array of ab initio methods. Along with the quasi-exact methods, chemists had employed approximate, but systematically improvable methods such as many-body perturbation theory (MBPT) and coupled-cluster (CC) theory [11, 22–28]. These methods employ various many-body truncations, where dynamic correlations beyond a certain level are omitted for reasons of computational feasibility. These truncations offer much more favorable polynomial scaling with system size at the expense of some precision. While these approaches were known in nuclear physics [29–31], their many-body truncations were ultimately their undoing in that field, as the highly nonperturbative interactions of 1 Figure 1.1 Taken from [3]. The progress of ab initio nuclear structure from 2005 to 2016. the time resulted in crippling many-body truncation errors. In contrast, descriptions of the electronic systems of interest in quantum chemistry can typically treat inter-particle interactions as a perturbation on top of the dominant mean-field. Owing to the difficulties associated with the interactions, the earliest attempts at ab initio nuclear theory primarily served as a means to draw loose connections between the more quantitative phenomenological models and the underlying inter-nucleon forces [32, 33]. Phenomenology such 2 as the empirical shell-model [34] and Skyrme energy density-functionals [35,36] have long offered a computationally elegant and numerically precise means to explain nuclear properties in regions of the nuclear landscape where experimental data can be used to constrain the models, but as the sights of experimenters turn to the exotic reaches of the nuclear chart, the utility of phenomenology is reduced drastically. These shortcomings stem from uncontrolled extrapolations beyond regions of known data; here artifacts of the fitting procedure are amplified, resulting in strongly model dependent predictions. Furthermore, phenomenological predictions for observables other than energies are a subtle matter, as there is no obvious way to consistently renormalize the operators with the fitted effective interactions of these methods. A path forward was revealed by two revolutionary developments, which brought forth a renaissance in ab initio nuclear theory. The first was the arrival of chiral effective field theory (EFT) [37, 38], where two- and three-nucleon interactions (and consistent electroweak current operators) are constructed systematically via a consistent power-counting scheme which connects with the symmetries of quantum chromodynamics (QCD). The second breakthrough came with the increased popularity of renormalization group (RG) methods [39, 40] in the field, which have brought about a unitary softening mechanism for the nuclear interactions. With chiral EFT providing a consistently constructed starting point for both the inter-nucleon interactions and observables, RG methods act as a resolution dial for nuclear interactions. Here we can tune the strength of shortrange correlations to construct a more perturbative interaction, where rudimentary approaches such as Hartree-Fock (HF) theory can then provide a reasonable starting point for approximate manybody methods. With these developments on the interaction front, the quasi-exact NCSM has been pushed to new frontiers, reaching as far as the oxygen chain [40]. In regards to the proliferation of ab initio methods, the most meaningful consequence of these developments was the reintroduction of the polynomial scaling methods which had taken refuge in quantum chemistry. Techniques such as CC theory have re-emerged in nuclear physics, along with a plethora of new approaches, such as the in-medium similarity renormalization group (IMSRG) and the self-consistent Green’s 3 function (SCGF) approach [41–60]. These methods, armed with the power of chiral EFT and RG softening, have pushed the boundaries of predictive ab initio nuclear structure to the entirety of the sd and p f -shells, and further beyond to the tin chain, as demonstrated in fig. 1.1. In this endeavor, significant progress has been made in including the effects of three-nucleon forces [60–64] and extending calculations to open-shell nuclei [44–46, 49, 51–53, 65, 66] through the clever use of various normal-ordering schemes and also through marriages of ab initio machinery with effective interaction ideas ported from phenomenology. Ab initio efforts of the past decade have largely targeted energies and scalar observables such as nuclear radii. In recent years however, there has been a significant push to develop a more general effective operator formalism for these approximate ab initio methods, which includes tensor observables such as transitions and moments. A particularly crucial frontier is that of neutrinoless double β -decay, where a consistent and reliable prediction of the nuclear matrix element [67] for the process would be invaluable to experimental efforts and analyses in the event of a confirmed observation. There are many hurdles to be surpassed before the ab initio community can contribute confidently on this front; to start, the development of a reliable theory for electromagnetic transitions between excited states in heavier open-shell nuclei would be a needed step forward. While there have certainly been efforts to explain excited states and strength distributions in mediummass nuclei [68–70], a versatile, lightweight framework for such explorations has not yet been formalized. In the IMSRG, extensions to excited states and observables come quite naturally, as the method offers a malleable and transparent Hermitian framework to renormalize various manybody correlations within the nuclear Hamiltonian. Of late, much of the IMSRG’s progress on this front has been through construction of effective valence-space interactions for shell-model calculations [46, 51, 52]. In the valence-space IMSRG (VS-IMSRG), ground states and excited states are treated on the same footing and observables such as transition strengths follow from existing shell-model machinery. This description is restricted however by those limitations which are endemic to any diagonalization method, where larger nuclei or multi-shell excitations will eventually be confronted with the factorial scaling of the method. 4 An alternative approach is offered by the equations-of-motion (EOM) formalism [71]. EOM methods are systematically improvable approximate diagonalization methods. Much like the shell model, ab initio EOM is greatly improved when used jointly with the IMSRG, which softens couplings between various excitations in the Hamiltonian, and thus facilitates approximate diagonalizations. The equations-of-motion IMSRG (EOM-IMSRG) [72] utilizes Hamiltonians which have been softened by IMSRG decoupling, where one may use less sophisticated transformations than those needed for the VS-IMSRG, as the approximate diagonalization in the EOM calculation allows one to keep all particles active in the full model space. As such, the two methods are excellent complements; the VS-IMSRG shell-model diagonalization allows for more detailed descriptions where valence-space collectivity is important, such as in deformed nuclei, and the EOM-IMSRG allows for less restricted explorations of states where low-order excitations dominate. Both methods are growing in the scope of their applicability, as valence space decoupling techniques continue to be perfected, and systematic improvements are built into the EOM-IMSRG in both the manner of improved approximate diagonalization and decoupling schemes. The EOM-IMSRG is an analog to the well established equations-of-motion CC (EOM-CC) theory [68, 69, 73–76], and thus we expect similar behavior from these methods. The promise of the EOM-IMSRG lies with the impending inclusion of generalized normal-ordering [77], which has been used to develop the multireference IMSRG (MR-IMSRG) [45, 47]. As of now, the MRIMSRG has shown great potential as a method to obtain open-shell ground state observables for even-even nuclei. Extension of this method to the MR-EOM-IMSRG will enable the exploration of spectra in open-shells without any of the inherent limitations of the shell model method. Additionally, the MR-EOM-IMSRG can be extended to odd-even and odd-odd nuclei with a generalized EOM formalism. The application of generalized normal-ordering is a non-trivial task, so we have first demonstrated the principle and produced some novel results in a single-reference formulation, which is limited to nuclei near closed shells. In this work, we introduce and explore the strengths and limitations of the EOM-IMSRG as a method for obtaining excited states, and introduce a general effective operator formalism for the IMSRG, which is explored and benchmarked in both the 5 VS- and EOM-IMSRG for numerous light and medium-mass nuclei. The rest of this dissertation is organized as follows. In chapter 2, we introduce the nuclear many-body problem, and much of the notation and terminology used in this work. In chapter 3, we formalize the IMSRG, introducing normal-ordering, the IMSRG equations, and the resummation of perturbation theory through IMSRG decoupling. In chapter 4, we discuss some of the major new developments brought about by this work, namely the direct computation of excited states methods in the EOM paradigm, first from the perspective of decoupling the 1p1h sector exactly, and second as a general formulation of the EOM-IMSRG. In chapter 5, we introduce a generalized effectiveoperator formalism for IMSRG, which was developed throughout the course of this work. This technical advance is applicable for all variants of the IMSRG, so we make comparisons between the VS-IMSRG and EOM-IMSRG. Here we seek to understand the strengths and weaknesses of the complementary methods, which are also benchmarked with NCSM and experiment where available. Finally we present conclusions in chapter 6. The technical advances which brought about the results in chapters 4 and 5 are showcased in the appendix. 6 CHAPTER 2 THE NUCLEAR MANY-BODY PROBLEM Nuclei are self-bound collections of protons and neutrons, held together by complicated interparticle interactions. If nucleons are taken to be the fundamental degrees of freedom for a nuclear structure problem, it is then necessary to take into consideration two-, three-, and up to A-body forces to account for the composite nature of the proton and neutron. Of course, such considerations are impractical for computation, so we must start from a heavily restricted approximation, and improve systematically from there. In this chapter we explore the basic facets of the nuclear many-body problem. First we introduce inter-nucleon interactions and briefly discuss chiral effective field theory, which is used to produce the input two- and three-nucleon interactions used in this work. We review the independent particle model, and from there motivate the nuclear many-body problem. We introduce HartreeFock theory as a means of generating a suitable reference state, and then summarize the formalism of the popular configuration interaction method, many-body perturbation theory, and coupled cluster theory. These methods serve both as motivations and benchmarks for the in-medium similarity renormalization group, which is the focus of this thesis. 2.1 Nuclear Interactions Perhaps the most daunting challenge for nuclear theory is to construct an accurate description of the inter-nucleon interaction. There are two classes of interactions used for the nuclear many-body problem; phenomenological interactions that are tuned to specific many-body methods and/or nuclei in a given mass range, and “realistic” interactions which accurately describe two- and threenucleon systems in free-space, and can be used to describe any nucleus within any many-body framework. Phenomenological interactions are designed to perform well for a restricted set of nuclei and/or observables, and are fit to data within that region of the nuclear chart. Most prominent 7 among these phenomenological interactions is the USDB interaction [78], which has performed excellently in the reproduction of nuclear observables in the sd-shell [34], and the Skyrme force, which in modern parlance is a phenomenological parameterization of the nuclear energy density functional [35, 36]. Beyond the specific region targeted by phenomenological interactions, their predictive power becomes extremely limited, often exhibiting model dependent extrapolation properties. Moreover, if one wishes to explore observables other than energies, such as electromagnetic moments or strengths, a phenomenological renormalization procedure for the relevant operators is often ad hoc and uncontrolled. For this reason, ab initio approaches, where one starts from realistic two- and three-nucleon interactions in free-space and then solves the many-body Schrödinger equation as accurately as possible, have attracted increased attention in recent years. Of course, the ab initio label raises a picture of interactions derived analytically from QCD, starting with quarks and gluons, and deriving from there the meson-exchange processes which govern the nuclear interactions. This is not possible at present, as the QCD Lagrangian is nonperturbative at low energies relevant to nuclear structure, in stark contrast to the lovely perturbative properties which have brought great success to quantum electrodynamics. While lattice QCD calculations offer a step forward, immense computational demands limit these explorations to 0s shell nuclei such as the deuteron and triton, where the pion mass is artificially large [79, 80]. A more practical starting point is from so-called “realistic” interactions, where all nucleons are treated as active, and the forces are fit to very light-mass nuclear data, such as nucleon-nucleon (NN) phase shifts, and binding energies from the triton and 4 He. Some early examples of these interactions are high-precision potentials [81–83] which use meson-exchange phenomenology to fit to elastic channel NN scattering. While these potentials have been extraordinarily successful in the description of light nuclei, they struggle as we enter the medium-mass region. Moreover, ab initio methods which use many-body truncations to access this region are subject to large errors resulting from the hard repulsive core featured in several of these interactions. These interactions also lack a formal consistency between two- and three-nucleon interactions, and offer no path to build consistent current operators for coupling to to external electroweak probes. 8 2N Force 3N Force 4N Force LO (Q/Λχ)0 NLO (Q/Λχ)2 NNLO (Q/Λχ)3 +... N3LO (Q/Λχ)4 +... +... +... Figure 2.1 Chiral EFT forces up to N3 LO (ν = 4). Solid and dashed lines represent nucleons and pions respectively. Vertex order ∆ is 0 for small dots, 1 for filled circles, 2 for filled squares, and 4 for filled diamonds. Figure taken from [4]. An ab initio paradigm which has gained significant traction in the past decade is to use chiral effective field theory (EFT) to construct the inter-nucleon interactions systematically starting from an effective Lagrangian which is modeled after QCD. Chiral EFT preserves the symmetries of QCD, in particular chiral symmetry breaking [4, 84]. Chiral symmetry is exhibited by massless particles, where right-handed helicity states are completely decoupled from left-handed states. In the two-flavor QCD Lagrangian, the constituent up and down quarks are nearly massless in comparison to the hadronic masses at around 1 GeV/c2 (mu = (2.5 ± 0.8) MeV/c2 , md = (5.0 ± 0.9) MeV/c2 [85, 86]). Therefore, the QCD Lagrangian exhibits approximate chiral symmetry, which is spontaneously broken in the axial-vector subgroup of the SU(2)L × SU(2)R chiral symmetry 9 group (see e.g. [38,87]). This spontaneous chiral symmetry breaking implies the existence of three massless Nambu-Goldsone bosons. In reality, the slight explicit breaking of chiral symmetry due to the non-zero quark masses results in massive, but unnaturally light psuedoscalar mesons in the place of the Nambu-Goldstone bosons, namely pions. The existence of pions implies a separation of scales in the hadronic degrees of freedom, as the pion mass mπ ≈ 140 MeV/c2 is significantly smaller than the mass of other vector mesons [86]. This separation is exploited to formulate chiral EFT. In contrast to the quarks and gluons of QCD, chiral EFT employs nucleons and pions as the fundamental degrees of freedom, LχEFT = Lππ + LπN + LNN + · · · . (2.1) Pions, being of significantly lighter mass than the other mesons exchanged in nuclear interactions, account for long-range interactions and are the only meson fields included in the Lagrangian. Short-range physics, the result of heavier meson exchange, is encoded in contact interactions with low energy constants (LECs) which must be fit to light-mass nuclear data. Treating short range physics in low-resolution such as this means that the theory is only valid for long-range physics whose characteristic energy scale Q is on the order of the pion mass. As such, the theory must be regulated by a cut-off at some momentum Λ, which suppresses high momenta on the order of the heavier meson masses. The EFT Lagrangian constitutes the most general expression which conserves the symmetries of QCD, and hence there are an infinite number of terms in the expansion, and with each term comes a myriad of LECs. Fortunately, the vertices may be organized by the number of terms proportional to the soft scale Q. This breaks down to the number of pion mass insertions or derivatives, d and nucleon fields n, ∆ = d + n/2 − 2 , (2.2) and the problem may then be treated in chiral perturbation theory, where interaction terms are grouped in powers of (Q/Λ), which is ostensibly small. Thus a chiral EFT is specified by the order 10 ν of (Q/Λ) kept in the perturbative expansion, as well as the fitting procedure used to constrain the forces. For a given chiral order ν, there are a finite number of diagrams which may contribute to the nuclear forces. As a result, the chiral forces can in principle be calculated to any order. Fig. 2.1 illustrates the chiral power counting for the EFT up to ν = 4 or next-to-next-to-next-to leading order (N3 LO). Here you can see the number of nucleons A, loops L, and the vertex orders ∆i factor into the chiral order of a given diagram, ν = −4 + 2A + 2L + ∑ ∆i . (2.3) i This figure also demonstrates the fact that many-body forces appear naturally as you increase the chiral order, with 3N forces appearing at N2 LO, 4N at N3 LO, and so on. Thus, we expect that for sufficiently small (Q/Λ), the importance of many-body forces decreases significantly with many-body operator rank. This is an important feature in ab initio calculations, as the diminished importance of these higher-body forces means that accurate computations are indeed feasible. The most popular interaction in the literature is the NN interaction of Entem and Machleidt, computed to N3 LO [5], with a cutoff at Λ=500 MeV. Often coupled with this is the 3N force of Navrátil, which is computed to N2 LO at Λ=400 MeV [13]. This is of course inconsistent with the order and cutoff of the NN force, but 3N forces at N3 LO have not been computed until very recently [88], and it has been demonstrated that using a 500 MeV cutoff for the 3N interaction leads to large induced 4N forces upon SRG softening [63]. The so called NN+3N(400) force has been quite successful as a means to describe ground state energies up to the light constituents of the sd-shell. A notable shortcoming is the interaction’s inability to produce ground states with proper radii. Radii are under predicted and significant overbinding is exhibited in heavier nuclei, but the effect seems to be filtered out by observables such as separation and excitation energies [55,89,90]. There have been numerous works in recent years to improve chiral EFT. Two main strategies have developed: a brute force approach to increase the chiral order [88, 91], and a re-evaluation of the fitting procedures at lower orders [92–94]. For this work we use almost exclusively the NN force of Entem and Machleidt, often including 11 the 3N force of Navrátil. We choose this interaction because of its preponderance in the literature. We refer to it as the standard NN+3N(400) force. 2.2 Independent Particle Model The most elementary picture of nuclear structure can be constructed by neglecting the inter-nucleon forces, and instead treating the problem as a collection of independent particles subject to some empirical mean-field; this reduces the original problem of solving the many-body Schrödinger equation to that of a single particle moving in a potential well, a problem that is familiar from introductory quantum mechanics. An approximation for the ground state of the A-body nucleus can be made simply by filling the lowest A orbitals, where the highest single-particle energy corresponds to the Fermi energy εF . We then construct an antisymmetric product state in accordance with the Pauli exclusion principle. The resulting theory of nuclear structure is often referred to as the independent particle model (IPM). A standard choice for the mean-field potential is the Woods-Saxon (WS) potential [95], shown schematically in fig. 2.2 and given by r − R0 −1 ) , V (r) = −V0 1 + exp( a (2.4) where V0 is the depth of the potential well, R0 is the empirical nuclear radius, and a controls the gradient of the potential near r = R0 . These parameters are chosen to match observed properties of nuclei. In particular, the approximately uniform density (saturation density) which occurs within the interior of the nucleus is ensured by the flat region of the WS potential, and the sharp gradient in the potential near R0 guarantees a strong binding force at the surface. The potential quickly goes to zero after R0 , which accounts for the finite range of the nuclear force. As the WS potential vanishes at infinity, the WS basis contains both discrete bound states in the potential well and scattering states within the continuum. 12 Energy continuum 0 bound states ƐF -V0 R0 Figure 2.2 Schematic diagram of the Woods-Saxon potential. Discrete levels exist in the potential well, and a continuum of scattering states exist above it. Orbitals are filled up to the Fermi level, εF . The constructed antisymmetric product state is called a Slater determinant, φ1 (r1 ) φ1 (r2 ) · · · φ1 (rA ) 1 φ2 (r1 ) φ2 (r2 ) · · · φ2 (rA ) Φ0 (r1 , . . . , rA ) = √ , .. .. .. .. A! . . . . (2.5) φA (r1 ) φA (r2 ) · · · φA (rA ) where φi are the WS orbitals. In the IPM, this is our approximation to the ground state wave function. Despite the IPM’s extreme simplicity, it gives a remarkable account of the gross structure of nuclei; the method is able to produce shell structure and magic numbers, as well as nuclear properties such as ground state spins and magnetic moments in the vicinity of shell closures. Conversely, open-shell nuclei have strongly correlated ground states due to the presence of quasi-degenerate orbitals near εF , and hence a single Slater determinant fails to describe them even qualitatively. Moreover, the method is not systematically improvable without the inclusion of correlations. 13 When we do add correlations by restoring the inter-nucleon interactions and solving the manybody Schrödinger equation, the IPM offers a convenient starting point for nuclei near shell closures, from which we may build in dynamic correlations in the form of excitations brought about by the interactions. To compute solutions to the nuclear many-body problem, we must construct a manybody basis from the single-particle basis. Such a task may be accomplished by enumerating all possible A-body Slater determinants allowed within the chosen single-particle basis. The groundstate approximation from the mean-field picture is now referred to as the “reference state”, and all other Slater determinants are particle-hole excitations on top of this reference. While there is a continued effort to solve the nuclear many-body problem using idealized single-particle bases where bound and scattering states are treated on equal footing [96–99], most many-body methods at present focus primarily on bound states using the harmonic oscillator (HO) potential 1 V (r) = mω 2 r2 2 (2.6) as a starting point. The HO potential, despite allowing only bound states, exhibits desirable properties owing to a clean factorization of the center-of-mass and relative coordinates. This allows for convenient transformations between momentum and coordinate space, as well as access to intrinsic properties without spuriosities from the violation of translational invariance necessarily suffered when using a spherical basis. Despite some striking differences from the WS potential, the IPM with the HO basis is nonetheless able to recover fundamental properties of nuclei such as shell structure arising from its evenly spaced degenerate orbitals. 2.3 The Many-Body Basis and Hamiltonian To include the dynamic correlations that are neglected by the independent particle model, we now formulate the interacting nuclear many-body problem in second quantization (see [100, 101]). The nuclear many-body problem boils down to solving the stationary Schrödinger equation, H|Ψν = Eν |Ψν , 14 (2.7) where Ψν and Eν are the wave function and energy for any stationary state of the system. The intrinsic A-body nuclear Hamiltonian H consists of a kinetic energy term, as well as two-, threeand up to A-nucleon forces, H = T 1b + A 1b − T 2b . ∑ V nb − Tcm cm (2.8) n=2 Here we have subtracted contributions from the COM kinetic energy, Tcm = A p2 1 A pi · p j P2 1b + T 2b , =∑ i + ∑ = Tcm cm 2mA i=1 2mA 2 i= j mA (2.9) which account for translation of the entire nucleus. In practice, finite computational power limits us to at most three-nucleon forces, so state-of-the-art many-body calculations use second-quantized Hamiltonians of the form 1 1 p ·p H = ∑(1 − ) p|T |q a†p aq + ∑ pq|V (2) − 1 2 |rs a†p a†q as ar A 4 pqrs mA pq + 1 pqr|V (3) |stu a†p a†q a†r au at as . ∑ 36 pqrstu (2.10) Here, we use antisymmetrized two- and three-body bras and kets. We will often use short-hand such as 1 Tpq ≡ (1 − ) p|T |q , A p ·p (2) Vpqrs ≡ pq|V (2) − 1 2 |rs , mA (3) Vpqrstu ≡ pqr|V (3) |stu , (2.11) (2.12) (2.13) for the antisymmetrized operators, where the subtraction of the center-of-mass kinetic energy has been absorbed into the interaction terms. a† and a are fermion creation and annihilation operators, which are used to construct the many-body basis of antisymmetrized A-body product states, or Slater determinants: A |Φ0 = { ∏ a†i }|0 . (2.14) i=1 Here each a†i corresponds to the addition of the ith single-particle orbital to the antisymmetrized product. |Φ0 is the reference state, consisting of the A single-particle orbits which comprise 15 the Fermi sea. The Pauli exclusion principle is encoded here by the fermion anti-commutation relations, [a†i , a j ]+ = δi j , [a†i , a†j ]+ = [ai , a j ]+ = 0 . (2.15) A complete many-body Hilbert-space can be constructed via the application of creation and annihilation operators, |Φai = a†a ai |Φ0 , † † |Φab i j = aa ab a j ai |Φ0 , etc . . . (2.16) Here we note that fermion creation and annihilation operators cannot create an already existing particle, or annihilate one which is already absent. Thus, when these cases arise, the creation and annihilation operators annihilate the state, which can be shown using eq. 2.15. In the definition 2.16, we see a single excitation and a double excitation from the reference state. The ellipses indicates that in order to construct the complete basis, we must include triples, quadruples and up to A-body excitations, corresponding to A creators and A annihilators acting on the reference state. To simplify the discussion of the many-body basis, we develop the particle-hole nomenclature, where particles are created above the Fermi level, and holes are destroyed below it. In this work (except where otherwise noted), the indices a, b, c, . . . will refer to particles and i, j, k, . . . to holes. Unspecified orbitals will be referred to as p, q, r, . . . In this terminology, the reference state corresponds to a Slater-determinant filled with all hole states. A single excitation corresponds to a particle-hole excitation (1p1h) from the reference, |Φai = a†a ai |Φ0 , (2.17) doubles to 2p2h, triples to 3p3h, and so on to ApAh excitations. This is shown schematically in fig. 2.3 As the reference state and all possible excitations constitute a complete basis, the eigenstates of eq. 2.7 can be written as a linear combination of these Slater determinants, A |Ψν = C0 |Φ0 + 1 a ,...,a ∑ Ci11,...,iNN a†a1 · · · a†aN aiN · · · ai1 |Φ0 . N! i1 ,...,iN N=1 ∑ a1 ,...,aN 16 (2.18) F 0p0h 1p1h 2p2h 3p3h Figure 2.3 Schematic depiction of the reference state (0p0h), single (1p1h), double (2p2h) and triple (3p3h) excitations. The goal of the nuclear structure theorist is then to solve for the coefficients Cα which describe the state of interest. The single particle basis used to construct the many-body basis is specified by a truncation on the single-particle energies in the HO basis, emax . It is not practical to solve the problem in mscheme, where each orbital has good spin projection quantum number m, as the necessary model spaces for converged calculations (emax > 10) would possess over 1000 orbitals, corresponding to billions of antisymmetrized two-particle states, and an astronomical number of three-particle states. We instead express the problem in J-scheme, exploiting the spherical symmetry of the Hamiltonian. For spherical nuclei, it is sufficient to only consider scalar matrix elements. To transition to J-scheme, all dependence on the spin projection m is summed out using the WignerEckart theorem (see appendix for details). In J-scheme, the number of orbitals is reduced by roughly an order of magnitude. Even with this simplification, many-body problems involving hundreds of orbitals are far from trivial. Thus the matrix must be stored in such a way as to exploit every symmetry available, namely total angular momentum J, isospin projection Tz , and parity Π. Additionally, we note that observables must be expressed as Hermitian operators, reducing the size of the matrix by roughly a factor of two. Even after these symmetries are exploited, it is often found that the number of unique non-zero three-body matrix elements requires many terabytes of storage even for light sd-shell nuclei. Thus an additional truncation E3max is introduced, where E3 = e1 + e2 + e3 . 17 (2.19) Here the summed single particle energies for a three-body basis state are restricted to E3max . Typical values are E3max = 12, 14, 16, which limit the size of the three-body interaction to 1-100 GB. Ground state energies are dependent on this truncation for low values of the HO single-particle basis frequency h¯ ω, but for sd-shell nuclei, E3max = 14 is sufficient after h¯ ω = 20 MeV. The difference between E3max = 12 and 14 is typically on the order of 1 − 2% for the optimal h¯ ω [102]. Solution of the many-body problem often requires the storage of several copies of the interaction, and therefore it is rarely practical to include the three-body force, even after E3max truncation. However, partial incorporation of the three-body force at the two-body level can be achieved through normal-ordering. Nuclear many-body operators are expressed in-medium through use of normal-ordering with Wick’s theorem. A normal ordered operator string is a string of N creation and annihilation operators (denoted generically by Ai ) which are ordered in such a way that the reference state expectation value is zero, Φ0 |{A1 A2 · · · AN }|Φ0 = 0 . (2.20) While this ordering may be achieved using the standard fermion anti-commutation relations, a more useful approach is through the use of Wick contractions a†i a j = a†i a j − a†i a j , (2.21) which are zero by construction for operator pairs which are already normal-ordered, and which correspond to the anti-commutator for pairs which are not. This is related to the convenient fact that all creation and annihilation operators within a normal-ordered string anti-commute, so exchanges only require a change of sign. For two strings of normal-ordered operators of length 2n and 2m, Wick’s theorem (see, e.g., [11]) relates their product to a linear combination of normal-ordered operator strings: a†i . . . a†in a jn . . . a j1 1 a†k . . . a†k alm . . . al = 1 m 1 (−1)m·n a†i . . . a†in a†k . . . a†k a jn . . . a j1 alm . . . al 1 m 1 1 18 (2.22) + (−1)m·n a†i al a†i . . . a†k a jn . . . al + singles 2 m 2 1 1 + (−1)m·n a†i al a†i al a†i . . . a†k a jn . . . al + doubles + triples + · · · . 3 m 3 1 1 2 2 (2.23) Here singles, doubles, and triples refer respectively to all possible single contractions, pairs and triplets of operator contractions in the two strings. The expansion continues until all terms are contracted in every way possible. The phase of each term is determined by the number of exchanges needed to bring the total operator string into the form shown, including the process of bringing contracted operators adjacent to one another. Note that we do not show contractions between two creation or two annihilation operators, as these anti-commute with one another, and thus their contractions vanish. Normal ordering has the effect of mapping the second quantized Hamiltonian, H = ∑ Tpq a†p aq + pq 1 1 (2) (3) Vpqrs a†p a†q as ar + ∑ ∑ Vpqrstua†pa†qa†r auat as , 4 pqrs 36 pqrstu (2.24) to a form where vertices contain density dependent elements of higher-body forces: H = E0 + ∑ f pq {a†p aq } + pq 1 1 Γ pqrs {a†p a†q ar as } + ∑ ∑ Wpqrstu{a†pa†qa†r auat as} . (2.25) 4 pqrs 36 pqrstu Here, “in-medium” vertices are determined by Wick’s theorem: E0 = ∑ Tii ni + i 1 (3) 1 (2) Vi ji j ni n j + Vi jki jk ni n j nk , ∑ 2 ij 6 (2) f pq = Tpq + ∑ Vpiqi ni + i (2) (3) 1 (3) Vpi jqi j ni n j , ∑ 2 ij Γ pqrs = Vpqrs + ∑ Vpqirsi ni , (2.26) (2.27) (2.28) i (3) Wpqrstu = Vpqrstu . (2.29) Here ni is the integer occupation number for the reference state (ni = Θ(εF − εi )). All normalordered vertices include in-medium contributions from higher order many-body forces. For example, the zero-body term E0 contains hole index summations over all of the free-space operators, and corresponds to the energy of the reference state, which has become the in-medium vacuum state. 19 Normal ordering of the Hamiltonian is injective, and thus exact solution of the many-body Schrödinger equation requires the same amount of work in either representation. However, in the in-medium formulation, one may truncate away the residual three-body force W , and still retain in-medium components of the free-space three-body force in the zero, one and two-body terms. This constitutes the normal-ordered two-body (NO2B) approximation, where three-body physics can be incorporated in a computationally practical two-body framework. With these tools, we have a foundation on which to build a many-body method. However, we will return briefly to the IPM in the context of Hartree-Fock theory, which we use to systematically construct a single-particle basis. 2.4 Hartree-Fock Theory The mean-field picture of the IPM can be put on a microscopic foundation through use of HartreeFock (HF) theory. This method again seeks to describe the ground state with only a single Slater determinant, but here we work with the full inter-nucleon interaction, constructing the mean-field through minimization of the energy functional E[Φ] = Φ|H|Φ , (2.30) where the ground state is taken to be a single Slater determinant. Using NN and 3N forces, the energy functional becomes E[Φ] = ∑ i|T |i + i 1 1 i j|V (2) |i j + ∑ i jk|V (3) |i jk , ∑ 2 ij 6 i jk (2.31) where the choice of indices indicates summation over hole states. We vary the single particle orbitals via Thouless’ theorem [103], which states that any two Slater determinants are related by an exponentiated one-body excitation operator, |Φ = eC1 |Φ . 20 (2.32) Taking the amplitudes of C1 to be small, we arrive at |Φ = (1 + ∑ δCai a†a ai )|Φ , (2.33) δ E[Φ] = ∑ δCai Φ|Ha†a ai |Φ . (2.34) ia and ai Setting δ E[Φ] to zero yields the condition Φ|Ha†a ai |Φ = 0 ∀ a, i such that εi ≤ εF and εa > εF . (2.35) Expanding in terms of the single-particle basis, we arrive at Φ|Ha†a ai |Φ = i|T |a + ∑ ik|V (2) |ak + k 1 ikl|V (3) |akl = 0 . 2∑ kl (2.36) this condition is known as the Brillouin condition, and is clearly satisfied by the eigenbasis of the operator (2) F = ∑ a†p aq Tpq + ∑ Vpkqk + pq k 1 (3) Vpklqkl . ∑ 2 kl (2.37) F is called the Fock operator. Its eigenfunctions correspond to the HF orbitals, and eigenvalues to the HF single-particle energies. F is the one-body piece of the normal-ordered Hamiltonian in eq. 2.25, highlighting the fact that in-medium operators contain mean-field components of higherorder free-space operators. Here we have computed the HF mean-field, (2) HF = Vpq ∑ Vpkqk + k 1 (3) Vpklqkl , 2∑ kl (2.38) a one-body potential constructed by summing over the hole orbitals in the reference state. This is the microscopic analog to the phenomenological construction of the IPM in sec. 2.2. The HF reference state |ΦHF is a Slater-determinant filled up to εF , and thus it is the energetically optimized reference state for a many-body calculation. The reference energy corresponds to the Hartree-Fock energy EHF = E[ΦHF ]. 21 One does not know the form of the HF basis a priori, so in practice, we start with a HO basis, and solve the eigenvalue problem self-consistently, as the construction of the mean-field requires knowledge of the HF orbitals. We construct F using the HO orbitals, diagonalize F, and use the new orbitals to restart the procedure. We iterate until convergence is obtained in the single particle energies and the HF basis has been established. As the HO basis is truncated to a finite single-particle energy emax , the HF orbitals will be dependent on the frequency of the HO basis h¯ ω. However, for large enough emax , we see convergence where h¯ ω dependence of computed quantities is negligible over a wide range. 2.5 The Configuration Interaction Method As previously stated, the principle task facing ab initio nuclear structure theorists is to solve for the coefficients Cα of the configuration expansion A |Ψν = C0 |Φ0 + 1 a ,...,a Ci 1,...,i N a†a · · · a†a aiN · · · ai1 |Φ0 . ∑ N 1 N 1 N=1 N! i1 ,...,iN ∑ (2.39) a1 ,...,aN If we have knowledge of the correlated wave function |Ψν , accurate observables such as energy, radii, and transition strengths follow from expectation values. The expansion of states in terms of the many-body basis suggests that the most straightforward way of doing this is to express the Hamiltonian as a matrix on the many-body Hilbert space. Solving the matrix eigenvalue problem yields the energies as eigenvalues and the coefficients Cα in the eigenvectors. Thus, for a given single-particle basis truncated at emax , we wish to construct all possible configurations of A filled orbitals, written as Slater determinants, and then represent the matrix in the resulting basis by computing individual matrix elements, e.g. Φab i j |H|Φ0 = Γabi j . (2.40) Subsequent solution of the eigenvalue problem yields the correlated wave functions for the ground and excited states. This approach is appropriately named the full configuration interaction (FCI) 22 approach. Frequently, we will refer to FCI as the standard of “exactness” within a given model space. It now appears that solving the many-body problem is a rather simplistic task, but of course, FCI is not practical for nuclear physics beyond the lightest nuclei, as construction of the full set of configurations scales factorially with the number of single particle orbitals. Nonetheless, a close relative of FCI has made a name for itself in ab initio nuclear structure, namely the no-core shell model (NCSM) [16–18]. Here COM factorization is ensured using a truncation on the excitation quanta allowed in the many-body basis rather than including the full set of configurations. In practice, the actual Slater determinants are never explicitly constructed in either of these approaches, as Krylov space approximate diagonalization methods [104, 105] can be used in place of conventional eigenvalue solvers, requiring only knowledge of the matrix-vector products, which may be computed fully in Fock-space with second quantization machinery. Regardless, computational demands grow rapidly with system size, so either of these exact methods are only applicable to light nuclei. Methods such as the FCI and NCSM belong to a class of ab initio approaches called “quasiexact” methods, meaning that they are exact within the accessible model spaces. Other examples include quantum Monte Carlo [19–21] and lattice effective field theory [106, 107], which are exact up to statistical errors. A common feature of any quasi-exact method is limited applicability, as the complex nature of the many-body problem demands significant computational power for an exact solution. While no ab initio method can proclaim universal applicability, vast improvements can be made here if we sacrifice the quasi-exact moniker by applying some systematically improvable manybody truncation. This can be accomplished by limiting the number of Slater determinants included in the expansion of the correlated state functions. The most obvious way to do this is to truncate the FCI expansion of eq. 2.39 up to a certain excitation rank from the reference state. The resultant configuration interaction (CI) method is specified by what remains. In this manner, we define the 23 configuration interaction with singles and doubles (CISD) [108]: 1 † † |ΨCISD = C0 |Φ0 + ∑ Cia a†a ai |Φ0 + ∑ Ciab ν j aa ab a j ai |Φ0 . 4 i,a i jab (2.41) This method is approximate of course, as the many-body Hamiltonian directly couples the reference state to single, double, and triple excitations, and indirectly to all excitation ranks. Any results expressed with such a method are then subject to so-called “many-body truncation errors”. We say this method is systematically improvable, as we can make things better by going to the next level of CISDT, or further to CISDTQ if we have the patience to implement such a method. The many-body truncation of the configuration interaction sheds some light on the importance of Hartree-Fock theory. As the CI expansion is limited to the reference state and a finite number of excitations built on top of that reference, it is critical that we use the best possible reference state. For most systems, the HF reference is a sufficient choice, as it minimizes the energy functional for Slater determinants. Truncated CI, despite having a desirable simplicity in its formulation, suffers from a fatal shortcoming; the method violates size-extensivity, a principle that states that a uniformly distributed system’s energy should be proportional to system size [11]. 2.6 Many-Body Perturbation Theory The lack of size-extensivity observed in truncated CI can be understood using many-body perturbation theory (MBPT), which is itself a method for constructing the correlated ground state wave function [11, 109, 110]. MBPT treats the non-interacting reference state as the order zero wave function, and the corresponding reference energy as the order zero energy. Generically, the Hamiltonian is partitioned into a diagonal and interaction piece H|Ψ0 = (H0 + HI )|Ψ0 = E0 |Ψ0 , (2.42) with the known order zero solution (0) H0 |Φ0 = E0 |Φ0 . 24 (2.43) Left multiplying eq. 2.42 by Φ0 | gives (0) Φ0 |HI |Ψ0 = E0 − E0 ≡ ∆E , (2.44) where we have invoked the intermediate normalization Φ0 |Ψ0 = 1. Treating this problem in Rayleigh-Schrödinger perturbation theory yields MBPT to infinite order ∞ ∑ [R0(HI − ∆E)]m|Φ0 |Ψ0 = , (2.45) m=0 with the resolvent operator R0 = Q (0) , (2.46) E0 − H0 where the complement space projector is given by Q = 1 − |Φ0 Φ0 |. Thus, MBPT can be computed up to a specified order m, and at each order, a finite number of terms contribute. For example, the first order wave-function is given by (1) |Ψ0 = |Φ0 + R0 HI |Φ0 , (2.47) and the corresponding second-order energy correction is ∆E (2) = Φ0 |HI R0 HI |Φ0 . (2.48) The energy and wave functions are often evaluated using Goldstone diagrams (for details, see [11]), and it can be shown that any unlinked diagrams arising from the HI [R0 HI ]m term in eq. 2.45 are canceled exactly at each order by so-called renormalization terms proportional to ∆E. This is a restatement of the famous linked-cluster theorem [111–114], and it results in a simplified expression for the MBPT energy corrections ∆E (m+1) = Φ0 |{HI [R0 HI ]m }C |Φ0 , (2.49) where the subscript C denotes connected diagrams. By the linked-cluster theorem, a method is size-extensive if the ground state energy can be expressed diagrammatically using only connected diagrams, hence MBPT is size-extensive at each order. 25 Starting from the NO2B approximation, we partition the Hamiltonian as H0 = E0 + ∑ f pp {a†p a p } HI = ∑ f pq (1 − δ pq ){a†p aq } + p pq 1 ∑ Γ pqrs{a†pa†qasar } . (2.50) 4 pqrs This is simplified in the HF basis, as the Brillouin condition sets the one-body piece of HI to zero. Employing this partitioning in the HF basis, the first-order energy correction vanishes and the second-order correction (MBPT(2)) is (2) E0 |Γi jab |2 | Φα |HI |Φ0 |2 1 = ∑ , =∑ E0 − Eα 4 i jab ∆i j α ab (2.51) ij where ∆ab = fii + f j j − faa − fbb is the Møller-Plesset energy denominator. This energy correction scales as O(N p2 Nh2 ) , where N p and Nh are the number of particle and hole orbitals in the single particle basis. The method is systematically improvable, with the MBPT(3) correction scaling as O(N p4 Nh2 ). In principle, truncated CI and MBPT are different routes to solving the same problem. In fact, CISD includes energy contributions from all orders of MBPT at a computational cost similar to MBPT(3). The resummation is incomplete however, in particular the renormalization terms which violate size-extensivity are not cancelled out at each order of CI. This manifests diagrammatically with the observation that the CI excitation operators C1 ≡ ∑ Cia a†a ai C2 ≡ ia 1 † † Ciab j aa ab a j ai , 4 i∑ jab (2.52) which are constructed by solving the CISD eigenvalue problem H|ΨCISD = ECISD |ΨCISD , 0 0 (2.53) contain unlinked diagrams composed of the bare interaction vertices. 2.7 Coupled Cluster Theory The resummation of perturbation theory exhibited by CI motivates a size-extensive counterpart, called coupled cluster (CC) theory [11, 25]. CC theory reorganizes the CI expansion into an expo26 nential expansion of cluster operators, exp[T1 + T2 + T3 + · · · + TA ] ≡ 1 + C1 + C2 + C3 + · · · + CA , (2.54) where TN are the N-body cluster operators defined by ∞ TN = m , ∑ {R0HI }C,N (2.55) m=1 where the subscript (C, N) denotes connected terms with 2N external lines. Rewriting the expansion in this way allows us to make a size-extensive many-body truncation on the cluster operators, denoted as CCSD, CCSDT, and so on. Thus the CCSD ground state is = exp[∑ tia {a†a ai } + |ΨCCSD 0 ai 1 ∑ tiabj {a†aa†ba j ai}]|Φ0 . 4 abi j (2.56) In coupled-cluster calculations, the exponential operator eT is a similarity transformation, ¯ 0 = E0 |Φ0 , H|Ψ0 = E0 |Ψ0 → e−T HeT |Φ0 = E0 |Φ0 → H|Φ (2.57) which is applied to the Hamiltonian and used to derive a system of algebraic equations in the intermediate normalization, ¯ 0 = E0 Φ0 |H|Φ ¯ 0 = 0. Φα=0 |H|Φ (2.58) Thus, CCSD calculations are carried out by solving eqs. 2.58 for the cluster amplitudes tia , tiab j and the energy E0 , with α ranging over all singly- and doubly-excited Slater determinants. Here it is possible to show that the MBPT energy through third-order is included completely in the CCSD energy, with incomplete but size-extensive contributions from all orders of perturbation theory included in addition to MBPT(3). This is a marked improvement over MBPT, as the time complexity is the same, yet significantly more components of the infinite MBPT energy expansion are included in the ground state energy and wave function. The many-body truncations used here lead to large errors when confronted with hard core potentials where HF is often unbound, but these methods have become popular in nuclear physics in 27 the past decade, as the number of soft perturbative potentials such as those from chiral EFT has been rapidly increasing. CCSD and perturbative approximations to CCSDT, such as Λ-CCSD(T) and completely renormalized CCSD(T) (CR-CC(2,3)), have ushered in a new era of ab initio calculations in nuclei, providing access to nuclei in the vicinity of closed shells throughout the mediummass region [41, 49, 50, 57, 58, 115, 116]. Along with CC theory, additional methods such as the self-consistent Green’s function (SCGF) method [53–55] and the in-medium similarity renormalization group (IMSRG) [43, 44, 46] offer alternative tactics to resum MBPT in a size-extensive framework. The IMSRG, which is of principle interest in this work, uses a modified exponential ansatz to produce a similar resummation to CC theory in a Hermitian framework, (s) = e−Ω(s) |Φ0 . |ΨIMSRG 0 (2.59) Here Ω(s) is an anti-Hermitian operator and consequently the similarity transformation is unitary. The Hermitian framework is extremely desirable as it allows for simplistic extensions of the method, such as the multireference IMSRG (MR-IMSRG) [45, 47], which targets open shell nuclei. The IMSRG offers Hermitian calculations at comparable computational cost to CC theory, but a consequence of this Hermiticity is that the IMSRG cannot construct the intuitive system of equations in 2.58. Instead, the IMSRG transformation is applied as a differential flow equation, ¯ d H(s) ¯ = [η(s), H(s)] , ds (2.60) where η(s) is the arbitrary anti-Hermitian generator of the transformation. Alternatively, one can solve for Ω(s) directly using the Magnus expansion [117]. This method will be explored in detail in chapter 3, and extended to the calculation of excited states and general observables in chapters 4 and 5, respectively. 28 CHAPTER 3 IN-MEDIUM SIMILARITY RENORMALIZATION GROUP As mentioned in previous chapters, much progress has recently been made in ab initio nuclear structure, largely due to the development of renormalization group methods to produce soft “lowmomentum” interactions from underlying “hard” interactions while preserving observables. Such potentials greatly facilitate many-body methods which expand wave functions in a finite basis of localized single-particle orbitals, as convergence is achieved with much smaller bases using softened potentials. Because softened interactions reduce the effects of short-range correlations, Hartree-Fock becomes a reasonable starting point for methods like many-body perturbation theory and coupled cluster theory, and consequently these methods have flourished in the presence of these innovations. In this chapter, we will flesh out the fundamentals of the IMSRG approach, which extends the RG concept of decoupling low- and high-momentum degrees of freedom to the decoupling of states in the many-body Hilbert space. Here we formulate “in-medium” flow equations; the solution of which is tantamount to the block-diagonalization of the many-body Hamiltonian [43– 48]. The IMSRG offers the ability to target ground and excited states in both closed- and openshell nuclei, and as it exhibits polynomial scaling with system size, the method is a powerful ab initio approach for medium-mass nuclei. We discuss recent results, as well as some advances in the implementation of the method. We review the Magnus expansion variant of the IMSRG, which enables the explicit construction of the unitary transformation. We also briefly review the perturbative resummation of the IMSRG(2), and discuss corrections to this resummation which are motivated by the availability of the unitary transformation via the Magnus generator. 29 3.1 The Similarity Renormalization Group The similarity renormalization group (SRG) can be viewed as a continuous sequence of unitary transformations, U †U = UU † = 1 , (3.1) which by construction preserves the inner product of two vectors, and consequently the eigenvalues of all operators. Renormalization groups (RGs) are designed to exploit separation of scales in physical problems. The SRG was developed independently by Wegner [118], for the solution of many-body problems in condensed matter physics, and Głazek and Wilson [119], who developed the procedure for Hamiltonians in light-front quantum field theory. The SRG was subsequently ported to nuclear physics, where it was used to decouple low- and high-momentum modes in the input inter-nucleon interactions [9,39,120–124]. This was motivated by the fact that realistic interactions which give a good description of low energy nuclear observables tend to feature strongly repulsive cores and strong tensor forces [21, 39, 125]. Both of these features produce strong couplings between low- and high-momentum states in the many-body Hamiltonian, which in turn leads to computationally demanding matrix diagonalizations due to the large couplings between energetically well-separated basis states. The SRG can be used as a means to soften this core, and thus reduce the off-diagonal couplings in momentum-space via a suitably chosen unitary transformation. 3.1.1 Formalism If we parameterize the SRG transformation by some arbitrary flow variable s, we can then take the derivative of eq. 3.1, dU † (s) dU(s) † U (s) +U(s) = 0, ds ds (3.2) which yields a differential equation for the explicit form of the unitary transformation, dU(s) = −η(s)U(s) . ds 30 (3.3) Here η(s) is the so-called generator of the unitary transformation η(s) ≡ dU(s) † dU † (s) U (s) = −U(s) = −η † (s) , ds ds (3.4) where the constraint of anti-Hermiticity immediately follows from eq. 3.2. The solution of eq. 3.3 is the s-ordered exponential, U(s) = S exp(− s 0 ds η(s )) ≡ lim N→∞ eη(sN )δ s eη(sN−1 )δ s · · · eη(s1 )δ s eη(s0 )δ s . (3.5) The s-ordering assures that the sequence of infinitesimal transformations, which are in general non-commutative, will be applied in the correct order. In practice, using this form of the transformation requires storage of the generator at each step in the s-evolution. One may circumvent the complications of ordered exponential operators by use of flow equations to determine the form of SRG evolved operators. For any operator O, the similarity transformation takes the form ¯ = U(s)OU † (s) . O(s) (3.6) Taking the derivative, ¯ d dU(s) dU † (s) d O(s) = [U(s)OU † (s)] = OU † (s) +U(s)O = [η(s),U(s)OU † (s)] , ds ds ds ds (3.7) we arrive at the SRG differential flow equation ¯ d O(s) ¯ = [η(s), O(s)] . ds (3.8) The power of eq. 3.8 is the ability to apply the SRG transformation numerically without retaining knowledge of η for all values of s. As the only constraint on η is that the operator must be anti-Hermitian, it can be chosen to construct Hamiltonians tailored for particular many-body methods. A general prescription for choosing an SRG generator was proposed by Wegner [118], η(s) ≡ [H d (s), H od (s)] , 31 (3.9) where H d and H od are arbitrarily defined diagonal and off-diagonal components of the Hamiltonian. H d need not be limited to the literal diagonal of the matrix, but rather describes the desired form of the rotated operator, which may be band- or block-diagonal. Such a choice of generator clearly produces a fixed point in the flow equation 3.8 if H od becomes zero for some value of s. If such a condition arises, the flow equation can be evolved to s → ∞, and the desired decoupling will be achieved. Coincidental commutation of H d and H od will also create a fixed point, but such an incidence has not been observed in practice. 3.1.2 A Practical Example As a pedagogical example, it is instructive to consider a two-body system in momentum-space, where we can express the problem in relative coordinates. The flowing Hamiltonian is H(s) = Trel +V (s) , (3.10) where the kinetic energy Trel is independent of s by construction, and is diagonal in momentum space. The residual s-dependence is absorbed into the potential V (s) and equation 3.9 becomes η(s) = [Trel ,V (s)] . (3.11) Plugging eq. 3.11 into eq. 3.8, we can show that for a given partial wave in momentum space [121], 2 ∞ 2 dV (k1 , k2 , s) = −(k12 − k22 )2V (k1 , k2 , s) + q dq(k12 + k22 − 2q2 )V (k1 , q)V (q, k2 ) . ds π 0 (3.12) The solution of eq. 3.12 for strongly off-diagonal terms is dominated by the linear term, V (k1 , k2 , s) ≈ V (k1 , k2 , 0)exp(−(k12 − k22 )2 s) . (3.13) Eq. 3.13 demonstrates proper RG behavior, where farther off-diagonal matrix elements decay at an increased rate, quantified by the lifetime λ (k1 , k2 ) = 1 (k12 − k22 )2 32 . (3.14) 0 s = 0.6 k (fm −1 ) 10 15 4 20 25 5 s = 1.1 s = 2.1 10 0 4 15 20 8 25 300 5 10 15 20 25 30 0 k (fm −1 ) s = 0.0 s = 0.6 8 10 15 4 20 25 30 0 5 k (fm −1 ) 30 0 k (fm −1 ) 5 8 Magnitude k (fm −1 ) 5 s = 0.0 s = 1.1 s = 2.1 10 4 15 20 8 25 300 5 10 15 20 25 30 k (fm −1 ) 0 Magnitude 0 5 10 15 20 25 30 0 k (fm −1 ) 5 10 15 20 25 30 k (fm −1 ) Figure 3.1 SRG evolution of the two-body Hamiltonian in momentum space. In the left panel, we show solution of eq. 3.8 for select values of s. In the right panel we show the non-unitary decay of matrix elements when given a half-life corresponding to eq. 3.14. A simple example is provided by the 1d mock-up of a local nuclear interaction suggested by Negele and Orland [126]: 2 2 V (k1 , k2 ) = V1 (e(−α1 (k1−k2) ) − e(−α1 (k1+k2) ) ) 2 2 −V2 (e(−α2 (k1−k2) ) − e(−α2 (k1+k2) ) ) . (3.15) Here V1 = V2 =12.0 fm−2 , α1 = 0.01 fm2 and α2 = 0.16 fm2 . We analyze this potential for the case of two particles, where constructing the many-body Hamiltonian is as simple as discretizing the relative momentum and evaluating V at each point. For purposes of demonstration, we use a modest 120 relative momentum states, building a momentum grid with spacing 0.25 fm−1 . Here we may demonstrate the SRG’s properties exactly, as at most two-body forces are needed. This potential does not induce extreme scattering from low to high momentum, but those modes are coupled through the band diagonal structure, shown in the s = 0.0 panels of fig. 3.1. While one could easily diagonalize this small matrix using a packaged solver, much insight is to be gained 33 by performing the SRG exactly. For reference, the right panel of fig. 3.1 shows the decay pattern if eq. 3.13 were the only contribution to the SRG evolution. The left panel demonstrates the exact SRG flow, where additional features are present. It is evident that the broad features of the flow are very similar in either case, but fine details are present in the SRG flow which account for unitary equivalence. This SRG softening allows for a significant reduction in storage requirements with minimal loss of accuracy in the ground state energy. For this system, diagonalization of the unevolved operator gives a ground state energy of -0.473 fm−2 . If the diagonalization is instead performed with only the first 90 basis states, the result is 46.000 fm−2 , clearly an unsuitable result. However, after SRG softening to s=2.1×10−4 fm4 , both the 90 and full 120 state diagonalizations yield -0.473 fm−2 . Additional softening further reduces the number of basis states needed for a converged ground state energy. For notational purposes, we note that SRG softened forces are often specified by a momentumcutoff λSRG = s−1/4 , in natural units. We can think of λSRG as a scale for the maximum size of momentum transfer between nucleons allowed after SRG softening [39, 121]. 3.1.3 SRG Softening of Interactions The primary use of the SRG is to pre-process nuclear interactions, making them more amenable to many-body calculations [18, 40, 45, 47, 48, 53–55, 63, 89, 102, 116, 124, 127–129]. NN and 3N interactions from χ-EFT are applied to many-body calculations using the second-quantized Hamiltonian: 1 k2 † ak ak + ∑ k1k2|V (2)|k3k4 a†k1 a†k2 ak4 ak3 2 4 k1 k2 k3 k4 k 1 k1 k2 k3 |V (3) |k4 k5 k6 a†k a†k a†k ak ak ak . + ∑ 1 2 3 6 5 4 36 k k k H =∑ (3.16) 1 2 3 k4 k5 k6 Here V (2) and V (3) in general connect two- and three-body kets with bras containing elements with drastically different momentum components, which is a hallmark of a hard-core interaction, and is 34 largely responsible for convergence difficulties in many-body methods based on diagonalization. Starting from unevolved quantities, the Wegner generator (eq. 3.9) is η= k72 ∑ 2 all k k1 k2 |V (2) |k3 k4 [a†k ak , a†k a†k ak ak ] 7 7 1 2 4 3 + k1 k2 k3 |V (3) |k4 k5 k6 [a†k ak , a†k a†k a†k ak ak ak ] , 7 7 1 2 3 6 5 4 (3.17) where the commutators of the first and second terms yield two and three-body operators respectively. Thus, η(s) is in principle a many-body operator. Solution of eq. 3.8 will necessarily induce higher order operators, as the commutator of two three-body operators includes up to fivebody operators. It is apparent that to retain an exact unitary transformation, the A-body nucleus will require up to A-body induced forces. Of course, this is intractable computationally, and in fact, including three-body forces is a demanding challenge which has only recently become feasible [9, 123, 124, 130, 131]. Nonetheless, it has been shown that inclusion of the three-body force is absolutely imperative [130], as truncation to two-body operators has been linked to strong sdependence in many-body observables. The errors introduced by truncating four-body forces and higher are ostensibly smaller than those of missing three-body forces, provided we do not make absurd demands of our SRG transformation. Softening beyond a certain point will almost certainly induce large many-body forces, so one must take care to stop at a reasonable value of λsrg . Typical choices are between λsrg = 1.8 and 4.0 fm−1 . 3.2 The In-Medium Similarity Renormalization Group The free-space SRG formalism has had great success as a method to soften input nuclear interactions for few- and many-body calculations, but the general SRG concept can be extended to serve as an ab initio method in and of itself when the evolution is performed directly in the A-body system. The in-medium similarity renormalization group (IMSRG) applies eq. 3.8 to the many-body Hamiltonian after normal-ordering with respect to an A-body reference state. In this context, the 35 application of the SRG seeks to renormalize couplings between A-body basis states, i.e. Slater determinants (SD). 3.2.1 The In-Medium Similarity Renormalization Group Equations With the power of normal-ordering, which was discussed in sec. 2.3, we may derive the in-medium variant of the SRG, by normal-ordering the generators and operators with respect to the A-body reference state, and evaluating eq. 3.8 using Wick’s theorem. The critical component of this derivation is the computation of the commutator of two normal-ordered two-body operators, [A, B](0b) = ∑(na − nb )Aab Bba + ab (1b) [A, B]qr 1 ∑ (Aabcd Bcdab − Babcd Acdab)nanbn¯cn¯d , 4 abcd (3.18) = ∑(Aqa Bar − Bqa Aar ) + ∑(na − nb )(Aab Bbqar − Bab Abqar ) a + ab 1 ∑ (nanbn¯c + n¯an¯bnc)(AcqabBabcr − BcqabAabcr ) , 2 abc (3.19) (2b) [A, B]qrst = ∑((Aqa Barst + Ara Bqast − Bqa Aarst − Bra Aqast ) a − (Aas Bqrat + Aat Bqrsa − Bas Aqrat − Bat Aqrsa )) + 1 (1 − na − nb )(Aqrab Babst − Aqrab Babst ) 2∑ ab − ∑(na − nb )(1 − Pqr )(1 − Pst )Abrat Baqbs . (3.20) ab Here ni is the reference state occupation number for the ith orbital, and n¯ i = (1 − ni ) and Pqr exchanges the indices q and r. The commutator of two-body operators will also have three-body components, but we will only concern ourselves with the two-body components, anticipating that this truncation will cause minimal error in the final results of computations in the NO2B approximation. Plugging these expressions into eq. 3.8, we arrive at the IMSRG(2) flow equations dE0 1 = ∑(na − nb )ηab fba + ∑ ηabcd Γcdab na nb n¯ c n¯ d , ds 2 abcd ab 36 (3.21) d fqr = ∑(1 + Pqr )ηqa far + ∑(na − nb )(ηab Γbqar − fab ηbqar ) ds a ab 1 + ∑ (na nb n¯ c + n¯ a n¯ b nc )(1 + Pqr )ηcqab Γabcr , 2 abc (3.22) dΓqrst = ∑((1 − Pqr )(ηqa Γarst − fqa ηarst ) ds a − (1 − Pst )(ηas Γqrat − fas ηqrat )) + 1 (1 − na − nb )(η12ab Γab34 − Γ12ab ηab34 ) 2∑ ab − ∑(na − nb )(1 − P12 )(1 − P34 )ηb2a4 Γa1b3 . (3.23) ab The name IMSRG(2) refers to the truncation of the commutators to two-body operators. While here we have only included two-body operators, taking the commutator of an N- and M-body operator results in an (N + M − 1)-body operator, thus an exact solution of the IMSRG equations will produce up to A-body forces for an A-body nucleus. IMSRG(A), where up to A-body operators are included in the commutator expressions and IMSRG equations, would then produce the same result as FCI, making it an exact method within a given model-space. Naturally, any implementation of this exact method would pose the same computational difficulties as the FCI calculation itself. The utility of the IMSRG lies in the ability to make systematically improvable, size-extensive truncations on the CI expansion, such as IMSRG(2) or the next level of IMSRG(3), which keep normal-ordered three-body operators in the flowing Hamiltonian and generator. In practice, the commutator expressions are often implemented in JJ-coupled scheme (see appendix), where spherical symmetry is exploited by summing magnetic substates out of the single-particle and many-body bases using the Wigner-Eckart theorem. 3.2.2 Choice of Generator A variety of generator choices have been explored for IMSRG calculations [3, 48]. As in the freespace SRG, the specific choice of generator reflects the desired decoupling scheme. For groundstate calculations, one seeks to decouple the reference state from the block of particle-hole excited 37 1p1h 2p2h 3p3h 0p0h 1p1h 2p2h 3p3h 3p3h 3p3h 2p2h 2p2h 1p1h 1p1h 0p0h 0p0h 0p0h ¯ i| H(0) |j ¯ i| H(∞) |j ¯ Figure 3.2 Schematic representation of the initial and ground-state-decoupled Hamiltonians, H(0) ¯ and H(∞), in the many-body Hilbert space spanned by particle-hole excitations of the reference state. Slater determinants. Fig. 3.2 demonstrates this schematically, where we have included up to 3p3h states in the visualization. In the left panel we see that the reference state is coupled to particle-hole excitations through the 1p1h and 2p2h channels. These matrix elements are induced by the f ph and Γ pp hh terms in eq. 2.25, p Φh | H |Φ0 = f ph , (3.24) pp hh (3.25) Φ | H |Φ0 = Γ pp hh , as well as their Hermitian conjugates. Here p and h denote particle and hole indices respectively. The generator of the IMSRG unitary transformation thus should be designed to suppress H od = ∑ fi j (n¯ i n j + n¯ j ni ){a†i a j } + ij 1 Γi jkl (n¯ i n¯ j nk nl − n¯ k n¯ l ni n j ){a†i a†j al ak } . ∑ 4 i jkl (3.26) With this definition, the Wegner choice can be computed using eq. 3.9, and the desired decoupling is achieved with leading off-diagonal behavior −(∆ Γ(s) pp hh ≈ Γ(0) pp hh e 38 pp 2 ) s hh . (3.27) Here ∆ pp hh = ( f pp + f p p − fhh − fh h ) is the Møller-Plesset (MP) energy difference. Analogous to the free-space variant, the decay rate of off-diagonals is mediated by this energy difference. As a result, the differential flow equations are moderately stiff when solved with this generator. This means that at different points in the flow, different terms will be emphasized for suppression in the differential equation, and thus numerical complications can arise. Under these circumstances, calculations will have difficulty converging, and often a more sophisticated and memory intensive ordinary differential equation solver will be required to solve the flow equations. More numerically practical generators can be devised to mitigate the stiffness by making the lifetimes of off-diagonal matrix elements less dependent on the single particle energy spectrum. An example is proposed by White, following from the theory of Jacobi canonical diagonalization [43, 132], (1) fiod j (2) Γod i jkl , (3.28) ij ∆kl where the ∆ may be either MP or Epstein-Nesbet (EN) denominators. The definition is chosen ηi j = ηi jkl = ∆ij to conceptually match the IMSRG rotation to a Jacobi rotation, by scaling individual off-diagonal matrix elements by the angle with which they must be rotated in order for suppression. At leading order, off-diagonals are suppressed like Γ(s) pp hh ≈ Γ(0) pp hh e−s , (3.29) which means all elements decay at a similar rate, and thus stiffness is relieved. The cost of this drastic reduction in stiffness is the presence of explicit energy denominators in eq. 3.28, which may potentially become small during the course of IMSRG decoupling. Such an event would certainly create pathological behavior in the flow and derail the computation. A reasonable compromise between the Wegner and White generators can be inferred by starting from the general form ηα = (∆α )γ Hαod . (3.30) Here α denotes a general set of indices, and γ an integer power. For the White generator, γ=-1, and for Wegner’s choice γ=1 at leading order in the interaction vertices. The compromise, known 39 as the imaginary-time generator [47], is found at γ=0, (1) (2) i ηi j = fiod j g(∆ j ) ij ηi jkl = Γod i jkl g(∆kl ) , (3.31) where g(x) is some odd regulator function which ensures an anti-Hermitian η. Here we get an off-diagonal decay of −∆ Γ(s) pp hh ≈ Γ(0) pp hh e pp s hh , (3.32) which is more stiff than the White generator, but less so than Wegner. In practical applications of the IMSRG, the White generator is often sufficient. However, in decouplings which strongly renormalize the single-particle spectra, it is useful to employ the imaginary-time generator to avoid vanishing energy denominators. For the remainder of this dissertation, it should be assumed that the White generator is applied, and use of any other generator will be specified explicitly. 3.2.3 Solving the IMSRG Equations We can analyze the IMSRG equations by writing them in terms of Hugenholtz diagrams. Diagrammatically, eq. 3.21 becomes d E= ds + . (3.33) Figure 3.3 Schematic zero-body flow. where η(s) is proportional to H(s), and MBPT energy denominators are included within the definition of η. The gray circles represent renormalized or dressed vertices, which have s-dependence. If we discretize the flow and take light gray circles to be Γ(2δ s), dark gray as Γ(δ s), and a small black circle as the bare vertex Γ(0), we may represent the integration step of eq. 3.23 for Γ(2δ s) in terms of the vertex of the previous step, and again in terms of the bare vertex: 40 = + + + = + + + + + + + + ... + ... + + ... + ... (3.34) Figure 3.4 Schematic resummation of MBPT diagrams. This representation illustrates an important feature of the IMSRG equations; solving them sums higher-orders of MBPT into the flowing vertices. In the first row, we see that IMSRG renormalizes short-range physics with the resummation of all particle-particle and hole-hole ladders into the dressed vertex, similar to Brueckner G-matrix approaches [112, 114, 133–136]. Additionally, long-range physics is consistently included with the resummation of particle-hole chains, which is a feature of the random phase approximation (RPA) [100,114,136]. On top of incorporating the resummations of both of these well known methods, the IMSRG incorporates interference diagrams between ladder and chain components. In the limit of IMSRG(A), an exact solution of the IMSRG equations would result in a complete infinite resummation of MBPT. As the IMSRG is truncated to two-body operators in practice, this resummation is only approximate, but can be shown to be exact up to third-order, while also containing extensive, but incomplete, content from all orders of perturbation theory [3]. 41 20.8 Eref Eref + EMBPT(2) Eref + EMBPT(2-3) FCI(H(0)) FCI(H(5.4)) Energy (Hartree) 20.7 20.6 20.5 20.4 20.3 0 1 2 s 3 4 5 Figure 3.5 IMSRG ground-state-decoupling for a 6-electron quantum dot, with ω = 1.0 and four major oscillator shells. Hartree-Fock energy, second-, and third-order MBPT energies are plotted as function of flow parameter s. Exact energies for the bare and dressed Hamiltonians are given for reference. This point is illustrated in practice by fig. 3.5, which demonstrates a typical IMSRG groundstate-decoupling for a 6-electron parabolic quantum dot in two dimensions. Parabolic quantum dots consist of A electrons confined by a harmonic oscillator potential in two dimensions. In atomic units, the Hamiltonian is given by: A H= 1 1 A 1 1 ∑ [ 2 p2i + 2 ω 2ri2] + 2 ∑ |ri − r j | . i=1 (3.35) i= j Quantum dots provide an excellent testing ground for approximate many-body methods, as the strength of many-body correlations can be controlled by varying ω, with smaller values corresponding to stronger correlations, and comparisons can be made to exact full configuration in- 42 teraction (FCI) calculations in sufficiently small bases. In the present work, all calculations are performed for the 6-electron system. The IMSRG(2) equations are solved here using a high-order adaptive step solver (see below), so numerical errors in the solution of the ODE should be negligible here. At s=0, the solid lines correspond to so-called bare quantities, which are yet to be renormalized. Thus the reference energy corresponds to the Hartree-Fock (HF) energy and the perturbative corrections represent standard MBPT. For reference, the dotted purple line gives the exact energy, Egs (0) = 20.4158 Hartree, computed with FCI using the bare Hamiltonian. The IMSRG equations are propagated forward to s=5.4, where numerical convergence has been obtained. At this point, some truncation errors due to the neglect of induced higher-body forces have accumulated; for reference the exact result using the dressed Hamiltonian, Egs (5.4) = 20.4070 Hartree, is given by the dotted blue line. The truncation error amounts to less than 10 mH, which is on the order of chemical precision. Note that as the IMSRG(2) equations are solved, the HF energy becomes equivalent to the correlated ground state energy of the dressed Hamiltonian. This is due to complete decoupling of the reference state from excitations, as illustrated in fig. 3.2. As the flow converges, the second- and third-order energy corrections are diminished to zero, consistent with the expectation of a resummation of MBPT. Note that the perturbative corrections have strong s-dependence, indicating that higher-order corrections are also being summed into the reference energy during the flow. While the MBPT(3) appears to capture the vast majority of the correlation energy here, this will not be the case for more strongly correlated systems such as nuclei. This indicates a clear preference for IMSRG, as the dominant contribution to the scaling of the IMSRG(2) equations goes as O(N p4 Nh2 ), which is the same as MBPT(3), while including higher-order terms such as ladder summations and bubble chains. The previous calculation, as well as the original implementation of the IMSRG [43–45], employed a high-order ordinary differential equation (ODE) solver to solve eqs. 3.21-3.23 directly. High-order predictor/corrector methods such as the Shampine and Gordan implementation of Adams-Bashforth [137] are necessary to solve the IMSRG equations accurately, as accumulated 43 numerical errors in low-order solvers rapidly degrade the unitarity of the IMSRG transformation to the point where computed eigenvalues can have sizable errors. High-order adaptive step solvers require anywhere from 10-25 copies of the ODE, and thus memory requirements can be extraordinary for computations with large model spaces. Additionally, the flow equation (eq. 3.8) must be solved individually for every operator of interest in order to compute observables consistently. This means that the memory requirements, which are already demanding for a single operator, scale linearly with the number of observables requested. Many of these complications were mitigated by the development of the Magnus expansion variant of the IMSRG [6]. The Magnus expansion [117] provides an expression for orderedexponentials in the form of a true exponential, U(s) ≡ exp(−Ω(s)) = S exp(− s 0 ds η(s )) , (3.36) where the components of the Magnus generator Ω(s) follow from the differential equation ∞ B dΩ j j = ∑ adΩ η(s) , ds j=0 j! (3.37) where B j are Bernoulli numbers (B0 = 1, B1 = −1/2 , B2 = 1/6, . . . ) and j 0 η(s) = η(s) , adΩ j−1 adΩ η(s) = [Ω(s), adΩ η(s)] . (3.38) This method is an exact reformulation of the IMSRG, provided that no truncations are made. Of course, the flow equation approach is truncated to two-body operators, and any practical implementation of the Magnus variant requires a similar truncation, as well as numerical termination of the expansion at a reasonable number of terms. As evaluation of eq. 3.37 depends on the same generator as eq. 3.8 and requires computation of only one commutator at at time, it can be solved with the same machinery that one might use in the flow equation method. The expressions 3.18j 3.20 can be used for the evaluation of adΩ η(s), where the only difference from the flow equation approach is that both operators are anti-Hermitian. The Magnus formalism avoids the complications of using a sophisticated ODE solver, as numerical step-size errors in the exponential operator do not degrade the unitary transformation, 44 because even if time-step errors are significant, upon exponentiation of the anti-Hermitian Ω(s), a unitary transformation is still realized. However, numerical errors will still accumulate in the exponentiation procedure, (i.e. applying the transformation to an operator), so care must be taken to assure that the desired transformation is achieved as quickly as possible. If we do not retain a reasonable amount of numerical accuracy in the solution of eq. 3.37, we will not achieve the desired decoupling and eventually errors from applying the transformation will destabilize the calculation. Consequently, the Magnus variant of the IMSRG may be solved with a rudimentary forward Euler step, but we cannot take arbitrarily large steps in s without changing the nature of the unitary transformation so far as to increase off-diagonal pieces rather than suppress them. The most useful property of the Magnus formalism is that eq. 3.37 need only be solved once, and all other observables follow from application of the explicitly constructed unitary transformation using the well known Baker-Campbell-Hausdorff expansion, ¯ = O(s) ∞ 1 j adΩ O . j! j=0 ∑ (3.39) Fig. 3.6 displays results for the 16 O ground-state energy for both approaches using a forward Euler step to solve the respective differential equations. For this system, we compute the ground state energies using both the direct ODE solver approach and the Magnus expansion method. Additionally the Adams-Bashforth method is used as a high-precision point of reference. Here we see that the IMSRG flow equation is subject to large errors when solved using a forward Euler step method, with the converged result strongly dependent on the step size δ s. The Magnus approach, on the other hand, does not exhibit dependence on δ s, but rather converges to approximately the same result as the Adams-Bashforth ODE solver, regardless of step-size. This is confirmation that the Magnus expansion minimizes the effect of step-size errors, and thus reduces the immense storage requirements of IMSRG which originate from the use of high-order ODE solvers. As mentioned above, the efficacy of the Magnus approach is contingent upon a reasonable numerical termination point for both the Magnus and the BCH expansions. For the kth term in 45 -120 16 O 3 -1 N LO (500 MeV) λ = 2.0 fm Energy (MeV) -140 IM-SRG(2) (Adams-Bashforth) Magnus(2) (Euler δs=0.5) Magnus(2) (Euler δs=0.1) IM-SRG(2) (Euler δs=0.5) IM-SRG(2) (Euler δs=0.1) -160 -180 0 1 2 3 4 5 s Figure 3.6 Flowing reference energy E0 (s) for 16 O at with the EM input NN potential, N3LO(500) [5] at λSRG = 2.0 fm−1 , emax = 8 and h¯ ω = 24 MeV. Computed with Magnus and flow equation methods using forward Euler step. Results are benchmarked against flow equation solved by high-order adaptive step solver [6]. eq. 3.37, we adjudicate numerical termination by the convergence criterion k η|| Bk ||adΩ < εderiv , k!||Ω|| (3.40) where double bars refer to the Frobenius norm. The convergence of eq. 3.39 is determined by k H} {adΩ 0b < εBCH . k! (3.41) This criterion is not as strict as the one used for the Magnus expansion, as the BCH expansion is used only to determine the ground state energy in this application. If one wishes to analyze the full content of the dressed Hamiltonian post-IMSRG decoupling, a more stringent criterion analogous to eq. 3.40 should be used. Fig. 3.7 displays the relative magnitude of each nested commutator term as a function of the flow parameter, in either expansion, computed for 16 O with a sample 46 10 10 -2 -4 -6 10 16 -8 5 10 10 k=1 k=3 k=5 k=7 k=9 2 0 -2 -4 -6 16 -8 O 5 10 s k=1 k=3 k=5 k=7 k=9 d) 10 10 10 2 0 -2 th -2 k=0 k=1 k=2 k=4 b) N3LO (500 MeV) λ = 3.0 fm-1 0 0 th 10 10 10 s c) 10 10 10 O 0 log10[ k term (MeV)] 10 Relative Magnitude 10 k=0 k=1 k=2 k=4 a) N3LO (500 MeV) λ = 2.0 fm-1 0 log10[ k term (MeV)] Relative Magnitude 10 10 -4 16 10 -6 3 O N LO (500 MeV) λ = 2.0 fm -1 10 -8 10 0 10 -4 -6 16 3 -1 O N LO (500 MeV) λ = 3.0 fm -8 5 s 10 0 10 5 s 10 Figure 3.7 The top panels show the relative magnitude of each term in the Magnus expansion, eq. 3.37, computed for 16 O with the EM input NN potential, N3LO(500) [5] at emax = 8 and h¯ ω = 24 MeV. The potential is SRG evolved to (a) λSRG = 2.0 fm−1 , and (b) λSRG = 3.0 fm−1 . The bottom panels show similar plots for the BCH expansion of eq. 3.39 [6]. NN interaction SRG evolved to either λSRG = 2.0 or 3.0 fm−1 . We see monotonic weakening of the magnitude of each successive term, indicating that these expansions numerically terminate at a manageable number of terms. In the Magnus expansion (top), the fourth nested commutator is sufficient for a numerical convergence criteria of 10−6 . The individual terms are larger for the harder λ = 3.0 fm−1 interaction since the system is more strongly correlated, but they systematically decrease with increasing order k. The convergence criteria for the BCH expansion (bottom) 47 is met at about nine nested commutators, with more commutators needed for harder interactions. Reassuringly, we find that the final results in our calculations are essentially independent of the convergence criteria provided εderiv 10−4 and εBCH 10−4 , where the latter is in units of MeV. For instance, raising both convergence criteria from 10−8 to 10−4 changes the ground state energy at the 1 eV level. 3.2.4 Center-of-Mass Factorization in the IMSRG The Magnus expansion offers a straightforward means to compute effective operators via the BCH expansion, and thus we can investigate the center-of-mass (COM) factorization using an effective operator Hcm (s). This problem is of critical importance in nuclear structure, as the structure of self bound nuclei is governed by a translationally invariant Hamiltonian, and thus we require factorization of the intrinsic and COM components of the wave function |Ψ = |ψ in ⊗ |ψ cm . (3.42) The many-body problem is often formulated in a spherical basis, as intrinsic conservation of angular momentum appears naturally here. A consequence of this is the loss of translational invariance due to localization of the center of mass. This is not a problem if the wave function factorizes properly, as we would only expect a degeneracy in the intrinsic spectra corresponding to each of the COM excitations. The expression of the problem in a truncated HO basis results in an artificial oscillatory behavior of the center of mass, which is often characterized by a harmonic trap ˜ Thus, COM excitations are evenly spaced by h¯ ω, ˜ and offsetting with characteristic frequency ω. the COM Hamiltonian (Hcm ) by (3/2)¯hω˜ guarantees that the COM ground state will have Ecm = 0 MeV. The COM excitations with Ecm = 0 MeV are considered spurious states, as we are only interested in the intrinsic structure of the nucleus, and the harmonic spectrum of the COM is an artifact of the truncated model space. There are two popular methods for preserving COM factorization with a spherical basis. The first is to formulate the problem in Jacobi coordinates, which are translationally invariant [138– 48 140]. This approach is limited in applicability due to factorial scaling of the antisymmetrization procedure. A second approach is to use the HO single particle basis with a truncation on the many-body basis which includes all excitations from the reference up to and including N h¯ ω. For this truncation, called the Nmax truncation, the model-space projection operator commutes with the COM Hamiltonian, and thus factorization is ensured. Again, this approach is not particularly useful for heavier nuclei, as the construction of the basis scales exponentially with the number of active nucleons. Thus, for medium-mass nuclei and beyond, we must be content with approximate center of mass factorization. The form of the COM Hamiltonian is taken to be that of a harmonic trap, ˜ = Hcm (ω) 1 3 P2 ˜ + mAω˜ 2 R2 − h¯ ω. 2mA 2 2 (3.43) For methods where factorization is analytically guaranteed, such as the NCSM, ω˜ is an arbitrary ˜ quantity. Here it can be shown that the model-space projection operator commutes with Hcm (ω) ˜ The IMSRG and similar methods are formulated in a general regardless of the choice for ω. basis with truncation on the single particle energies (emax = 2n + l), and as a result we cannot ensure rigorous factorization of the COM and intrinsic wave functions. Nevertheless, we often see oscillatory COM behavior if we probe the intrinsic wave functions with COM coordinate space ˜ where approximate COM operators. From these observations we may extract an optimized ω, factorization is typically observed. Hcm is evolved consistently with the intrinsic Hamiltonian in the IMSRG unitary transformation [6]. Calculation of ω˜ demonstrates that equation 3.43 is indeed the approximate form of Hcm , as the observed trapping frequency is approximately independent of the underlying frequency of the single particle basis. This is demonstrated in fig. 3.8, where results are shown for 14 C at two frequencies, ω˜ ± , which are solutions of a quadratic equation [141]. 2 h¯ ω˜ = h¯ ω + Ecm (ω) ± 3 4 4 (Ecm (ω))2 + h¯ ωEcm (ω) . 9 3 (3.44) In the top panel of figure 3.8, Ecm (ω) is the center of mass energy calculated for trapping frequencies equal to that of the underlying basis. It is apparent that Ecm (ω) exhibits quadratic dependence 49 14 C 10 Ecm (MeV) 8 6 Ecm(ω) Ecm(˜ω + ) Ecm(˜ω− ) NN+3N(400) emax=14 4 2 0 2 50 20 25 15 20 25 ω˜ + ω˜ − ω˜ (MeV) 40 15 30 20 10 ω (MeV) Figure 3.8 Center-of-mass diagnostics for IMSRG(2) calculations of 14 C starting from the NN+3N(400) interaction, SRG evolved to λSRG = 2.0 fm−1 . See the text for details. The calculations were done in an emax =14 model space. 50 on ω, indicating that the center of mass behaves approximately as an isotropic harmonic oscillator ˜ which corresponds to the vertex of the parabola. of frequency ω, The lower panel displays the calculated frequencies from eq. 3.44, which show approximate independence of ω, with an approximate value of 15.7 MeV/¯h, consistent with the vertex of Ecm . ˜ is approximately independent of ω, at an energy of Returning to the top panel, we see that Ecm (ω) 0 MeV, which indicates that our wave function is properly factorized, and in the ground state of the COM trap. Verification of factorization must be performed for every computed state, but the full frequency sweep and computation of ω˜ is often unneeded. ω˜ is obtained using the intrinsic ground state wave function, and as we expect all non-spurious states to have the same COM ground state, it suffices to demonstrate that Ecm ≈ 0 for each excited state. States which exhibit Ecm = N h¯ ω˜ for non-zero N are clearly spurious excitations of the center of mass. ˜ This is an indicator of strong Difficulty arises when Ecm is non-zero, nor a multiple of h¯ ω. COM contamination of an intrinsic state. In other words, the wave function does not factorize for these states. For such states, it is best to augment the intrinsic Hamiltonian with a scaled variant of eq. 3.43, constituting the Lawson-Gloeckner method [142], ˜ . Hin → Hin + β Hcm (ω) (3.45) Here the energy from COM excitations is explicitly included, but as the ground-state is defined to have Ecm = 0 MeV, this should only scale spurious states. Thus, β can be taken to arbitrarily large values, which will scale spurious states from the spectrum and increase the perturbative energy denominators for contaminated states, improving their factorization. 3.2.5 Results for Ground States of Closed Shell Nuclei This dissertation focuses primarily on the single reference IMSRG, which is most appropriate for the description of nuclei in the vicinity of shell closures, which are subject to correlations of primarily dynamic nature. The IMSRG(2) exhibits computational scaling O(N p4 Nh2 ) and is thus able to work with very large single particle bases, on the order of 15 major harmonic oscillator 51 shells. In these model spaces, even bare chiral NN interactions will show reasonably converged results for doubly magic nuclei. Fig. 3.9 demonstrates the convergence properties for the three doubly-magic nuclei 4 He, 16 O, and 40 Ca. In the left column, results are shown for the bare (λsrg = ∞) Entem and Machleidt (EM) NN interaction at N3 LO with cutoff Λ = 500 MeV [5]. In the right column are results for the same potential softened with the free-space SRG to λsrg = 2.0 fm−1 . These calculations were originally published in the review by Hergert et. al. [3]. They use a Hartree Fock reference state, and the White generator for IMSRG(2) ground-state-decoupling in the flow equation approach. The exhibited convergence properties are desirable in either case, where for each nucleus, independence from basis frequency has begun to appear by emax = 14. For each case an extrapolated result is given by the dashed line, using the prescription described in [3]. For the bare interaction, the extrapolated result is matched nicely by actual calculations for 4 He and 16 O at optimal basis frequencies h¯ ω ≈ 32 MeV. For 40 Ca, convergence in emax is not quite obtained, but the difference between actual calculations and extrapolation is minimal. SRG softening clearly improves the convergence properties of 16 O and 40 Ca, which are essentially converged outside of the high-frequency infrared regime. The convergence of 4 He with the soft interaction does not show much improvement from the bare interaction, as this system is small enough to have significantly converged calculations in either case. Some non-variational convergence patterns are evident in the softened interaction, which is a result of the IMSRG(2) truncation having invalidated the variational principle. The accelerated convergence in heavier nuclei is indicative of the fact that SRG softened potentials can be used with IMSRG to describe much heavier nuclei and also more strongly correlated open shell nuclei with multi-reference approaches. An important point is that the SRG softened interaction produces a significantly stronger binding than that of the bare interaction. As alluded to in section 3.1.3, this is because using the SRG to soften interactions shifts significant repulsive content into the induced three-body force, which is ignored here. In order to properly describe binding in these nuclei, we must at least keep the three-body forces induced from the free-space SRG. The results of doing so are shown in fig. 3.10, 52 emax -18 λ =∞ 4 He -27.9 . E [ MeV] E [ MeV] -20 -22 -24 -28.2 -28 . -28.3 -156 16 O λ = 2.0 fm−1 -158 E [ MeV] -80 -100 -160 -162 -164 -120 . . -150 λ = 2.0 fm−1 -28.1 -26 λ =∞ E [ MeV] -28 ●8 10 12 14 (c) λ =∞ 40 Ca -166 -540 λ = 2.0 fm−1 E [ MeV] E [ MeV] -200 -250 -300 -560 -580 -350 . . 16 20 24 28 32 36 40 44 48 52 h¯ ω [ MeV] -600 16 20 24 28 32 36 40 44 48 52 h¯ ω [ MeV] Figure 3.9 Convergence of 4 He, 16 O, and 40 Ca IMSRG(2) ground-state energies w.r.t. single-particle basis size emax , for a chiral N3 LO NN interaction with λ = ∞ (left panels) and λ = 2.0 fm−1 (right panels). Notice the significant differences in the energy scales between the left and right panels. Gray dashed lines indicate energies from extrapolation the emax ≥ 10 data sets to infinite basis size (see text and Refs. [7, 8]). 53 which was taken from the work of Hergert et. al., [64]. For the time being, we will focus only on the IMSRG(2) results, shown in blue. In the left column, two nucleon forces plus 3N forces induced by SRG softening are included. In the right column, both induced and SRG softened forces from a local chiral 3N potential [13] are included. While including induced forces significantly corrects for the errors in binding energy introduced by SRG evolving the NN potential, significant dependence on λsrg is still present, and for reliable binding energies, one must use large values of λsrg , which do not exhibit the pleasant convergence properties of more softened potentials (e.g. λsrg = 2.00, 1.88 fm−1 ). Inclusion of the full 3N force mostly rectifies this difficulty, where the difference between λsrg = 2.24 and 1.88 fm−1 has been reduced to roughly 1-10% of the total binding energy. Returning to the additional methods plotted in fig. 3.10, we see that IMSRG(2) is compared with coupled-cluster (CC) theory. Two variants of CC theory are shown. CC with singles and doubles (CCSD) [11, 58, 116, 143] is a truncation analogous to IMSRG(2), where correlations are included through exponential cluster operators, truncated at two-body operators, |Ψ0 = exp[T 1b + T 2b ]|Φ0 . (3.46) Here cluster amplitudes are solved for by applying the exponential operator to the Hamiltonian and exploiting the intermediate normalization of the ground-state (see sec. 2.7). Λ-CCSD(T) [116,144] includes an additional perturbative step to correct for the missing three-body cluster operator T 3b . For reference, quasi-exact importance-truncated no-core shell model (IT-NCSM) [145, 146] calculations are included where available. We note that IT-NCSM includes the full 3N force, where in IMSRG and CC theory, the 3N force is included in the NO2B approximation. Interestingly, IMSRG(2) tracks closer to Λ-CCSD(T) than CCSD. This point will be discussed in the next section. It’s evident that all of these approximate approaches produce overbinding in comparison with IT-NCSM. This is largely attributable to three components, which are the many-body truncation, the NO2B approximation and the E3max truncation, which does not contribute to the error in ITNCSM as the model-space truncation incorporates this. The NO2B approximation has been shown 54 N N + 3N -induced N N + 3N -full −25 E [ MeV] −28 −25.5 −28.5 4 . He −26 −29 −121 −128 −123 −130 E [ MeV] −119 −126 16 E [ MeV] . −125 O −132 −355 −365 −360 −370 −365 . −370 40 2.24 −375 Ca 2.00 1.88 −1 λ [ fm ] 2.24 2.00 1.88 −1 λ [ fm ] −380 Figure 3.10 Ground-state energies of stable closed-shell nuclei as a function of the resolution scale λ for the SRG-evolved NN + 3N-induced (left column) and NN + 3N-full Hamiltonians (right column), using different many-body methods: IMSRG(2) (●), CCSD ( ), Λ-CCSD(T) ( ), and IT-NCSM ( ). IMSRG and CC results are obtained with emax = 14, and the IT-NCSM results have been extrapolated to infinite model space, with error bars indicating the uncertainties of the importance truncation and the model space extrapolation (for 4 He, the error bars are smaller than the symbols). The gray solid line for 4 He in the top left panel is the result of a converged NCSM calculation with the bare NN Hamiltonian, E = −25.39 MeV (see e.g. [9]). Dashed black lines are experimental values from [10]. 55 to add about 1.0-2.5% additional binding, and is likely responsible for the bulk of the error in the approximate methods shown here. 3.2.6 Perturbative Corrections As mentioned in section 3.2.5, IMSRG(2) ground state results, despite having a more restrictive many-body truncation, track with the many-body method Λ-CCSD(T). IMSRG(2) is third-order exact in MBPT, meaning that the resummation of MBPT into the reference state energy includes all terms up to MBPT(3), along with additional incomplete resummations of all orders of MBPT [3]. The same is true of CCSD. The difference comes from the fact that the missing terms at fourth order and above are different between the two methods. Fig. 3.11 shows all fourth order Hugenholtz energy diagrams, grouped by intermediate states. CCSD contains all fourth order energy diagrams except those with intermediate triples states, whereas IMSRG(2) lacks all intermediate triples and also undercounts a class of asymmetric intermediate quadruples diagrams (Q1 , Q2 , Q6 , Q7 ) by a factor of two [3, 147, 148]. The different counting of these quadruples diagrams is responsible for the bulk of the difference between the CCSD and IMSRG(2) ground state energies. Λ-CCSD(T), on the other hand, is fourth-order exact, and thus includes all energy diagrams from MBPT(4). Errors from the undercounting of repulsive asymmetric quadruples in the IMSRG(2) apparently cancel those from missing attractive triples diagrams, as IMSRG(2) has been observed to reproduce Λ-CCSD(T) results robustly for a variety of single-reference systems. The Magnus variant of the IMSRG offers a convenient framework to formulate perturbative corrections to the IMSRG(2) [147], as the unitary transformation is explicitly constructed and then applied via the BCH expansion. Explicitly writing the first several terms of eq. 3.39 1 1 ¯ H(s) = H + [Ω(s), H] + [Ω(s), [Ω(s), H]] + [Ω(s), [Ω(s), [Ω(s), H]]] + · · · , 2 6 (3.47) we see that the induced three-body forces in this case may be approximated by Wpqrstu = [Ω(s),Y (s)] pqrstu , 56 (3.48) Singles (1p1h) S3 S1 S2 S4 Doubles (2p2h) D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 Triples (3p3h) T1 T2 T9 T10 Q1 Q2 T3 T5 T4 T11 T6 T13 T12 T14 Quadruples (4p4h) Q3 Q4 Q5 T7 T8 T15 T16 Q6 Q7 Figure 3.11 Connected Hugenholtz diagrams for the fourth-order energy correction E (4) (cf. [11]). 57 where Y(s) can be as limited as Y (s) = H or as sophisticated as 1 1 Y (s) = H + [Ω(s), H] + [Ω(s), [Ω(s), H]]] + · · · , 2 6 (3.49) where the commutators in expansion of eq. 3.49 are limited to two-body operators, similar to the BCH expansion used in the Magnus implementation of the IMSRG(2). Y (s) and Ω(s) appear at first order in MBPT, and thus the leading order component of eq. 3.48 is second-order in MBPT. From this we may immediately construct a fourth order triples correction, using Rayleigh-Schrödinger perturbation theory. The induced three-body forces from eq. 3.48 can be treated as a perturbation on the IMSRG(2) ground-state-decoupled Hamiltonian, and the first non-zero correction to the energy is |Wabci jk |2 1 , δ E{3} = ∑ i jk 36 abci ∆ jk (3.50) abc where ∆ are EN energy denominators. This correction is just an MBPT(2) correction in the dressed vertices, but constitutes a fourth order correction in the bare vertices. Diagrammatically, one can show that this expression encapsulates all fourth-order triples expressions, in the approximation that W is the full three-body force. We will refer to the IMSRG(2) with this triples correction as IMSRG({3}), where brackets indicate the approximate nature of the triples inclusion. The undercounting of antisymmetric fourth-order quadruples diagrams in IMSRG(2) originates from the neglect of induced three-body forces which feed into renormalized two-body forces later in the flow. Fig. 3.12 shows the relevant dressed two-body forces which contribute to the asymmetric quadruples diagrams. Looking at the portion of the diagrams above the dashed blue line, we see two two-body vertices which are singly contracted and thus form an intermediate three-body force at second-order in MBPT. The intermediate force is triply contracted with an another two-body vertex, which creates a dressed two-body force at third-order in MBPT. The diagram can be closed at fourth order to form an asymmetric quadruples energy diagram. These diagrams are included in CCSD due to the presence of (T2 )2 in the exponential expansion. Inclusion of these terms in the IMSRG would require the IMSRG(3) formalism, but their effect can be approximated in the spirit of eq. 3.48. In CCSD, these diagrams would first appear in 58 Figure 3.12 Induced two-body force with intermediate three-body force. Above the dashed blue line is the intermediate three-body force. The portion to the left of the dashed red line is an effective one-body insertion term. the third term (k = 2 or two nested commutators) of the BCH expansion. We can add a correction for the missing fourth order quadruples by adding them to the k ≥ 2 terms in eq. 3.39, Xk → Xk + Xk∗ , (3.51) k (H)] and X ∗ contains the missing quadruples contributions. In this context, the where Xk = k[adΩ k square vertices in fig. 3.12 refer to Ω(s) and the black circle vertex to the Hermitian intermediate, or previous (k − 1) nested commutator. We note here that there are two additional diagrams which must be computed, where the black circles and squares are exchanged in the intermediate three-body term. These additional terms are the result of the commutators in the BCH expansion. While one might think that a three-body vertex must be computed in this case, this is not actually required, because the vertices on the left side of the dashed red line constitute a one-body operator at second order in MBPT, and can be computed as an intermediate state, and then contracted with the remaining two-body vertex in the form of a one-body insertion. 59 -27 Ground State Energy (MeV) 4 He 3 -1 N LO(500) λ=2.0 fm -27.5 emax= 3 (Mag(2*)) emax= 5 emax= 7 emax= 9 emax= 9 (Mag({3}*) ) emax= 9 (Mag(2) ) CCSD Λ-CCSD(T) -28 -28.5 20 25 _ h ω (MeV) 30 Figure 3.13 Convergence of IMSRG(2*) ground state energy for 4 He with the input EM N3 LO NN potential, SRG evolved to λSRG = 2.0 fm−1 . IMSRG ground state energies are computed to emax =9 for IMSRG(2), IMSRG(2*), and IMSRG({3}*). Results from CCSD and Λ-CCSD(T) are also shown. The problem becomes the evaluation of 1 (Xk∗ )abi j = (1 − Pi j ) ∑ nk [X(k−1) ]abk j Gki − Ωabk j Dki 4 k 1 − (1 − Pab ) ∑ n¯ c Gac [X(k−1) ]cbi j Dac − Ωcbi j , 4 c (3.52) where the intermediate one-body operators are given by G pq = 1 Ωt prs Ωrstq (n¯ r n¯ s nt − nr ns n¯t ) 2∑ rst (3.53) and D pq = 1 [X(k−1) ]t prs Ωrstq n¯ r n¯ s nt − Ωt prs [X(k−1) ]rstq nr ns n¯t 2∑ rst 60 . (3.54) Ground State Energy (MeV) 16 O 3 N LO(500) λ=2.0 fm -1 -145 -150 -155 emax= 3 (Mag(2*)) emax= 5 emax= 7 emax= 9 emax= 9 (Mag({3}*)) emax= 9 (Mag(2)) -160 -165 -170 20 _ 25 hω (MeV) -50 16 Ground State Energy (MeV) -140 3 N LO(500) λ = 3.0 fm -1 -100 -150 30 O 20 30_ h ω (MeV) emax= 3 (Mag(2*)) emax= 5 emax= 7 emax= 9 emax= 9 (Mag({3}*) emax= 9 (Mag(2)) 40 50 Figure 3.14 Convergence of IMSRG(2*) ground state energy for 16 O with the input EM N3 LO NN potential, SRG evolved to λSRG = 2.0 (3.0) fm−1 in the left (right) panel. IMSRG ground state energies are computed to emax =9 for IMSRG(2), IMSRG(2*), and IMSRG({3}*). This method is denoted as IMSRG(2*) or IMSRG({3}*) in the case where it accompanies the perturbative triples correction. Fig. 3.13 shows the convergence of the IMSRG(2*) method as a function of emax and h¯ ω for 4 He using the E.M. N3 LO NN potential. The calculation uses up to emax =9, where convergence is satisfactory for purposes of demonstration. The IMSRG({3}*) and IMSRG(2) curves for emax =9 are also included. For reference, converged CCSD and Λ-CCSD(T) calculations are provided [43]. Consistent with our expectations, IMSRG(2) falls nearer to Λ-CCSD(T) than CCSD. IMSRG(2*) closely reproduces CCSD and IMSRG({3}*) closely reproduces Λ-CCSD(T). Thus, we conclude that for this case, IMSRG(2*) and IMSRG({3}*) capture much of the same perturbative content as CCSD and Λ-CCSD(T), respectively. In fig. 3.14, the same calculation is shown for 16 O with the E.M. N3 LO interaction softened to 2.0 and 3.0 fm−1 in the left and right panels respectively. Here we do not have CC results for comparison, but the same pattern is evident. With the softer interaction, the IMSRG(2*) correction removes about 2 MeV of binding energy, which is promptly reintroduced by the triples correction. For the harder interaction, convergence is not as clear, but nonetheless the same trend is observed, 61 albeit at a somewhat larger scale. 62 CHAPTER 4 EXCITED STATE EXTENSIONS After ground-state-decoupling, the IMSRG rotated Hamiltonian is block diagonal, where the reference state has been completely decoupled from all excitations. Ground state properties can then be extracted using the reference expectation value of any consistently transformed effective operators. However, if we wish to explore excited states, we must further process the remaining block of the Hamiltonian matrix, which still exhibits correlations between all levels of excitation, despite complete isolation from the reference state. While the couplings between various excitation ranks have been softened in the ground-state-decoupling, 1p1h excitations are still indirectly coupled to as much as ApAh excitations, and thus the problem of solving for excited states requires essentially the same effort as an FCI calculation using the bare Hamiltonian. There are two different strategies for the computation of excited states in the IMSRG formalism. One approach is to devise a secondary transformation to decouple blocks of different particle-hole excitation rank. If such a transformation is possible with the IMSRG(2) truncation, then diagonalizations of the particle-hole sub-blocks would provide tractable access to low lying excited states. We will show that this method is viable for a limited set of physical systems, which includes doubly magic nuclei. For these nuclei, the method performs exceedingly well for low lying 1p1h dominant odd-parity states, but poorly for states with higher-order content in their wave functions. As such, the method is often formulated in a valence 1p1h space, and is hence named the valence Tamm-Dancoff approximation IMSRG (vTDA-IMSRG). A more robust, but less elegant approach may be formulated without a second decoupling. Here we work with the ground-state-decoupled Hamiltonian, where softened couplings between the various excitation ranks facilitates the use of approximate diagonalization methods such as the second-RPA [149, 150]. The general method, named the equations-of-motion IMSRG (EOMIMSRG), is introduced in this chapter. We show that this method provides more consistent access to 1p1h states, with slightly diminished accuracy due to the approximate nature of the diagonaliza- 63 tion. However, the EOM-IMSRG is systematically improvable through the many-body truncations of both the IMSRG and EOM stages. The implementation of EOM-IMSRG is similar to what is done in CC theory [68, 69]. While the EOM-IMSRG potentially offers some technical simplifications due to the Hermiticity of the transformed Hamiltonian (e.g., no need to solve a separate left-eigenvalue problem when calculating properties other than energy), the practical limitations of the single-reference formulation should be comparable to the analogous EOM-CC calculations, limiting the method to nuclei within 1 or 2 nucleons of a closed shell. To remove these limitations, one possibility is to merge EOM techniques with the multireference IMSRG (MR-IMSRG) formulation recently developed for ground-state calculations of openshell even-even nuclei [45, 47]. In principle, spectroscopy for the target nucleus and its even-odd, odd-even, and odd-odd neighbors could then be accessed using suitably generalized EOM excitation operators. Since the full implementation of the MR-EOM-IMSRG is a significant undertaking, we first develop the single-reference EOM-IMSRG to calculate excited states in closed-shell systems as a “proof-of-principle”, before taking on the much more challenging multireference formulation. In the following we will show that the EOM-IMSRG is indeed a viable approach to target excited states, giving good agreement with analogous EOM-CC calculations for the 16 O and 22 O nuclei considered, and exhibiting systematic improvement towards the exact full configura- tion interaction (FCI) results in 2d quantum dots when perturbative triple-excitation corrections are included in our EOM calculations. We demonstrate improvement in nuclear spectra when computed with the perturbative triples correction. We also introduce extensions to the EOM-IMSRG in the form of particle removed and attached variants for neighboring isotopes and isotones, and a charge-exchange variant to compute properties of neighboring isobars. 64 4.1 4.1.1 Sequential Decoupling TDA-IMSRG At the end of the IMSRG ground-state decoupling, Hartree-Fock is made exact for the ground state of the evolved Hamiltonian. One might propose a similar treatment for excited states, supposing that a decoupling can be designed to make some simple approximation for excited states exact at the end of the evolution. We start with the Tamm-Dancoff approximation (TDA), where excited states are approximated as linear combinations of 1p1h excitations of a reference Slater determinant [95], |ΨTν DA = ∑ Xia a†a ai |Φ0 . (4.1) ai With this excitation operator, the Schrödinger equation becomes: ∑( fabδi j − f jiδab + Γa jib)X bj = ωνT DAXia (4.2) bj where ωνT DA = (EνT DA − Ere f ). For a Hartree-Fock reference state, the TDA is equivalent to diagonalizing H on the subspace spanned by |Φ0 and the singly-excited |Φai Slater determinants. The TDA neglects all ground-state correlations and higher-rank particle-hole excitations in the excited states, making it a poor approximation for Hamiltonians that feature significant coupling between the reference state and the higher particle-hole sectors, or between the 1p1h and higher excitation blocks. The initial nuclear Hamiltonian (treated in the NO2B approximation) certainly falls into this class, as indicated by the left panel in Fig. 3.2. For the ground-state-decoupled Hamiltonian in the right panel of Fig. 3.2, we note a blockdiagonal form, where couplings between individual particle-hole excitation ranks have been considerably weakened. The ground-state is completely decoupled from the excited states, and direct couplings between the 1p1h and 3p3h blocks have been suppressed to zero. Thus, we expect a ¯ TDA calculation using H(∞) to be more accurate than an analogous calculation for the initial ¯ Hamiltonian H(0) ≡ H. In fact, as ground-state decoupling removes all correlations between the reference state and 1p1h excitations, the TDA becomes equivalent to the random phase approx65 imation (RPA) [95] in this reference frame, as the only difference between the methods is the incorporation of ground-state correlations in the RPA. A secondary transformation can be devised to make the TDA exact, up to IMSRG(2) truncation errors, for a subset of excited states. Such a transformation necessarily decouples the 1p1h block from all higher rank particle-hole excitations, while leaving the decoupling from the ground state intact. In fact, such a transformation would bring the Hamiltonian to the block diagonal form seen in the left panel of Fig. 4.1. As the transformations are applied sequentially, we label the groundstate decoupling as U1 , and the secondary transformation to decouple the different particle-hole sectors as U2 . We seek to construct H¯ 21 (s) ≡ U2 (s)H¯ 1 (∞)U2† (s) = U2 (s)U1 (∞)HU1† (∞)U2† (s) , (4.3) with the relevant off-diagonal terms for U2 given in the NO2B approximation by a ¯ H¯ od : Φbc jk |H21 (s)|Φi ∼ Γic jk , Γbcak (4.4) and their Hermitian conjugates. Assuming the second IMSRG evolution converges, the transformed Hamiltonian becomes block-diagonal in particle-hole excitations a ...am a ...a Φi 1...inn |H¯ 21 (∞)|Φ 1 =0 1 i1 ...im (n = m) , (4.5) taking the schematic form shown in the left panel of Fig. 4.1. It is often useful to use the imaginarytime generator for the sequential decoupling, as the secondary stage strongly renormalizes the single particle spectra. Hereafter, we refer to this sequential decoupling as TDA-IMSRG, since the TDA is exact when applied to H¯ 21 (∞). 4.1.2 vTDA-IMSRG The simple block-diagonal structure of H¯ 21 (∞) enables the exact calculation of eigenvalues by diagonalization of each npnh block separately, with the TDA being the simplest case. However, in 66 1p1h 2p2h 3p3h 0p0h 1p1h 2p2h 3p3h 3p3h 3p3h 2p2h 2p2h 1p1h 1p1h 0p0h 0p0h 0p0h i| H¯ 21 | j i| H¯ 31 | j Figure 4.1 Schematic representation of the sequentially-decoupled Hamiltonians in the many-body Hilbert space spanned by particle-hole excitations of the reference state. The left panel corresponds to the use of Eq. 4.4 where the entire 1p1h sector is decoupled, and the right panel corresponds to Eq. 4.10 where just the valence 1v1h excitations are decoupled. The latter corresponds to the small block in the upper left corner of the full 1p1h-block. practice we find that the second U2 transformation does not always converge with respect to the flow parameter s. Even if the U2 evolution converges, the IMSRG(2) truncation errors are typically found to degrade the unitary equivalence between the initial H and H¯ 21 (∞) beyond acceptable tolerances. The breakdown of unitarity is a result of the large number of off-diagonal terms driven to zero in the second transformation, which can lead to large induced three- and higher-body terms which are neglected in the IMSRG(2). We can mitigate the effects of truncation errors in the second transformation by decoupling a smaller portion of the 1p1h configuration space in which the particle orbital is restricted to the lowest valence shell, as shown in the right panel of Fig. 4.1. This valence-space approach is denoted vTDA-IMSRG, and we denote the new transformation by U3 with H¯ 31 (s) ≡ U3 (s)H¯ 1 (∞)U3† (s) = U3 (s)U1 (∞)HU1† (∞)U3† (s) . 67 (4.6) To determine the form of H¯ od for the U3 transformation, let us denote low-lying valence particle states as av , bv , cv , . . . and high-lying non-valence particle states as aq , bq , cq , . . .. If we don’t distinguish between valence and non-valence particle states, we use the labels a, b, c, . . . as before. Performing TDA in the valence space alone modifies Eq. 4.1 to |ΨVν T DA = ∑ Xiav a†av ai |Φ0 , (4.7) av i and hence condition 4.5 is reduced to av ¯ Φbc jk |H31 |Φi = 0 , (4.8) aq Φ j |H¯ 31 |Φai v = 0 . (4.9) with the additional requirement These two conditions are met if we choose H¯ od : Γaq i jav , Γic jk , Γbcav k , faq av . (4.10) This definition of the off-diagonal terms is significantly reduced in scope from that of Eq. 4.4, so we expect that the loss of unitarity caused by the IMSRG(2) truncation should be less severe. The right panel of Fig. 4.1 shows the schematic form of a successful vTDA-IMSRG(2) decoupling. A vTDAIMSRG(2) calculation will not leave the Hamiltonian block diagonal for all excitation ranks, and will limit the number of states accessible to the calculation. However, if we are interested in only low-lying states, this calculation is much more stable than the full TDA-IMSRG(2). We note here that both TDA-IMSRG(2) and vTDA-IMSRG(2) are conceptually similar to the similaritytransformed EOM-CC method, reviewed recently in [151]. 4.1.3 Full Configuration Interaction Analysis of TDA-IMSRG(2) and vTDA-IMSRG(2) The sequential TDA-IMSRG decoupling is designed so that FCI in the space of 1p1h configurations is exact for a set of low lying eigenvalues. As both IMSRG decouplings are carried out using 68 the IMSRG(2) truncation, we expect some deterioration of the unitary equivalence between the initial, ground-state-decoupled, and sequentially-decoupled Hamiltonians due to the neglect of induced three- and higher-body forces. The breakdown of unitarity can be understood and quantified by performing FCI calculations at various values of s during the flow. In the limit of an IMSRG(A) calculation with no truncations, we expect energies computed with FCI to be s-independent. Thus, in an IMSRG(2) calculation, we can measure the truncation errors by analyzing the s-dependence of each energy level as computed by FCI. In the left panel of fig. 4.2, FCI behavior is demonstrated as a function of s for a few lowlying energy levels of a 6-electron quantum dot, with single-particle basis truncated to four major oscillator shells. The FCI calculations use the evolved Hamiltonian at intermediate steps in the sequential decoupling, defined by eq. 4.3. The first stage of the IMSRG evolution is the groundstate-decoupling, and the second stage decouples the particle-hole blocks. The vertical black line in Fig. 4.2 indicates the endpoint of the ground-state decoupling U1 , and the beginning of the secondary 1p1h decoupling U2 . The ground state and first (ML , MS ) = (1, 0) excited state show weak dependence on s in either transformation, indicating that the loss of unitarity from the IMSRG(2) is small. A more pathological behavior is uncovered by zooming in on the second and third excited states, as seen in the right panel of Fig. 4.2. For reference, FCI results computed with the initial Hamiltonian are indicated by horizontal dotted lines. In the second stage, dashed lines show FCI results for the flowing H¯ 21 (s) during TDA-IMSRG(2), and solid lines correspond to the analogous vTDA-IMSRG(2) calculations using H¯ 31 (s) where the valence-space used for the vTDA-IMSRG calculation includes all 1p1h excitations into the third oscillator shell. The U3 evolution shows limited s-dependence, with small errors induced at the beginning of the flow. The calculation is converged numerically around s ≈ 11.5. Conversely, the U2 evolution induces larger errors from the outset, and continues to show s-dependence as the flow continues, exhibiting undesirable features such as erratic convergence patterns and level crossings. These results are consistent with our intuition that IMSRG(2) truncation errors should be smaller for the valence 1p1h decoupling, 69 21.5 21.3 FCI (H) FCI (H21) FCI (H31) FCI (H21) 20.5 Ground State (0,0) Energy (Hartree) Energy (Hartree) (1,0) 21 (1,0) 21.28 21.26 (1,0) 21.24 U1(s) 20 0 U2(s) 10 U1(s) 20 Flow Parameter (s) 30 0 Ux(s) 10 20 Flow Parameter (s) 30 Figure 4.2 (ML , MS ) = (1, 0) excited states of a 6-electron quantum dot in an ω = 1.0 trap model space composed of four major oscillator shells. Three FCI calculations are shown; two using the flowing TDA-IMSRG(2) and vTDA-IMSRG(2) Hamiltonians corresponding to each choice of the secondary decoupling Ux (s) (where x = 2 or 3), and one using the bare Hamiltonian. as fewer matrix elements are driven to zero in this framework. In fact, the behavior of the U2 transformation in this specific case is relatively pleasant; we find that in larger systems, the pathological behavior of the TDA-IMSRG(2) decoupling typically leads to numerical instabilities and thus non-convergent energies. It is for this reason that we consider the vTDA-IMSRG(2) to be the preferred sequential decoupling method. In fig. 4.3, we show the practical benefits of the vTDA-IMSRG(2), where sequential decoupling is used to make TDA exact in the rotated frame. This calculation is performed for the first (1, 0) state of the 6-electron quantum dot. For comparison, we have plotted the flowing and bare FCI calculations for both the ground and first excited state, as well as the 0-body component of the Hamiltonian Ere f (s). At s = 0, Ere f (0) corresponds to the Hartree-Fock energy and ET DA (0) to a traditional TDA calculation limited to the third oscillator shell. These are rather poor approximations to reality, as they fail to incorporate couplings to higher order configurations, which are critical in correlated systems such as quantum-dots and nuclei. As s increases, successively more many-body correlations are resummed into the relevant pieces of the flowing Hamiltonian, and thus the corresponding Ere f (s) and ET DA (s) become better approximations to exact results. At 70 21.5 Eref(s) ETDA(s) FCI (H31) FCI (H) Energy (Hartree) 21.25 21 (1,0) 20.75 20.5 20.25 0 (0,0) U1(s) U3(s) 2 4 6 8 Flow Parameter (s) 10 Figure 4.3 FCI and TDA calculations of the first (ML , MS ) = (1, 0) excited state using the flowing vTDA-IMSRG(2) Hamiltonian for a 6-electron quantum dot with N = 3 and ω = 1.0. For reference, the flowing 0-body part of the Hamiltonian Ere f (s) and the FCI results for the bare Hamiltonian are also shown. the completion of ground-state-decoupling, Ere f (s ≈ 5.4) is in excellent agreement with the exact ground state, while ET DA (s ≈ 5.4) is significantly improved compared with its initial counterpart. As the secondary decoupling progresses, ET DA (s) approaches the flowing FCI calculation, which at this point has deviated little from the bare result. Thus ET DA (s) is now a very good approximation of the exact ground state, with a much more modest computational demand. This method is powerful because of its scalability. The time-complexity of vTDA-IMSRG(2) is dominated by the IMSRG(2) equation, and thus behaves as O(N p4 Nv Nh ), which is far more manageable than the factorial scaling of FCI. Another important feature of this sequential decoupling is that Ere f (s) is independent of s during the second stage, indicating that the secondary decoupling does not re- 71 introduce couplings between the reference state and particle-hole excitations. The computational benefits of this approach come at a cost, as the valence space severely limits the number of excited states which one may describe with this method. Thus we turn to the equations-of-motion method for a less elegant but more general approach to excited states. 4.2 Equations-of-Motion Method The sequential decoupling seeks to make simple methods exact, by suppressing the matrix elements which are responsible for the errors in these methods. While this strategy is intuitively pleasing, methods such as the TDA- and vTDA-IMSRG(2) cannot circumvent the problems associated with maintaining unitary equivalence, particularly when they are tasked with decoupling a large space of excited states from the rest of the Hamiltonian. We showed that one can decrease the number of matrix elements we must suppress by defining a valence space for vTDA-IMSRG, but if we want to describe higher energy excited states, vTDA-IMSRG(2) quickly becomes unpractical because of the increased size of the valence space required to encapsulate these states. These limitations can be avoided if we pursue a third strategy where we apply higher-order equations-of-motion techniques to approximately diagonalize the ground-state-decoupled Hamiltonian without the need for a second transformation. Equations-of-motion (EOM) methods [71] describe a class of systematically improvable methods, which include the TDA and RPA. The key observation in the formulation of EOM is that any excited state can be exactly rewritten in terms of a ladder-operator Xν† and the correlated ground state |Ψν = Xν† |Ψ0 . (4.11) Xν† is formally given by the dyad |Ψν Ψ0 |, and can be written as a linear combination of 1- to Abody excitation and deexcitation operators. The energy eigenvalue problem can then be expressed in terms of the commutator of H and Xν† : [H, Xν† ]|Ψ0 = ων Xν† |Ψ0 , 72 (4.12) where ων = Eν − E0 is the excitation energy. The strength of EOM methods, when applied to the bare Hamiltonian, lies in the ability to make controlled, computationally feasible approximations on the form of Xν† . Given some approximation of the correlated ground state, the amplitudes of Xν† can be solved for in a generalized eigenvalue problem [95]. In principle, the approximate ground state can then be improved iteratively using the Xν† and its Hermitian conjugate, which can then be used to get an improved Xν† , and so on. We can eliminate the need to solve the equations-of-motion self-consistently with the IMSRG. After ground-state-decoupling, the reference state |Φ0 corresponds to the true ground state of H¯ 1 ≡ U1 (∞)HU1† (∞), and thus need not be improved. Multiplying Eq. 4.12 by U1 (∞) and using the rotated ground state U1 (∞)|Ψ0 = |Φ0 gives [H¯ 1 , X¯ν† ]|Φ0 = ων X¯ν† |Φ0 , (4.13) where X¯ν† ≡ U1 (∞)Xν†U1† (∞) only contains excitation operators since the reference state is annihilated by deexcitation operators. We recover the TDA equations for the ground-state-decoupled Hamiltonian if we choose X¯ν† = ∑ X¯ia a†a ai . (4.14) ai Alternatively, we may use a more complete ladder operator which includes up to 2p2h excitations, 1 † † X¯ν† = ∑ X¯ia a†a ai + ∑ X¯iab j aa ab a j ai . 4 ai abi j (4.15) The consequence of the increased precision introduced with eq. 4.15 is a more complicated eigenvalue problem than that of the TDA, but it significantly reduces the need for a second transformation as it includes a large portion of the correlations which are suppressed by U2 or U3 . If we return to fig. 3.2, we see that in addition to the complete decoupling of the reference state, IMSRG(2) ground-state-decoupling also achieves a significant softening of the excited state block, and thus an EOM method such as that specified by eq. 4.15 will be much more successful here than with the bare operator, which includes direct couplings between the 1p1h and 3p3h channels. Note that the EOM calculation with this ladder operator is equivalent to diagonalizing H¯ 1 on the 73 space of singly- and doubly-excited Slater determinants. This is a configuration interaction calculation with singles and doubles (CISD) in disguise. However, this is only due to the fact that the true ground state is a simple Slater determinant in this rotated frame. In general, the EOM ladder operator may have any excitation rank up to ApAh, which would constitute an exact diagonalization of H¯ 1 . Similarly, the truncation rank of the IMSRG equations can in principle be increased to the IMSRG(A) level, where the unitary equivalence of H¯ 1 and H is exact. Therefore, EOM-IMSRG approximations are systematically improvable, allowing for EOM(m)-IMSRG(n) calculations, which will simply be referred to as EOM-IMSRG(m,n). The calculations in the present work are carried out in the EOM-IMSRG(2,2) approximation. As a result of the vanishing de-excitation piece of X¯ν† , Eq. 4.13 has the advantage that it may be solved as a traditional eigenvalue problem using power-iteration methods such as the Lanczos algorithm. Such methods only require knowledge of matrix-vector products. If X¯ν† is taken to be an eigenvector, the corresponding matrix-vector product is given by [H¯ 1 , X¯ν† ] = {H¯ 1 X¯ν† }C (4.16) where the subscript C denotes connected terms. Thus, pre-packaged Lanczos algorithm solvers can be used with the EOM-IMSRG, where the matrix-vector product is just the evaluation of the commutator expressions, and thus the method scales the same as IMSRG(2) ground-state-decoupling: O(N p4 Nh2 ). Note that for nuclei, ladder operators are spherical tensors of rank J with definite parity, as they must connect the ground state to excited states of any desired spin J π . For this reason, EOM-IMSRG calculations are more computationally demanding than TDA-IMSRG and vTDAIMSRG calculations. However, the relatively small rotation of the ground-state decoupling U1 makes the EOM-IMSRG equations far more numerically stable in comparison to the sequential decoupling approaches, which require a large secondary rotation U2,3 . 74 4.2.1 Improving the EOM-IMSRG(2,2) By the nature of the ground-state-decoupled Hamiltonian, we expect some coupling between the 1p1h and 3p3h channels simply because those blocks are transitively related through the 2p2h block. Thus, description of 1p1h dominant states can be largely improved by going to the next truncation level in EOM, the EOM-IMSRG(3,2). Additionally, inclusion of triples excitations will be imperative for the description of 2p2h dominant states, which are still directly coupled to the 3p3h block in the ground-state-decoupled frame. The appropriate ladder operator is 1 1 † † † X¯ν† = ∑ x¯ia a†a ai + ∑ x¯iabj a†a a†b a j ai + ∑ x¯iabc jk aa ab ac ak a j ai . 4 36 ai abi j abci jk (4.17) The solution of eq. 4.13 using eq. 4.17 can be achieved with a modest increase in computational scaling, O(N p5 Nh3 ), but doing so requires the storage of a three-body operator for each desired excited state. This is not practical computationally, particularly in the case of nuclear excited states with large total angular momentum, where a significant number of matrix elements can be non-zero. We can instead treat the problem with Rayleigh-Schrödinger perturbation theory. The order zero wave function is taken to be the solution of the EOM-IMSRG(2,2), (0) ˜ν |Ψ ¯ ν = X¯ν† |Φ0 . = |Ψ (4.18) Using Epstein-Nesbet partitioning of the Hamiltonian, the zero order energy is (0) Eν = E0 + ων , (4.19) and the first order energy correction is zero by definition. The second order energy correction is then given by Qˆ (2) ˜ (0) ˜ (0) ¯ H¯ 1 |Ψ Eν = Ψ ν , ν |H1 (0) (0) ¯ Eν − H (4.20) where Qˆ is the complement space projector Qˆ = |Φ0 Φ0 | + ¯µ Ψ ¯ µ | + 1 ∑ |Φabc Φabc | + · · · |Ψ i jk 36 i jkabc i jk µ=ν ∑ 75 (4.21) ¯ν Ψ ¯ ν | projects onto the particular solution of the EOMˆ where Pˆ = |Ψ Note that Qˆ = 1 − P, IMSRG(2,2) for which we are calculating the perturbative correction. Since couplings between ¯ ν and the reference state or npnh excitations with n ≥ 4 are zero in a ground-state-decoupled |Ψ ¯ µ=ν vanish due to the approximate diagonalization perframework, and since couplings with |Ψ formed in the EOM-IMSRG(2,2) calculation, the triply-excited terms of Eq. 4.21 give the only non-vanishing contribution and Eq. 4.20 becomes (2) Eν abc ¯ ¯ † Φ0 |X¯ν H¯ 1 |Φabc 1 i jk Φi jk |H1 Xν |Φ0 . = ∑ (0) 36 i jkabc ¯ 1 |Φabc | H Eν − Φabc i jk i jk (4.22) Eq. 4.22 gives a perturbative energy correction that approximates the full EOM-IMSRG(3,2) energy. In practice, we write Eq. 4.22 as: (2) Eν |Wabci jk |2 1 = , ∑ i jk 36 i jkabc D (4.23) abc where i jk ¯ abc Dabc = ων − Φabc i jk |H|Φi jk , (4.24) ¯ X¯ν† ]abci jk . Wabci jk = [H, (4.25) and Storage of three-body matrix elements is not needed as Eqs. 4.23-4.25 need only be calculated once for each excited state with manageable N p4 Nh3 scaling. In the following, the inclusion of perturbative triples on top of EOM-IMSRG(2,2) will be referred to as the EOM-IMSRG({3},2) approximation. 4.2.2 Comparison of the EOM-IMSRG and vTDA-IMSRG via Quantum Dots Fig. 4.4 shows vTDA-IMSRG(2), EOM-IMSRG(2,2) and EOM-IMSRG({3},2) (labeled TDA31 , EOM1 , EOM{3}1 , respectively) spectra for two different quantum dots, along with FCI calculations performed for the bare Hamiltonian (FCI0 ) and FCI calculations using the ground-statedecoupled Hamiltonian H¯ 1 (FCI1 ). The length of the lines indicate the 1p1h content of a given 76 2.2 (b) ω = 0. 5 (a) ω = 1. 0 1.0 (ML ,MS ) (0,0) (1,0) 1.6 (2,1) (3,0) Excitation Energy (Hartree) Excitation Energy (Hartree) 1.9 1.3 1.0 0.8 0.6 0.4 0.7 TDA 31 FCI 0 FCI 1 0.2 EOM 1 EOM{3} 1 TDA 31 FCI 0 FCI 1 EOM 1 EOM{3} 1 Figure 4.4 Selected excitation spectra of 6-electron quantum dots for ω = 1.0 (left) and ω = 0.5 (right) performed in an N = 3 single-particle basis. The quantum numbers of the various states are color-coded as (ML ,MS ) = (0,0), (1,0), (2,1), (3,0). The calculated spectra are displayed for five different many-body approaches, where the subscript indicates which Hamiltonian the respective method is applied to. For example FCI0 and FCI1 denote FCI calculations on the bare and ground-state-decoupled Hamiltonians respectively, TDA31 denotes a TDA calculation on the vTDA-decoupled Hamiltonian, etc. The lengths of the plotted energy levels indicate the 1p1h content of the state as defined in Eqs. 4.26-4.29. state which we define as p n(1p1h)FCI0 = ∑ |Ch |2 (4.26) ph p n(1p1h)FCI1 = ∑ |(C¯1 )h |2 (4.27) p n(1p1h)EOM1 = ∑ |(X¯1 )h |2 (4.28) n(1p1h)T DA31 = ∑ |(X¯31 )vh |2 = 1 . (4.29) ph ph vh 77 Note that this quantity is defined differently depending on the particular unitary transformation, so a direct 1-to-1 comparison can be misleading. For instance, in the vTDA-IMSRG calculations, the excited states are completely 1p1h in the unitarily-transformed frame, hence all of the lines in the TDA31 column are the maximum possible length. Since FCI1 and EOM1 are performed for the same operator H¯ 1 , a direct comparison of Eq. 4.28 and 4.27 is consistent. We note that EOM-IMSRG({3},2) partial norms are corrected by normalizing with the wave function corrected to first order in perturbation theory, resulting in a slight decrease. In Fig. 4.4 we show four sets of states with the indicated quantum numbers chosen to demonstrate the robustness of the EOM-IMSRG method. For odd parity states such as (ML ,MS ) = (1,0) and (3,0), we see that those which are strongly of 1p1h nature are well-described by both vTDAIMSRG and EOM-IMSRG methods. We also note that the EOM-IMSRG reproduces the FCI1 partial norms nicely for these states, indicating that the EOM-IMSRG(2,2) is a good approximation to the full diagonalization of H¯ 1 . The EOM-IMSRG spectra degrades somewhat for even parity states, since the sizable shell gap at the Fermi level tends to suppress the 1p1h dominance for such states, and at higher excitation energies. However, it bears repeating that the EOM-IMSRG is significantly more flexible than the vTDA-IMSRG, as the latter is intrinsically unable to access even parity and/or higher excited states without expanding the model space to include the entire 1p1h configuration space, which often leads to numerical instabilities and/or erratic convergence. While the EOM-IMSRG is only an approximate diagonalization of the evolved Hamiltonian, it can be systematically improved. EOM-IMSRG({3},2) corrections significantly reduce the errors in the EOM-IMSRG(2,2) approximation at a manageable computational cost. Excitation energies, which are consistently overestimated by the EOM-IMSRG(2,2) calculation, are consistently reduced by the perturbative triples correction, bringing results into better agreement with the FCI1 and FCI0 spectra. The quality of EOM-IMSRG({3},2) energies is still dependent on higher excitation rank content, but 1p1h states are described quite sufficiently by this approximation. The error in 2p2h dominant states is significantly reduced, but this block couples directly to the 3p3h block, so a perturbative treatment may be uncontrolled here. 78 Difference from FCI (Hartree) Difference from FCI (Hartree) TDA31 EOM1 EOM{3}1 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 n(1p1h)FCI 0.8 0.2 0.15 0.1 0.05 0 0 1 EOM1 EOM{3}1 0.25 0.2 0.4 0.6 n(1p1h)EOM 0.8 1 1 0 Figure 4.5 Absolute difference between quantum dot excitation energies calculated via IMSRG methods and those calculated with FCI on the bare Hamiltonian. Each point corresponds to an EOM or TDA energy level in Figure 4.4. The left panel is plotted against FCI 1p1h content, while the right is plotted against the EOM 1p1h amplitudes. The quality of IMSRG results degrades as the importance of correlation in the system increases. This is seen clearly in the right panel of Fig. 4.4 for the smaller trap frequency ω = 0.5. Nevertheless, the perturbative triples correction still gives substantial improvement. A strong correlation between the errors of either method and the bare FCI 1p1h amplitudes is apparent. The right panel of fig. 4.5 shows the absolute difference between the FCI0 excitation energy and those calculated via EOM-IMSRG(2,2) or vTDA-IMSRG(2), plotted against the bare FCI partial norm for each state. A clear inverse proportional relationship can be seen. Accessible vTDA-IMSRG(2) results are generally more accurate than EOM-IMSRG(2,2). While a successful vTDA-IMSRG calculation should fully decouple the relevant excited states from truncated terms in the configuration expansion, where EOM-IMSRG(2,2) ignores some non-zero couplings by definition. Despite full decoupling, the vTDA-IMSRG(2) should induce additional errors due to the secondary transformation. Evidently, the errors from the EOM(2) truncation are typically greater than those of the vTDA-IMSRG(2) secondary transformation. However, this difference is for the most part erased by the EOM-IMSRG({3},2) triples correction. The root-mean square deviations from the FCI results are 0.095 Hartree for EOM-IMSRG(2,2) and 0.031 Hartree for EOM-IMSRG({3},2). 79 In larger spaces, FCI calculations are not feasible, so we should also consider the relationship between the error and the EOM1 partial norm of Eq. 4.28. The right panel of fig. 4.5 demonstrates this relationship, using the same states displayed in Fig. 4.5. Here it is evident that EOM-IMSRG overestimates the 1p1h content of calculated states, however there is a modestly linear relationship between the error and the EOM partial norm. This is a useful tool to gauge the reliability of EOMIMSRG calculations in larger spaces, as EOM amplitudes are immediately available after solution of Eq. 4.12. 4.2.3 Results in Finite Nuclei Applying both the vTDA-IMSRG(2) and EOM-IMSRG(2,2) to finite nuclei, we find a clear preference for the latter method. Unfortunately, the promising results of vTDA-IMSRG(2) calculations in quantum dots do not carry over to nuclei, as uncontrolled numerical instabilities in the secondary transformation render the vTDA-IMSRG unusable for systems with strong correlations. Until these instabilities are better understood and overcome, sequential decoupling appears to be appropriate only for computations in doubly-magic nuclei. Figure 4.6 depicts the lowest excitation energies of 16 O calculated at several different angular momenta and parities. We find that the vTDA-IMSRG(2) tracks well with EOM-IMSRG(2,2) for low-lying 1p1h dominant states, but is non convergent for all others. In the left column, the 1p1h partial norm (Eq. 4.28) of the EOM wave function is listed above the corresponding energy for that state. While the 0+ state has strong 1p1h content in the EOM calculation, the vTDA-IMSRG(2) fails to converge beyond the three lowest excited states1 . Strongly 1p1h even parity states in doubly-magic nuclei require the valence space to contain two-major shells above the Fermi level, hence the convergence difficulties. In nuclei with sub-shell closures such as 22 O, the vTDA-IMSRG(2) fails to converge even for most low-lying 1p1h dominant states. This is most likely due to the presence of stronger correlations, 1 We are not attaching any physical meaning to states at such unphysical high excitation energies. Rather, our point is to illustrate that obtaining converged, stable calculations in the EOMIMSRG is relatively foolproof for a wide range of states, whereas the vTDA-IMSRG calculations are fraught with numerical difficulties. 80 45 3+ 40 1+ Energy (MeV) 35 2+ 30 0+ 0.5175 0.7875 0.7659 0.8842 25 20 15 10 0.9190 20.9198 10.9323 3- EOM 1 TDA 31 Figure 4.6 Lowest 16 O excitation energies plotted for various quantum numbers, calculated with EOM-IMSRG(2,2) and vTDA-IMSRG(2) starting from the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 2.0 fm−1 . The single-particle basis is given by h¯ ω = 24.0 MeV and emax = 8. Above each plotted energy level from the EOM-IMSRG(2,2) calculation is the 1p1h partial norm of Eq. 4.28. which exacerbate the issue of unitary breakdown. For this reason, we will restrict ourselves to the EOM-IMSRG(2,2) formalism in the remainder of this chapter. As we restrict the single-particle basis by orbital energy rather than excitation quanta, there is no a priori factorization of center of mass components of the wave function for self-bound systems such as nuclei. Nonetheless, we expect approximate factorization and thus spurious COM excitations are treated via the Lawson-Gloeckner method [142], with an augmented intrinsic Hamiltonian ˜ , H = Hint + β HCM (ω) 81 (4.30) where ω˜ is determined with the method of Hagen et. al. [6, 152]. Assuming that the intrinsic and COM wave functions factorize, the use of the Lawson term in Eq. 4.30 should remove spurious excitations from the low-lying spectrum as β is increased. An example is shown in Figure 4.7 for the 1− spurious state in 22 O, which gets shifted rapidly out of the spectrum for non-zero values of β . The weak residual β dependence of the remaining states indicates that the COM factorization 12 1+ Energy (MeV) 10 8 6 3+ 4 0 2+ 1- 2 0.0 1.0 2.0 3.0 βCM 4.0 5.0 Figure 4.7 Low lying states of 22 O at h¯ ω = 28.0 MeV and emax = 11 for several values of the Lawson parameter β , using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 3.0 fm−1 . The COM frequency h¯ ω˜ = 17.28 MeV. is approximately satisfied for these states. We expect this factorization to improve (weaker β dependence) as we go to higher excitation levels and larger bases, as has been empirically observed in [152, 153]. An important litmus test for the EOM-IMSRG(2,2) method is the ability to produce results that are comparable to analogous EOM-CC calculations. As with ground-state coupled cluster theory, EOM-CC methods originated in nuclear physics [73, 74, 154], but were largely ignored for many years due to convergence issues arising from the “hard cores” found in most NN potential models, 82 (a) 32- 10 (b) 1+ 10 0+ 8 Energy (MeV) 12 1+ 8 Energy (MeV) 12 6 H = Hint 4 6 H = Hint + βHCM (ω˜) 4 3+ 3+ 2 2 2+ 1- 2+ 0 EOM-IMSRG(2,2) EOM-CCSD EOM-IMSRG(2,2*) 0 EOM-IMSRG(2,2) EOM-CCSD EOM-IMSRG(2,2*) Figure 4.8 Selected excitation spectra of 22 O at h¯ ω = 20.0 MeV and emax = 11 using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 2.0 fm−1 . The left frame shows the excitation energies calculated with the intrinsic Hamiltonian, and the ˜ with right frame shows the result of adding a Lawson center-of-mass term H = Hint + β HCM (ω), Π β = 5.0. Different colors indicate different J . while going on to enjoy great success in quantum chemistry [11, 76]. In recent years, EOM-CC methods have had a resurgence in nuclear physics due to the development of softer chiral EFT interactions and RG methods to soften them further, providing unprecedented access to ab initio calculations of ground and excited state properties for medium-mass nuclei in the vicinity of closed shells [42, 59, 68, 69, 93, 144]. Due to similar truncations being made, we expect the EOM-IMSRG(2,2) to produce results that are comparable to EOM-CCSD calculations. As mentioned in section 3.2.6, IMSRG(2) ground state calculations tend to more closely reproduce the more sophisticated CCSD(T), due to an 83 12 12 (a) 10 (b) 10 1+ 1+ 8 Energy (MeV) Energy (MeV) 8 6 H = Hint 4 6 H = Hint + βHCM (ω˜) 4 3+ 2 3+ 2 2+ 2+ 10 EOM-IMSRG(2,2) EOM-CCSD EOM-IMSRG(2,2*) 0 EOM-IMSRG(2,2) EOM-CCSD EOM-IMSRG(2,2*) Figure 4.9 Selected excitation spectra of 22 O at h¯ ω = 28.0 MeV and emax =11 using the N3 LO (500 MeV) NN interaction of EM [5], softened by free-space SRG evolution to λ = 3.0 fm−1 . The left frame shows the excitation energies calculated with the intrinsic Hamiltonian, and the right frame shows the result of adding a Lawson center-of-mass term H = Hint + β HCM , with β = 5.0. Different colors indicate different JΠ . undercounting of asymmetric quadruple diagrams at MBPT(4). Understanding how each specific resummation of MBPT affects excited states would require an analytical treatment in quasidegenerate perturbation theory, which is beyond the scope of this work. We will content ourselves with empirical investigations, using both the IMSRG(2) and IMSRG(2*) ground-state-decoupled Hamiltonians in our exploration. In Figure 4.8 we show calculations of the low-lying spectra of 22 O performed on the intrinsic Hamiltonian, as well as the Lawson Hamiltonian with β =5.0. In each panel, the left column shows the excited states calculated in the standard EOM-IMSRG(2,2) approximation, while the right column shows the quadruples corrected EOM-IMSRG(2,2*). 84 On a technical note, COM frequencies ω˜ are calculated independently for the IMSRG(2) and IMSRG(2*) methods, and corresponding Lawson terms are constructed. The Lawson term is constructed in the CCSD calculations using the frequencies calculated in the IMSRG(2*) formalism, which we expect to be a good approximation given the similar perturbative content of both methods. The relevant frequencies are given in table 4.2.3. The observed removal of spurious centerof-mass excitations is consistent in all three approaches. Figure 4.9 displays a similar comparison for a “harder” interaction at λ = 3.0 fm−1 . Differences between the EOM-IMSRG(2,2*) and CCSD are more notable here, but qualitative agreement is still intact. The EOM-IMSRG(2,2) excitation energies are generally shifted up from their CCSD counterparts. This is indicative of the increasing effect of missing 4th-order quadruples for harder interactions and is consistent with observations of increasing differences between IMSRG(2) and CCSD ground-state energies for interactions with larger λ values [3]. To reproduce experimental spectra, we include an N2 LO 3N force fit at Λ = 400 MeV [13], which has had some success in the ab initio description of excited states [46, 51, 155]. Fig. 4.10 shows the computed spectra for closed shell nuclei 14 C and 22 O, using the above interaction. Results are compared with experiment. Here spurious states have been identified and removed from the spectra. Here much of the experimental spectrum is poorly reproduced by EOM-IMSRG(2,2). We see many of the states in 14 C are missed, and the calculated 2+ 1 state appears to align with the exper+ imental 2+ 2 state. We are unable to rule out the possibility that we miss the first experimental 2 state entirely due to low 1p1h content, as we only have energetics to draw comparisons with. This Method IMSRG(2) IMSRG(2*) CCSD λ = 2.0 fm−1 18.19 18.05 18.05 λ = 3.0 fm−1 17.28 17.50 17.50 Table 4.1 Values of h¯ ω˜ in MeV, used in center-of-mass Hamiltonian for Lawson calculations of energy spectra. 22 O 85 10 Energy (MeV) 8 2+ 6 1+14+32+ 14 2+ 2+23- 00+ 1- 10 4 2 0 EOM-IMSRG(2,2) 10- 12 Energy (MeV) 11+ 8 1+ 2+ 3+ 4+ (0-,1-,2-)? 6 4 3+ 2+ 2 14 C 0 EOM-IMSRG(2,2) Exp 4+ 2+ ? 0+ 3+ 2+ 22 O Exp Figure 4.10 Low lying spectra of 14 C and 22 O at h¯ ω = 20.0 MeV and emax =14 using the standard input NN+3N(400) interaction, softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. calculation can surely benefit from the inclusion of a triples correction and analysis of observables. Matching with experimental states will be discussed in more detail in chapter 5. The picture is less obscure in 22 O, where given the assumption that the nucleus is appropriately described by a single-reference state, parity arguments give us confidence that the low-lying 2+ 1 and 3+ 1 are largely composed of single excitations within the sd-shell. Accordingly, EOM-IMSRG(2,2) reproduces experiment for these states fairly well. Higher energy spectra are not produced particularly well, which is expected as the higher energy states certainly contain more 2p2h and higher content than the first two excited states. Again, this study can be greatly facilitated by inclusion of a triples correction, which we would expect to be small for the first two excited states. 86 18 8 16 2- 1+14+32+ 2+ 14 4+ 1012 2+ 2+23-00+ 1- 6 Energy (MeV) Energy (MeV) 2312 011+ 10 10 8 1+ 2+ 3+ 4+ 6 4 4 3+ 2+ 2 14 C 2 0 EOM-IMSRG(2,2) ({3},2) 0 EOM-IMSRG(2,2) ({3},2) Exp (0-,1-,2-)? 4+ 2+ ? 0+ 3+ 2+ 22 O Exp Figure 4.11 Triples correction for low lying spectra of 14 C and 22 O at h¯ ω = 20.0 MeV and emax =8 using the N3 LO (500 MeV) NN interaction of EM [5] with N2 LO 3N force (400 MeV) [13], softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. 4.2.4 Perturbative Corrections for Nuclei While the EOM-IMSRG({3},2) correction is a rather straightforward computation for manageable systems such as parabolic quantum dots, the correction is much more computationally challenging for a strongly correlated nucleus. The correction must be implemented in J-scheme (see appendix), and thus requires numerous sums over intermediate couplings to total angular momentum quantum number, along with the already taxing O(N p4 Nh3 ) scaling. For this reason we are unable to perform extensive EOM-IMSRG({3},2) calculations with the large model spaces used in the previous calculations. However, as a demonstration of the method, results for an emax =8 model space are shown in Fig. 4.11. Here we compute the same states as in fig. 4.10, but we use a smaller model space for the purposes of computational feasibility. As calculations are performed at the 87 optimal basis frequency (¯hω = 20 MeV) for these systems, EOM-IMSRG(2,2) calculations are reasonably converged despite the small model space. The EOM-IMSRG({3},2) correction shifts all of the states down, as constituents of the excluded-space of triply excited Slater-determinants do not exhibit any near degeneracies with the low-lying EOM-IMSRG(2,2) spectra. Strongly 1p1h + 22 states such as the 2+ 1 and 31 states in O undergo smaller corrections than higher energy states, which couple more strongly to the 3p3h block through their higher-order many-body content. The 14 computed 2+ 1 state in C experiences a shift which moves it closer to its experimental counterpart, whereas previously it aligned more closely with the experimental 2+ 2 state. Likewise, numerous states which appear at high energy are moved down significantly by the inclusion of the triples correction, in better correspondence with the experimental spectra. The larger shifts seen in oddparity states indicate significant content beyond 1p1h, and thus it is likely that for a perturbative triples treatment, additional IMSRG evolution will be needed to soften couplings to higher-order content. A curious case is presented by the 0− state for both nuclei. The shift is quite large compared with the other odd-parity states, and in 22 O, the triples result (not shown to reduce clutter) gives a negative excitation energy for the 0− state. Another issue is the observation that in 22 O, the EOM-IMSRG(2,2) results for the two lowest states, seemingly consistent with experiment, are shifted further from experiment by their triples corrections. At the moment, these results are not thoroughly understood, but the discrepancy may be attributable to several sources. The interaction may contribute to the overbinding in these states, leading to slightly lower energies. Work is in progress to understand the nature of these errors. Overall, the EOM-IMSRG({3},2) is inconsistent in that it treats the IMSRG calculation with a two-body truncation and the EOM portion at an approximate three-body level. We could certainly construct the EOM-IMSRG({3},{3}) by shifting all of the absolute energies by the triples correction of eq. 3.50, however a full inclusion of correlations with the triples-corrected ground state would require additional perturbative terms involving the first-order wave functions of the ground and excited states, which we will denote EOM-IMSRG({3,3}) when it comes to fruition. 88 4.2.5 Extensions of EOM-IMSRG The EOM-IMSRG formalism presented is sufficient to describe the excited states of closed shell nuclei only. This stems from the fact that the flowing vertices in eqs. 3.21-3.23 are normal-ordered with respect to a single Slater determinant reference state. The most obvious solution to this limitation would be to merge multireference IMSRG (MR-IMSRG) [45,47] with EOM techniques. This method uses generalized normal-ordering to include static correlations in the reference state [77], and as a result augments the IMSRG(2) equations with terms proportional to one-, two- and higher-body irreducible density matrices. Ultimately the full MR-EOM-IMSRG will allow for the description of many additional open-shell nuclei, but the current formulation of the MR-IMSRG with scalar operators in J-scheme limits the description to spherical even-even nuclei. Single reference EOM-IMSRG can be extended to open shell nuclei using particle attached (PA) and removed (PR) operators, Yν† = ∑ ya a†a + a Zν† = ∑ zi ai + i 1 † † ∑ yab i aa ab ai 2 abi 1 zaij a†a a j ai , 2∑ ai j (4.31) (4.32) where particle number breaking operators are used to connect that A-body reference state to the (A ± 1)-body spectra. Solving eq. 4.13 using eq. 4.31 or eq. 4.32 in place of X † constitutes the PA-EOM-IMSRG(2,2) or PR-EOM-IMSRG(2,2) respectively, where the first “2” now refers to the truncation to 2p1h or 2h1p operators. Analogously, we can construct methods for two-particle attachment or removal. It’s not foolhardy to think that the MR-IMSRG precludes the practical use of PA-EOM-IMSRG, as it provides direct access to open-shell nuclei. However, the particle number conserving formalism will only be able to describe even-even nuclei, as the MR-IMSRG can only provide a 0+ reference state. Here the PA and PR formalism will be needed to fill in the gaps. At the moment, no MR-EOM-IMSRG implementation exists, but we have performed PA/PR-EOM-IMSRG for cylindrical quantum dots, as a proof of efficacy. The left panel of Fig. 4.12 shows results of particle addition energies for the six particle quan- 89 5.5 Hartree-Fock PA-EOM-IMSRG(2,2) PA-EOM-IMSRG(2,2*) PA-EOM-CCSD 6.46 Energy (Hartree) Addition Energy (Hartree) 6.47 6.45 Hartree Fock PR-EOM-IMSRG(2,2) PR-EOM-IMSRG(2,2*) PR-EOM-CCSD 5 6.44 6.43 8 10 Harmonic Oscillator Shells 4.5 12 8 10 Harmonic Oscillator Shells 12 Figure 4.12 Calculated (ML , MS ) = (0, 1) particle addition and removal energies for a 6-electron quantum dot with ω = 1.0 state with EOM-IMSRG, EOM-CCSD and Hartree-Fock theory. Results are plotted as a function of oscillator shells used for the single-particle basis. tum dot system. Results are computed using the IMSRG(2), IMSRG(2*), and CCSD as a base for EOM methods, and the HF single particle energy is given for reference. The effect of correlations here is rather small, but this system offers an excellent testing ground to compare methods. The IMSRG and CCSD methods show similar convergence properties. The discrepancy between PAEOM-IMSRG(2) and PA-EOM-CCSD accounts for less that 0.01% of the total addition energy, with that discrepancy being reduced by roughly a factor of two when the IMSRG(2*) approximation is used. In the right panel, removal energies are computed for the same system. Despite showcasing a larger overall discrepancy than the addition energies, the difference between EOMIMSRG and EOM-CCSD (1% of the removal energy) is much smaller than the effect of correlations (10%). This stands in contrast to the addition energies, where the discrepancy is only about a factor of two smaller than the discrepancy caused by correlations. In nuclear systems, an additional extension of the single-reference EOM-IMSRG(2,2) can be made through use of isospin breaking charge-exchange operators, which can be used to describe isobaric neighbor nuclei to closed shells. The formalism is exactly the same as the particle-number conserving EOM-IMSRG derived from eq. 4.15, with the only difference being that neutrons are converted to protons or vice versa. For example, we may construct a ladder operator for the 1+ 90 ground state of 14 N , using the 14 C reference state, |Ψ0 (14 N) = X † (∆T z = −1)|Φ0 (14C) X † (∆T z = −1) = ∑ xia p†a ni + ia 1 xiabj δtz tz p†a a†b a j ni b j 4 i∑ jab (4.33) (4.34) where p† and n are proton creation and neutron annihilation operators, respectively. Results for 14 N are shown in fig. 4.13. Similar to the charge conserving EOM-IMSRG(2,2), the 1p1h-dominant low-lying 0+ 1 state is described well by this method. The remaining states have excitation energies which are too high, indicating some admixture of 2p2h and higher-order content. We have not implemented a triples correction here, but again we expect that such a correction would have a noticeable effect on the odd-parity states, which likely have more balanced compositions of 1p1h excitations into the sd-shell and 2p2h excitations where only one particle crosses the major shell-gap. We also consider the accuracy of ground state energy, as the wave function is built here upon the 14 C reference state. Its binding energy is calculated to be 7.430 MeV/nucleon, while the experimental value is 7.475614 MeV/nucleon [156]. We will use the charge-exchange formalism for both energetics and electromagnetic observables in chapter 5. We have seen that with a single reference EOM-IMSRG(2,2) code, closed shell nuclei and their four nearest isobaric, isotonic and isotopic neighbors should be within reach. However, we caution that next-nearest neighbors will be described rather poorly in this way, as EOM-IMSRG(2,2) is a fairly limited approximation for nuclei which have no 1p(1h) components with respect to the closed-shell reference state. 91 1210 1- Energy (MeV) 2+ 8 1+ 1- 2- 4 2+ 1+3+ 1- 3201+ 2 0+ 0+ 6 14 N 0 EOM-IMSRG(2,2) Exp Figure 4.13 Low lying spectra of 14 N at h¯ ω = 20.0 MeV and emax =14 using the standard NN+3N(400) softened by free-space SRG evolution to λ = 2.0 fm−1 . Results are compared with experiment [12]. Experimental spectra beyond the 1− 2 state are omitted for clarity. 92 CHAPTER 5 CONSISTENT COMPUTATION OF OBSERVABLES In the previous chapters, we have discussed the computation of both ground and excited state energies within the IMSRG framework. A complete theory, capable of guiding and analyzing experiment, should also be able to describe general observables which are of interest in experiment such as electromagnetic transition strengths and moments. A general and consistent framework for observables in ab initio methods will help address longstanding problems in nuclear physics, such as the nature of the observed quenching of the axial vector coupling constant gA , or the source of effective charges in medium. A critical property of physical observables is that they are invariant under unitary transformation. In other words, when a change of resolution scale is implemented through some renormalization group transformation or otherwise, the effects of the scale change should be applied consistently to the operators of interest. The consistently transformed operators, ¯ = U(s)OU † (s) , O(s) (5.1) are called “effective” operators. This highlights the fact that interactions and operators themselves are not observables [39], but rather tools used to probe observables. These tools are most useful when tuned to the proper degrees of freedom. In this chapter, we will introduce a general effective operator formalism for the IMSRG, where the bare operators are transformed consistently alongside the Hamiltonian. Non-scalar effective operators have been implemented in both the EOMIMSRG and the recently developed valence-space IMSRG (VS-IMSRG). We explore the effects of consistent operator evolution and convergence properties for both methods, benchmarking with no-core shell model (NCSM) and experiment when available. 93 5.1 Effective Operators in the IMSRG The IMSRG transformation may be straightforwardly applied to any operator using eq. 3.8 where the generator η is defined in terms of the flowing Hamiltonian. Prior to the Magnus formulation, large scale effective operator calculations were unpractical. As discussed in section 3.2.3, the traditional IMSRG approach is computationally unfeasible because of the many copies of each operator needed for a high-order ODE solver. In the Magnus formulation, we may solve first for the Magnus generator Ω, and subsequently apply the transformation to each additional operator via the Baker-Campbell-Hausdorff expansion of eq. 3.39. Until recently, these calculations were limited to scalar observables such as radii and monopole moments. Ab initio calculations of non-scalar observables have been possible in the NCSM and QMC, where no renormalization of the operators is needed [157–160], but these explorations are limited to light nuclei and often lack sufficient convergence due to the restricted model spaces necessary for quasi-exact approaches. The present work extends IMSRG calculations in the medium-mass region to transitions and moments, through the derivation and efficient implementation of commutator expressions involving arbitrary rank spherical tensor operators. The relevant equations are given in the appendix. As a result, consistent calculation of all tensor observables such as electromagnetic strengths, moments, and Gamow-Teller strengths are now accessible. In principle, this formalism also paves the way for consistent predictions of quantities on the cutting edge of physics, such as the nuclear matrix elements for neutrinoless double β -decay [67] and structure factors of weakly interacting massive particles (WIMPs) scattering from nuclei, which are relevant in searches for dark matter [161, 162]. 5.1.1 Transition Operators In this work, we will consider three electromagnetic multipole operators, the M1, E2, and E3 operators. The latter, known as the electric octupole operator, allows for odd-parity transitions up to ∆J = 3, which are relevant to the low lying 3− 1 states observed in doubly magic nuclei. The M1 94 and E2 operators are responsible for the even-parity transitions which are most prolific in nuclear physics, namely the magnetic dipole and electric quadrupole transitions. Observables discussed here include transition strengths, given by B(σ λ ; Ψi Ji → Ψ j J f ) = 1 |M |2 , 2Ji + 1 f i (5.2) where M f i is the reduced transition matrix-element M f i = Ψi Ji ||Oσ λ ||Ψ f J f , (5.3) and Oσ λ are the transition operators (σ is a dummy variable for M or E, and λ is the tensor rank). We also compute moments of the M1 and E2 operators, λ M σ λ = ΨJM = J|Oσ 0 |ΨJM = J , (5.4) where the transition matrix element is no longer reduced, and we take the M = J projection of the state Ψ. We take the bare operators as our starting point, using only the leading-order one-body piece for each operator, written in the lab frame. The operators are derived from the electric and magnetic multipole expansions [163], OEλ µ = A ∑ e( j)rλj Yµλ (Ω j ) , (5.5) j=1 OMλ µ = µN A 2 ( j) ( j) gl l( j) + gs s( j) · ∇ j [rλj Yµλ (Ω j )] , λ + 1 j=1 (5.6) ∑ ( j) ( j) where µN is the nuclear magneton. The electric charge e( j) and gyromagnetic ratios gs and gl are given by their bare values e(p) = 1 (p) gs = 5.586 (p) gl =1 (5.7) for protons, and e(n) = 0 (n) gs = −3.826 (n) gl =0 (5.8) for neutrons. Phenomenological computations of electromagnetic observables typically require effective values for these parameters to account for in-medium effects, which can be thought of as 95 arising from the polarization of the core by valence nucleons through the inter-nucleon interactions. As the valence nucleons change their state, the polarized core produces a motion of charge beyond that of only the valence protons [95, 164–167]. In the spirit of ab initio nuclear structure, we start with the bare nucleon values for e, gs and gl , and expect them to be dressed appropriately by renormalization from consistent operator evolution. While a detailed analysis is beyond the scope of this dissertation, the IMSRG provides an excellent environment to study the renormalization of these parameters in-medium. Of critical importance is the axial-vector coupling constant gA , which mediates the Gamow-Teller strength of nuclear β -decay; a strong quenching of gA is observed in medium [168, 169]. While some attempts to describe this effect have been made in an ab initio framework [170], the origin of this quenching is very much an open question in nuclear physics. Theoretical explanations attribute the quenching either to configurations missing from the model spaces used in the computations, or to non-nucleonic degrees of freedom. The former explanation pertains to phenomena such as shortrange correlations or multi-phonon states [171–173], where the latter refers to missing physics such as ∆-isobar excitations [172, 174]. The effective operator formalism present here, coupled with the versatility of a Hermitian method such as the IMSRG, offers a promising framework to investigate such ideas for a wide range of nuclei. 5.1.2 General Observables in the IMSRG At present, two distinct flavors of the IMSRG offer viable frameworks to study transitions and moments in nuclear structure. The EOM-IMSRG is a capable vessel, but observables may also be accessed by the valence-space (VS)-IMSRG [46, 52]. In the EOM-IMSRG, reduced matrix elements between excited-states and the ground state must be computed by the coupled product of spherical tensor operators, M0ν = Φ0 ||{Oλ ⊗ Xν† (JνΠ , ∆Tz )}0 ||Φ0 96 (5.9) J1 J2 X˜abi X 1 J1 J2 j ai = δλ Jν (−1)Jν ∑ √ Oai + ∑ ∑ √ O˜ abi j . 4 abi j J J 2J + 1 ai 2J + 1 (5.10) 1 2 Here Oλ represents an operator of rank λ , and Xν† (JνΠ , ∆Tz ) is an EOM ladder operator whose tensorial rank is given by the total angular momentum Jν , with parity Π and charge-exchange character given by ∆Tz . For moments or transitions between two EOM excited states, a more involved calculation is needed: Mµν = Φ0 ||{Xµ (JµΠ , ∆Tz ) ⊗ {Oλ ⊗ Xν† (JνΠ , ∆Tz )}Jµ }0 ||Φ0 . (5.11) Knowledge of the full tensor product J ≡ {Oλ ⊗ X † (J Π , ∆T )}J = YM z M ν ν ∑ Mν µ λ Jν J CµM Oλ X † (J M ) , νM µ ν ν ν (5.12) is required for computation of eq. 5.11. The expressions for this are given by eqs. 108 and A.6.2 in the appendix. The VS-IMSRG is not the focus of this dissertation, but nonetheless we include results for the sake of benchmarking and for a more complete interpretation of effective operator evolution in the IMSRG. The VS-IMSRG approaches spectroscopy and related observables via a different paradigm in which a valence space is decoupled for use with existing shell-model machinery. In this context, the IMSRG serves as a means to construct nonperturbative effective valence space interactions and corresponding effective operators. After decoupling the shell-model core by the ground-state IMSRG, a secondary decoupling, similar to the vTDA-IMSRG decoupling of section 4.1.2, is used to suppress dynamic correlations between the two valence particle-attached space and the core and non-valence orbitals. The relevant off-diagonal is given by H od : f ph , f pp , f pp , Γ pp hh , Γ pp vh , Γ pq vv (5.13) and their Hermitian conjugates [46]. Here p and h are particle and hole indices, respectively, v are valence space orbitals, and q non-valence particle orbitals. In typical calculations, the Hamiltonian 97 is normal ordered with respect to an ensemble reference state specified by the density matrix ρ = ∑ Cα |Φα Φα | , (5.14) α where |Φα are Slater determinants of varying particle number Aα , and the Cα are chosen so that the ensemble-averaged particle number is equal to the mass of the target nucleus. This alters the Wick contractions (see sec 2.3) such that they are proportional to a fractional occupation number, rather than an integer occupation. Ensemble normal ordering (ENO) has the effect of properly including valence three-body forces in the NO2B approximation, and is crucial for the reproduction of binding energies far from shell-closures. With these improvements in normal-ordering, the method has had great success in the reproduction of binding energies and spectra throughout the sdand p f -shells [51, 52]. Despite this progress, the VS-IMSRG has heretofore been limited to scalar operators. However, the same machinery required for effective operator evolution in the EOMIMSRG can be used for the VS-IMSRG. The actual computation of observables is accomplished in a shell model framework using the consistently transformed operator together with the wave functions that result from diagonalizing the valence Hamiltonian. 5.2 5.2.1 Results Center-of-Mass Treatment Spurious center-of-mass-excited states manifest as nearly degenerate intrinsic states in nuclear spectra. As discussed in sec. 3.2.4, these states are removed via the Lawson-Gloeckner method [175], where the intrinsic Hamiltonian is augmented with a scaled center of mass trap of the form of eq. 3.43, H = Hin + β Hcm . (5.15) Here, the scale factor β can be taken to arbitrarily large values if sufficient factorization is achieved in calculations using Hin only. This process effectively scales spurious states out of the spectrum by adding a large COM excitation energy. 98 15 20 25 15 20 25 20 25 10 β=0 β=1 β=3 β=5 9 8 7 B(E2; 2 1+ → 0 gs+ ) (e 2 fm 4 ) E(2 1+ ) (MeV) Egs (MeV) 102 103 104 105 106 6 5 4 3 2 15 ω (MeV) Figure 5.1 Observables calculated at several values of the Lawson scaling parameter β , for 14 C, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 , at emax = 14. 99 β 0 1 Ecm (0+ gs ) (MeV) 0.099 0.068 Ecm (2+ 1 ) (MeV) 1.298 0.046 Table 5.1 Ecm for intrinsic ground state and first 2+ state of 14 C, computed at emax =14 and h¯ ω = 20 MeV using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 with EOM-IMSRG(2,2). Values are given for calculations using Hin (β =0), and Hin + Hcm (β =1). Figure 5.1 demonstrates this procedure with 14 C, for the ground state, first 2+ excited state, and corresponding B(E2) value. Quantities are calculated with the EOM-IMSRG(2,2) method. The energies are approximately independent of β ; this may be taken naively as evidence of factorization for these states. However, the B(E2) value undergoes a sudden downward shift as the Lawson term is introduced. The value saturates and displays factorization as we go to higher β . This shift indicates that the approximate factorization observed in the energies is not sufficient to describe observables whose operators are not expressed in a translationally-invariant manner, such as the electric quadrupole operator discussed here. A common resolution for this problem is the inclusion of a cumbersome recoil correction for the transition operator [164]. This is a two-body operator which may require approximate or exact IMSRG(3) machinery to minimize truncation errors. A more elegant solution is to force the COM wave function into the ground state via explicit trapping with the Lawson-Gloeckner method discussed above. Table 5.2.1 gives the computed Ecm for calculations with and without explicit inclusion of a center of mass trap via the Lawson term. We expect a perfectly factorized wave-function to have Ecm =0 MeV, as Hcm is written so that the COM ground state has zero energy. For either case, the ground state wave function demonstrates limited contamination from spurious COM excitations, with Ecm < 100 keV. The 2+ 1 state of Hin does not exhibit this level of factorization, with Ecm =1.298 MeV, indicating a relatively small admixture of spurious states, as the first spurious excited state has energy h¯ ω˜ = 15.7 MeV. This level of contamination is evidently negligent for excitation energies, but has important effects when the state is probed by the quadrupole operator. When the COM trap is explicitly added this issue is rectified, and as a result, we see a shift in the 100 B(E2) strength, which corresponds to a recoil correction. 5.2.2 The Deuteron As a first illustration, we consider some ground state properties of the deuteron. This is useful for a few reasons. First, the system consists of only two particles and so induced three body forces are irrelevant. Further, the reference is taken to be the true vacuum, so the neglected three body forces do not feed back into the two body terms. We should therefore expect the IMSRG(2) to be exact. Second, full configuration interaction (FCI) calculations are easily performed for modest model spaces, allowing a direct evaluation of the precision of the IMSRG calculation. Finally, we may treat the deuteron in the 0s valence space where the bare quadrupole moment is identically zero. In this case, the calculated quadrupole moment is entirely due to effects of the IMSRG decoupling. Fig. 5.2 shows the ground-state energy, root-mean-square charge radius, quadrupole moment and magnetic moment of the deuteron, computed both with FCI and using IMSRG to decouple the 0s valence space, followed by a trivial diagonalization. We can see that the IMSRG calculation indeed reproduces the FCI. Here again we see the effect of COM spuriosities in the deuteron wave function. While the energy and dipole moment converge to the exact values with little alteration from COM contamination, the charge radius overshoots it drastically. Although we have not reached convergence for the charge radius, it is evident that Lawson scaling significantly reduces its value. Furthermore, the quadrupole moment, which is also non converged at emax =12, shows a more agreeable convergence pattern when treated with the Lawson scaling. We note that the convergence properties and COM sensitivity of the charge radius and quadrupole moment are quite different, despite the latter being proportional to the square of the radius. These discrepancies are not fully understood at the moment. 101 ( µN ) gs FCI βCM = 0 FCI βCM = 3 VS IM-SRG βCM = 0 VS IM-SRG βCM = 3 µ Egs (MeV) 5 4 3 2 1 0 1 2 exact 3 0 2 4 6 emax 8 10 12 0.885 0.880 0.875 0.870 0.865 0.860 0.855 0 2.4 Rch (fm) (efm2 ) 2.2 exact 2.0 1.6 1.4 0 2 4 6 emax 8 Charge radius Q gs 1.8 10 12 Magnetic moment exact 2 0.30 exact 0.25 0.20 0.15 0.10 0.05 0.00 0 2 4 4 6 emax 6 8 10 12 Quadrupole moment emax 8 10 12 Figure 5.2 Ground state properties of the deuteron calculated with a full diagonalization (labeled FCI), compared to the same properties calculated in the 0s valence space using operators transformed with the IMSRG. The line labeled “exact” is obtained by diagonalizing in the Jacobi basis with emax = 200. Here we use the EM input N3 LO nucleon-nucleon interaction SRG evolved to λSRG = 2.0 fm−1 . 5.2.3 p shell Nuclei: Comparison to NCSM and Small-Space FCI The deuteron is, of course, an exceptionally simple case, due to the fact that there is not really a “medium”, and so the IMSRG is really a free-space SRG. Once additional particles are considered, the normal-ordered two-body approximation is used, and the IMSRG(2) should no longer be exact. To test this approximation, we consider p-shell nuclei which may also be treated exactly in the nocore shell model (NCSM). As a test in the p shell, we consider 14 C. Because this is a closed-shell nucleus, we may employ the EOM-IMSRG as well as the VS-IMSRG, while 14 particles is still feasible with the NCSM. + + 14 Fig. 5.3 displays results for the 2+ 1 excitation energy and B(E2; 21 → 00 ) for C. Here, we find excellent agreement between NCSM and both variants of the IMSRG. We remind the reader that the EOM-IMSRG calculation is performed with an explicit center-of-mass trap, as in eq. 5.15, 102 12 Exp E(2 +1 ) (MeV) 11 NCSM ω=20MeV Valence-space IMSRG emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 10 EOM IMSRG 9 8 B(E2; 2 +1 → 0 +0 ) (e 2 fm 4 ) 7 5 4 3 2 14 0 2 4 6 8 12 Nmax 16 20 24 ω (MeV) 28 12 16 20 24 ω (MeV) C 28 14 Figure 5.3 Convergence of the 2+ 1 excitation energy and B(E2) to ground state of C with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. where β = 1.0. This treatment only serves to remove spurious COM contamination of the 2+ 1 state. Of note are the excellent convergence properties of the IMSRG calculations. For the EOMIMSRG, observables are nearly independent of the specified h¯ ω for the single-particle basis. VSIMSRG calculations have not used the exhaustive model spaces of the EOM-IMSRG, but they too demonstrate desirable convergence features. The NCSM has begun to show convergence at Nmax =8, but extrapolation methods must be used to reveal fully converged values. Hence the utility of the IMSRG; for light mass nuclei such as 14 C, convergence is obtainable without extrapolation because the IMSRG cannot only reach higher model-spaces, but also indirectly treats the full configuration space within the emax truncation on the single-particle basis. On the other hand, the quasi-exact NCSM is restricted to the subset of Slater determinants with total excitation quanta 103 14 Exp NCSM E(1 +1 ) (MeV) 13 Valence-space IMSRG emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 12 EOM IMSRG 11 10 ω=20MeV B(M1; 1 +1 → 0 +0 ) (µN2 ) 9 ω (MeV) 2 1 0 14 0 2 4 6 8 12 Nmax 16 20 24 ω (MeV) 28 12 16 20 24 ω (MeV) C 28 14 Figure 5.4 Convergence of the 1+ 1 excitation energy and B(M1) to ground state of C with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. up to and including Nmax . Correspondingly, we expect extrapolation procedures in heavier nuclei to be relatively painless in the IMSRG, compared with exact methods such as NCSM. As a further test in 14 C, we analyze the first 1+ state and corresponding M1 transition to the ground state in figure 5.4. Again, fantastic convergence properties are exhibited by both IMSRG methods, and consistency with NCSM is observed. The 1+ 1 excitation energy is consistent between methods and shows reasonable reproduction of experiment. While B(M1) values computed by the many-body methods are in excellent agreement, these overestimate experiment by a significant amount. A possible source of this discrepancy are contributions from two-body currents which originate from chiral-EFT. These currents can in principle be incorporated in our calculations, but the normal-ordered current operators have yet to be implemented in this formalism. 104 4.5 Exp NCSM 4.0 emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 EOM IMSRG E(0 1+ ) (MeV) Valence-space IMSRG 3.5 3.0 2.5 2.0 1.5 1.0 B(M1; 1 1+ → 0 1+ ) (µN2 ) 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.55 12 16 20 24 ω (MeV) 28 12 16 20 24 28 16 20 24 28 ω (MeV) µ(1 1+ ) (µN ) 0.50 0.45 0.40 0.35 0.30 0.25 0.20 14 0 2 4 6 12 Nmax 16 20 24 ω (MeV) 28 N 12 ω (MeV) Figure 5.5 Convergence of 0+ 1 excitation energy, B(M1) to ground state, and magnetic dipole 14 moment of N with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. 105 Additionally, we analyze the isobaric neighbor nucleus 14 N. Here the EOM-IMSRG requires charge-exchange formalism discussed in sec. 4.2.5, i.e. ladder operators which exchange one neu14 tron for a proton. Figure 5.5 displays the 0+ 1 excitation energy for N, the strength of the dipole transition to the 1+ 0 ground state and the ground state magnetic dipole moment. Here we see a trend of VS-IMSRG overpredicting observables. The EOM-IMSRG(2,2) underpredicts experiment for the energy and B(M1) value, while slightly overpredicting the magnetic moment. The VS-IMSRG(2) drastically overpredicts both experiment and NCSM for the B(M1) value. We do not yet understand the source of this discrepancy, but the EOM-IMSRG(2,2) also struggles here, underpredicting the B(M1) value by roughly an order of magnitude. While the lack of discernible convergence for the B(M1) value in the NCSM calculation seems to demonstrate that this nucleus is challenging for ab initio theory, the IMSRG methods certainly should share a fair portion of the blame; in sec. 5.2.7, we will discuss potential sources of error in the VS- and EOM-IMSRG approaches. Ultimately, the power of IMSRG approaches for excited states and effective operators will be the ability to describe these properties in medium- to heavy-mass regions where exact methods are not computationally tractable. In the next sections, we investigate the quality of these calculations for several medium-mass nuclei, again using the electric quadrupole and magnetic dipole operators as case studies. 5.2.4 Electric Quadrupole Observables in sd and f p shell Systems Figure 5.6 displays the first 2+ energy and B(E2) strength for transitions to the ground state of 22 O. While we see excellent convergence and agreement with experiment for the excitation energy, the transition strength is underpredicted by an order of magnitude. These results are strikingly consistent between the two methods. A tentative explanation is provided by the lack of valence protons in the sd-shell. The VS-IMSRG treats this nucleus as six neutrons sitting above an 16 O core; the low-lying 2+ 1 state is composed solely of excitations within the sd-shell, and thus the wave func- 106 7 6 Exp Valence-space IMSRG EOM IMSRG E(2 +1 ) (MeV) 5 4 3 2 1 B(E2; 2 +1 → 0 +0 ) (e 2 fm 4 ) 0 6 5 4 3 2 1 0 22 12 16 20 24 ω (MeV) 28 emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 O 12 16 20 24 ω (MeV) 28 22 Figure 5.6 Convergence of the 2+ 1 excitation energy and B(E2) to ground state of O with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. tion is is composed purely of neutron excitations. As a result, valence neutrons must be dressed consistently as quasi-neutrons possessing an effective charge in order to properly produce transition strengths. A similar situation manifests in the EOM-IMSRG(2,2), where by conservation of parity, the only proton excitations allowed across the major shell gap are 2p2h, and these are of significantly higher energy than the 1p1h neutron excitations within the sd-shell. The absence of any appreciable strength in the two IMSRG calculations is a somewhat convincing argument that IMSRG decoupling does not appropriately renormalize the neutron charges. However, this discrepancy is evident in many other nuclei, regardless of whether the protons are semi-magic or not. Figure 5.7 shows a similar situation for 32 S, which has numerous valence protons to account for the transition strength. Here we see reasonable reproduction of the 2+ 1 experimental excita- 107 7 Exp Valence-space IMSRG 6 EOM IMSRG E(2 +1 )(MeV) 5 4 3 emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 2 1 Q(2 +1 )(fm 2 ) B(E2; 2 +1 → 0 +0 )(e 2 fm 4 ) 0 70 60 50 40 30 20 10 0 2 4 6 8 10 12 14 16 18 32 12 16 20 24 ω(MeV) 28 S 12 16 20 24 ω(MeV) 28 Figure 5.7 Convergence of the 2+ 1 excitation energy, corresponding B(E2) to ground state and 32 quadrupole moment of S with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [1]. 108 10 9 Exp 48 Ca Exp 56 Ni Exp 60 Ni E(2 +1 ) (MeV) 8 7 6 5 4 3 B(E2; 2 +1 → 0 0+ ) (e 2 fm 4 ) 2 1 emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 150 100 50 0 10 15 20 25 ω (MeV) 30 10 15 20 25 ω (MeV) 30 10 15 20 25 ω (MeV) 30 Figure 5.8 First excited 2+ states and corresponding B(E2) transition strengths for 48 Ca and 56,60 Ni, using the standard input NN+3N(400) interaction softened by the SRG to λ SRG = 2.0 −1 fm . Convergence patterns are shown, computed with EOM-IMSRG(2,2). Results are compared with experimental points in the left panel [1]. tion energy, but the B(E2) value is drastically underpredicted by roughly an order of magnitude. Correspondingly, the quadrupole moment’s magnitude is underpredicted by a factor of about 3.5 for the valence-space IMSRG and about 6 for EOM-IMSRG. Much like 14 N, this nucleus exhibits marked disagreement between the two IMSRG approaches. We will forego discussion here and discuss this in detail in section 5.2.7. As we investigate heavier nuclei, we find that underprediction of B(E2) values is far more common than the stunning success seen with 14 C. Figure 5.8 shows convergence of 2+ 1 energies and E2 transition strengths in 48 Ca and 56,60 Ni. Again, calculated strengths are roughly an order of magnitude smaller than experiment, while energies are reasonably consistent with observed values. Table 5.2.4 compiles the results from several of the calculations presented here, where 109 Nucleus 14 C 22 O 32 S 48 Ca 56 Ni 60 Ni B(E2↓)exp 3.6(6) 4.2(1.6) 59(1) 17(2) 1.2(3)×102 186(3) B(E2↓)EOM 4.1 0.5 7.2 2.6 30.7 16.2 B(E2↓)V S 3.9 0.4 11.3 BW (E2↓) 2.0 3.7 6.0 10.4 12.7 14.0 Table 5.2 Computed E2 transition strengths from first excited 2+ state to 0+ ground state for even-even nuclei, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experiment [1] and Weisskopf [2] single particle estimates are compared with IMSRG calculations. + B(E2↓) corresponds to B(E2;2+ 1 → 00 ) value. In the far right column, the Weisskopf units (W.U.) for the transition are included [2]. Weisskopf units, given by 9r04 4/3 2 4 A e fm , BW (E2) = 100π (5.16) take the transition to be a single proton excitation from a hard-core with the empirical nuclear radius r0 A1/3 , where r0 = 1.2 fm. Strongly 1p1h transitions will yield experimental B(E2) values near the Weisskopf estimate. This picture certainly falls short of describing those nuclei with magic proton numbers, such as 22 O, but it is instructive to consider what the single particle estimates are for this nucleus nonetheless, as they describe neutrons with an effective charge in this case. We find that computed B(E2) values track with Weisskopf estimates rather than actual experimental values, except in the case of a magic proton shell closure, where computations are significantly smaller than the Weisskopf units, suggesting that indeed, the renormalization of neutron effective charges may not be sufficient in our IMSRG calculations. Importantly, the fact that many of the experimental B(E2) values are significantly larger than the single particle estimates indicates that collectivity which is neglected by IMSRG(2) calculations may be more critical to E2 transition strengths than it is to excitation energies. 110 15 Exp Valence-space IMSRG 14 EOM IMSRG E(1 +1 ) (MeV) 13 12 11 10 9 emax = 4 emax = 6 emax = 8 emax =10 emax =12 emax =14 B(M1; 1 +1 → 0 +0 ) (µN2 ) 8 4 3 2 1 0 22 12 16 20 ω (MeV) 24 28 O 12 16 20 ω (MeV) 24 28 22 Figure 5.9 Convergence of the 1+ 1 excitation energy, B(M1) to ground state of O with the VSand EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . 5.2.5 Magnetic Dipole Observables in sd and f p shell Systems We turn to M1 observables, where the Weisskopf unit has no dependence on A and thus we expect the transition to have similar properties from nucleus to nucleus, unlike E2 observables. As an analog to the E2 observables, we first investigate 22 O. Here we do not have a measurement of the 1+ 1 state, so this analysis serves only to compare EOM- and VS-IMSRG. Here we see excellent consistency between the two methods, with convergence patterns for the energy being nearly identical, and those for the M1 transition very similar in character. This matches expectations derived from the agreement we saw for the E2 transition. As expected from the mass independence of the Weisskopf estimates, the B(M1) for this nucleus is predicted around 1 µN2 , which is similar to that of 14 C. 111 7 Exp Valence-space IMSRG 6 EOM IMSRG E(1 +1 )(MeV) 5 4 3 2 emax = 6 emax = 8 emax =10 emax =12 emax =14 1 ×10 −3 B(M1; 1 +1 → 0 +0 )(µN2 ) 0 8 6 4 2 0 µ(2 +1 )(µN ) 1.5 1.0 0.5 32 0.0 12 16 20 24 ω(MeV) 28 S 12 16 20 24 ω(MeV) 28 + Figure 5.10 Convergence of 1+ 1 excitation energy, B(M1) to ground state and 21 magnetic dipole moment of 32 S with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. 112 E(1 +gs ( 32 Cl)) − E(0 +gs ( 32 S)) (MeV) 16 Exp Valence-space IMSRG EOM IMSRG 15 14 emax = 6 emax = 8 emax =10 emax =12 emax =14 13 12 µ(1 +1 ) (µN ) 1.5 32 Cl 1.0 0.5 0.0 12 16 20 ω (MeV) 24 28 12 16 20 ω (MeV) 24 28 Figure 5.11 Computed ground state energy and magnetic dipole moment of 32 Cl with the VS- and EOM-IMSRG methods, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Experimental values are taken from [14]. + 32 We also compute M1 observables for the 1+ 1 and 21 states in S. Figure 5.10 shows results for these calculations. Here we see that IMSRG calculations are within an order of magnitude of experiment for all observables considered. Again, some notable discrepancies between VS- and EOM-IMSRG are evident, stemming from the fact that the VS-IMSRG uses a full valence space diagonalization, and the EOM-IMSRG uses an approximate diagonalization in the full active space. Nonetheless, some consistency is observed, particularly for the dipole moment of the 2+ 1 state. We don’t expect agreement with experiment for the M1 observables, as we neglect meson-exchange currents, which are understood to be important to reproducing experiment [95, 176]. However, as the leading order effects are included in our calculation, it is promising to see qualitative agreement within an order of magnitude. A more pleasant picture surfaces in fig. 5.11, where we have computed ground state properties of 32 Cl. The ground state energy is given here as an excitation energy from the 32 S ground state, 113 as it is calculated in the EOM-IMSRG as an excited state of 32 S with a charge-exchange excitation operator. Disagreement between EOM- and VS-IMSRG is far less prominent here than in the isobaric neighbor 32 S, and the methods again show qualitative agreement with experiment. This agreement is somewhat surprising considering that 32 Cl is situated near the center of the sd-shell. The observed consistency indicates strong 1p1h character in the ground state. We have computed the magnetic dipole properties of five nuclei: 32 S, 32 Cl, 22 O, 14 C and 14 N, with reasonable consistency between VS-IMSRG and EOM-IMSRG achieved for 14 C, 22 O, and 32 Cl. Considering the closed shell cases only, this corresponds to what is seen for E2 observables. In order to compare with experiment, we would need to add higher-order mesonic currents which occur within the nucleus during the transition. Computationally, these manifest as a two body correction to the M1 transition operator. This is beyond the scope of this work, as normal-ordering has yet to be implemented for two-body tensor operators (normal-ordering is trivial for one-body tensor operators). IMSRG truncation errors are more important for two-body operators than onebody, so additional considerations must be made before exploring the addition of mesonic currents. 5.2.6 Electric Octupole Transitions The electric octupole transition offers an additional test of the EOM-IMSRG. We investigate the transition strengths from the first 3− state to ground state for the doubly magic nuclei 16 O and 40 Ca. The left panel of figure 5.12 shows the convergence of this calculation for 16 O. This is an 16 interesting case study, as the 3− 1 excitation energy has been shown to correlate with the O charge radius and thus depends on saturation properties of the interaction [177]. For this reason, we compare calculations with the NN+3N(400) interaction to those using N2 LOsat , which is fit to the 16 O charge radius [177]. We see an improvement of the excitation energy when using N2 LOsat , moving from 9.03 MeV with the NN+3N(400) interaction to 6.90 MeV, in significantly better agreement with the experimental value at 6.13 MeV. Regardless, the gains from switching to the saturating interaction are not seen in the B(E3) value, as both interactions underpredict experiment 114 10 Exp 7 N2 LOsat 6 8 Exp N2 LOsat E(3 1− )(MeV) E.M. NN+3N(400) 4 B(E3; 3 1− → 0 0+ )(e 2 fm 6 ) 3 2 emax = 6 emax = 8 emax =10 emax =12 emax =14 2 0 250 150 100 16 50 12 16 20 ω(MeV) 24 emax = 6 emax = 8 emax =10 emax =12 emax =14 1 200 0 E.M. NN+3N(400) 4 12 O 16 20 24 ω(MeV) 28 B(E3; 3 1− → 0 0+ )(e 2 fm 6 ) E(3 1− )(MeV) 5 6 0 30 25 20 15 10 5 ×10 2 0 40 12 16 20 ω(MeV) 24 12 Ca 16 20 24 ω(MeV) 28 Figure 5.12 Excitation energies of the first 3− state of 16 O and 40 Ca, along with corresponding B(E3) strength for transition to ground state. These values are computed using the EOM-IMSRG(2,2) with N2 LOsat and the standard NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 .. Results are compared with experiment [15]. by roughly an order of magnitude. Despite EOM partial norms indicating 90% 1p1h content in the 3− wave-function, higher order correlations may play a significant role in the structure pertinent to the E3 transition, as α-clustering may be important to the structure of the 3− 1 state [178]. A similar picture presents itself for 40 Ca in the right panel of figure 5.12, where again, N2 LOsat improves excitation energy but underpredicts the B(E3) strength. For either interaction, the discrepancy is less striking than that seen in 16 O, but the deviation is significant nonetheless. Notable is the poor convergence features exhibited by N2 LOsat , where results are seemingly dependent on the basis frequency h¯ ω for both energy and B(E3) value. At the moment we are working to understand these issues. Computed E3 strengths suffer from largely the same shortcomings as E2 strengths, where we see a significant reduction of the strength from that of experiment. The Weisskopf single-particle estimates for 16 O and 40 Ca are 15.21 and 95.04 e2 f m6 respectively. The immense size of the experimental values compared with these estimates indicates a strong level of collectivity in these 3− 1 states, which is apparently missed by our calculations, though computed E3 strengths are indeed 115 larger than the single-particle estimates. 5.2.7 Comparing and Contrasting Methods We return now to 32 S, where noticeable differences between the EOM- and valence-space variants of IMSRG were demonstrated. This discrepancy is the result of some combination of two sources of error. The VS-IMSRG(2) introduces a second decoupling, and as such, unitarity is degraded beyond truncation errors from ground-state-decoupling. On the other hand, the EOM-IMSRG(2,2) lacks the ability to describe higher-order correlations in states with minimal 1p1h character. This underscores the fact that the two methods are complimentary, and a different class of states fall into the set that is best described by either method. The VS-IMSRG takes into account all possible valence-particle configurations within the specified valence space. States that are described well by phenomenological shell-model approaches should then be described appropriately by the VS-IMSRG. However, states with significant contributions from multiple major shells, in particular unnatural parity states, are unreachable by this method in its current state. Methods to decouple a multi-shell valence space are still under investigation. EOM-IMSRG is not restricted by any valence space, but rather derives its computational simplicity from a restriction of the configurations included in the diagonalization. Thus, unnatural parity states are treated on the same footing as natural. However, any state will be poorly described if the dominant configurations of particle-hole excitation are left out of the definition of the ladder operators. For the EOM-IMSRG(2,2), we work with a space of 1p1h and 2p2h configurations. In this case, states with strong 1p1h content are best described by the method. States with 2p2h dominant wave functions are described reasonably well, but the ground-state-decoupled Hamiltonian still introduces strong correlations between these states and 3p3h excitations. For these states, additional IMSRG softening may be needed to make a perturbative treatment valid. For the specific case of 32 S, the differences between the two IMSRG methods appear in calculations of both the energy and transition strength. The 1p1h partial norms for the 2+ 1 state are 116 284 Energy (MeV) 286 32 EOM-IMSRG(2,2) S 288 E(21+ ) 290 292 294 296 E(00+ ) EOM-IMSRG(2,2) 298 300 [0,0] [1,1] [2,0] [0,2] [0,4] [4,0] [2,1] [1,2] [Np , Nn ] allowed in d3/2 [2,2] [3,3] [4,4] 32 Figure 5.13 Computed ground-state and 2+ 1 energies for S, using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Here we use VS-IMSRG with a variety of truncations on particles allowed into the 0d3/2 shell. Results are compared with EOM-IMSRG values. around 0.87 for 32 S, while they are at 0.92 for the analogous state in 22 O. This difference, while small, could be responsible for the noticeable discrepancy between EOM and VS-IMSRG. In the sd-shell diagonalization, 32 S has eight active protons and an equal amount of active neutrons. This allows for the inclusion of many sd-shell configurations not accounted for by EOM-IMSRG(2,2), which at most examines two particle excitations from the 32 S core. On the other hand, EOMIMSRG includes excitations of nucleons into much higher shells, and from lower lying shells. While VS-IMSRG compensates for this through decoupling of the valence space, this additional decoupling introduces IMSRG truncation errors beyond those inflicted by the EOM-IMSRG. Figure 5.13 demonstrates the effect of restricting various particles from entering the 0d3/2 shell during the VS-IMSRG diagonalization. The ground and 2+ 1 state are computed at various [N p , Nn ], which corresponds to the number of protons and neutrons allowed in the 0d3/2 shell. EOM-IMSRG(2,2) energies are given for reference. The EOM-IMSRG(2,2) treats the ground state as a Slater determinant which is filled through the 1s and 0d5/2 shells. Any excited states are 117 constructed of at most 2p2h excitations into the 0d3/2 shell and higher. The comparison made here is not rigorous, as the two methods use different decoupling schemes, but nonetheless it sheds light on some of the differences between these methods. It is evident from fig. 5.13 that the full active valence-space is needed to describe the absolute energies with VS-IMSRG, which are similar to that of the single-reference EOM-IMSRG(2,2) for lower truncations, and have shifted downward by 4-5 MeV once the full valence-space becomes active. The discrepancy between the single-reference ground state and VS-IMSRG is due to IMSRG(2) truncation errors related to the differing decoupling schemes. EOM-IMSRG(2,2) allows for all one- and two-particle excitations of neutrons or protons. While limitations of the shellmodel code [179] used here do not allow for an analogous truncation in the VS-IMSRG, we can compare to truncations which include a portion of the EOM-IMSRG(2,2) content; these are [1,1], [2,0], and [0,2]. We can also analyze truncations which contain EOM-IMSRG(2,2) content as well as extra valence-space configurations, namely [1,2], [2,1], and [2,2]. We see that the 2+ 1 excitation energy of EOM-IMSRG(2,2) is produced reasonably well by the [1,1] truncation, using just one proton and neutron in the 0d3/2 shell. For the [0,2] and [2,0] truncations, we find that the VS-IMSRG produces a larger excitation energy than EOM-IMSRG(2,2). The [1,2], [2,1] and [2,2] truncations allow for significantly more configuration mixing relevant to the EOM-IMSRG(2,2), but also introduce 3p3h and 4p4h content that EOM-IMSRG(2,2) neglects. Nonetheless, we see that the EOM-IMSRG(2,2) excitation energy is still closely mirrored by the VS-IMSRG(2) for these truncations, with a shift in the absolute energies. As one adds more active particles in the [3,3] and [4,4] truncations, the excitation energy reduces to the result shown in fig. 5.7. It is apparent from these results that the full valence space needs to be active here to produce the smaller excitation energy of VS-IMSRG(2), which is somewhat illuminating when considering the noticeably larger excitation energy of EOM-IMSRG(2,2), where configurations of more than two 0d3/2 nucleons are ignored. The discrepancies exhibited in 32 S are ostensibly the result of its midshell character, where for nuclei with fewer valence nucleons, such as 22 O, discrepancies are minimized. While this analysis 118 has certainly clarified some of the differences between EOM- and VS-IMSRG, it is generally difficult to clearly determine which method performs best for any given state because, at present, there is no prescription to assign accurate theoretical error bars to these calculations. However, one can make inferences about which method will perform best based on the 1p1h partial norms, the number of valence nucleons, and the size of the valence-space decoupling. Here it will be important to develop a reliable measure of unitarity to quantify IMSRG(2) truncation errors associated with additional decoupling. 5.2.8 The Effects of Consistent Operator Evolution We now seek to understand the effects of consistently applying the IMSRG unitary transformation to the operators we’ve discussed thus far. Bare operators are expressed in the Hartree Fock basis, and have not been consistently evolved along with free-space SRG softening. As a result, the operator evolution cannot be exactly consistent, but free-space softening transformations are understood to have little effect on long-range operators such as the electromagnetic multipole operators discussed here. The principle effect of SRG softening is to renormalize short-range physics. Nonetheless, the problem is currently being given ample attention in the nuclear physics community [180, 181]. The IMSRG transformation is expected to have a noticeable effect on transition operators, as it renormalizes dynamic correlations in the nucleus which are crucial to transition behavior. Table 5.2.8 contains a few examples of transition matrix elements computed with and without consistent 1b refers to the matrix element computed without consistent evoluevolution of the operator. Mbare 1b,2b tion, (the wave functions are computed in the evolved frame however), and Mdressed refer to the same calculation with consistently evolved operators. It is evident from those values included in the table that IMSRG evolution transforms a one-body operator into a many-body operator. Here we truncate this operator to two-body terms, consistent with the IMSRG(2) truncation. For EOM-IMSRG(2,2) calculations, The induced two-body term generally contributes less than 10% of the total magnitude. This indicates that the current level of truncation is sufficient. 119 Nucleus 14 C operator M1 14 C E2 22 O M1 22 O E2 32 S M1 32 S E2 method EOM VS EOM VS EOM VS EOM VS EOM VS EOM VS 1b Mdressed 1.816 1.724 4.257 4.345 2.154 2.124 1.383 1.499 0.110 0.049 5.403 7.688 2b Mdressed 0.155 0.051 0.297 0.094 0.134 -0.025 0.117 -0.071 0.004 0.004 0.594 -0.174 1b+2b Mdressed 1.931 1.775 4.554 4.439 2.288 2.099 1.500 1.428 0.114 0.053 5.997 7.514 1b Mbare 2.396 1.889 3.870 4.933 2.620 2.351 0.845 0.000 0.126 0.063 4.990 5.773 Table 5.3 Operator matrix elements computed for select nuclei with the EOM-IMSRG(2,2), using the standard input NN+3N(400) interaction softened by the SRG to λSRG = 2.0 fm−1 . Calculations are performed for bare operators and operators dressed by consistent IMSRG evolution. Values are expressed in e f m2 and µN for E2 and M1 operators respectively. In regards to the total magnitude, consistently evolved M1 transition matrix elements exhibit a 1020% decrease in magnitude compared with the bare operator, and the equivalent comparison for E2 transitions show an increase in magnitude of 20%, except in the case of 22 O, where the magnitude increases by 77.5%. Regardless of consistency with experiment, these results demonstrate that the use of effective operators is crucial in IMSRG calculations of electromagnetic observables. As the decoupling schemes of the VS- and EOM-IMSRG are different, we include results for both methods. We note that the VS-IMSRG(2), despite having a two-stage decoupling, produces a smaller two-body contribution to the matrix elements for all cases studied here. The patterns demonstrated by the EOM-IMSRG are not observed for the VS-IMSRG, where the one- and twobody terms do not have consistent signs, and the E2 matrix element is reduced rather than increased by consistent evolution. This suggests that the renormalization of operators in the EOM-IMSRG is more predictable than it is in the VS-IMSRG. This is naively expected, as the EOM-IMSRG uses a more intuitive decoupling of the reference state for the specific nucleus of interest, where for the VS-IMSRG, the reference state used for normal ordering is not typically decoupled from excitations. The VS-IMSRG results demonstrate the critical effect of charge renormalization in 120 22 O, which has no sd-shell protons, and thus vanishing strength when using the bare operator. From the results shown here, it is evident that consistent operator evolution is of integral importance to ab initio nuclear structure calculations, as it is responsible for dressing the effective parameters of the transition, such as e, gl and gs . Furthermore, even when the relevant nucleons are protons, renormalization effects are important for an accurate description of transition matrix elements. 121 CHAPTER 6 SUMMARY AND CONCLUSIONS The present thesis focused on novel extensions and applications of the IMSRG to excited states and observables related to excited states. The particular vehicle chosen for this task was the newly formulated equations-of-motion IMSRG (EOM-IMSRG), which was demonstrated to be successful for describing low-lying, single excitation dominant excited states. The results here were focused on closed-shell nuclei and isobaric neighbors, which were accessed by generalized chargeexchange ladder operators. In our quest to describe excited states with the IMSRG, we also have introduced the vTDAIMSRG, which is a special case of the EOM-IMSRG, where explicit decoupling of the valence 1p1h block can in principle be achieved to make the method exact up to IMSRG(2) truncation errors. While this latter method has shown some promise in doubly magic nuclei as a means to describe odd-parity 1p1h states, the nature of the secondary valence space decoupling renders it impractical for more correlated systems. Thus, the generic EOM-IMSRG, which exchanges errors from the secondary decoupling for errors in approximate diagonalization, is championed as the preferred and more robust approach for excited states computed directly in the IMSRG. We demonstrated that EOM-IMSRG(2,2) produces results comparable to the well established EOMCCSD, and also that the method can be systematically improved using perturbative corrections to include triple-excitation components in the wave functions. With these corrections we demonstrated general improvements in spectra when compared with experiment, but we note that these corrections have yet to be analyzed rigorously. We have also introduced an effective operator formalism for the IMSRG, where now it is possible to consistently evolve operators of arbitrary spherical tensor rank. Thus, this work has introduced to the IMSRG the capability to handle electromagnetic transitions and moments, and in principle, we can also tackle Gamow-Teller strengths and various form factors. We benchmarked results for energies and electromagnetic observables using the EOM-IMSRG, the recently estab- 122 lished VS-IMSRG, and NCSM. General agreement between all three methods was observed for energies, B(E2), and B(M1) values in 14 C. As we go to heavier nuclei where NCSM is not applicable, we see a varied picture of consistency between the VS- and EOM-IMSRG. Here the VS-IMSRG is presumably more accurate for midshell nuclei such as 32 S, where the EOM-IMSRG neglects many-particle configurations within the valence space. For nuclei nearer to the extrema of the shell, such as 22 O, excellent agreement is exhibited. Despite this agreement, we note striking disagreement with experiment for B(E2) values in all but 14 C, which is ostensibly due to significant many-particle collectivity which is neglected by the truncation to two-body operators in both the IMSRG and EOM calculations. The EOM-IMSRG offers a path forward here, as the triples corrections are straightforward, and their wave function components can be used to add collectivity to the calculations of observables. The broad conclusion from this work is that EOM-IMSRG is a viable framework for which to compute nuclear excited states and observables, despite some notable limitations. The method is straightforwardly extended to a multireference formalism which will make it applicable to openshells. The MR-EOM-IMSRG, which we anticipate to arrive within a year, will be a powerful method to compute spectra, transitions, and response in open-shell nuclei. Response functions follow directly from the Lanczos method used to evaluate the equations of motion. We note that this is not straightforward in the shell-model formalism which has become popular in the ab initio community, as the required method of moments will certainly excite states outside of the valence space. The MR-EOM-IMSRG will derive the bulk of its utility from its lightweight computational requirements, as even in the multireference formalism, polynomial scaling will be observed. This will provide a simple tool to perform large computational sweeps, crossing the nuclear landscape in an attempt to analyze generic issues in nuclear physics, such as the quenching of gA and the enigma of neutrinoless double β -decay, which is tied intimately with the former point. Moreover, the methods discussed here are important to the nuclear community because it is desirable to have numerous approaches to ab initio nuclear structure problems, where we can make predictions and benchmark them against methods starting from the exact same fundamental 123 assumptions. For example, multiple ab initio methods producing consistent results for the neutrinoless double β -decay matrix element would mark a triumph for nuclear theory. Consistency is not the only goal for our methods however; here we return to the oft repeated distinguishment between precision and accuracy. As alluded to in this work, we have achieved precise predictions for observables and energies which are reproduced from method to method. However, we lack accuracy with experiment. It is crucial that we use these precise methods to facilitate the quest to achieve accuracy through improved interactions and currents, and for this, intricate understanding of the strengths and weaknesses of each method is paramount. 124 APPENDIX 125 A.1 Definitions for J-scheme Formulation Shorthand: (ama ) ≡ (ξa , ja , ma ) , (1) (a) ≡ (ξa , ja ) , √ Jˆ ≡ 2J + 1 , (2) Prs A...r...s... ≡ A...s...r... (4) Prs (J) ≡ (−1)( jr + js −J) Prs , (5) ∑ : sum over all m , (3) (6) {m} m-scheme antisymmetrized matrix elements: λµ A pq ≡ pm p |Aλµ |qmq , (7) λµ (8) A pqrs ≡ pm p , qmq |Aλµ |rmr sms , Clebsch-Gordan coefficients: j p jq J Cm p mq M ≡ j p m p ; jq mq |( j p jq )JM j p j¯q J Cm p m¯ q M ≡ (−1)( jq −mq ) j p m p ; jq − mq |( j p jq )JM , (9) (10) J-scheme matrix elements: Aλpq ≡ p||Aλ ||q , (J J )λ 1 2 A pqrs ≡ (pq)J1 ||Aλ ||(rs)J2 , (11) (12) Unnormalized matrix elements for practical use: (J1 J2 )λ A˜ pqrs ≡ (N pq(J ) )−1 (Nrs(J ) )−1 (pq)J1 ||Aλ ||(rs)J2 , 1 2 (13) A˜ Jpqrs ≡ (N pq(J) )−1 (Nrs(J) )−1 (pq)J0|A0 |(rs)J0 , (14) (N pq(J) )−1 ≡ (1 + δξ p ξq δ j p jq ) 1 + (−1)J δξ p ξq δ j p jq 126 (15) Pandya transformed matrix elements:      j p js J1        (J3 J4 )λ (J J )λ (J +J + j + j ) 1 2 q s ¯ ˆ ˆ ˆ ˆ 2 4 = − ∑ (−1) A psr J1 J2 J3 J4 jq jr J2 V˜ pqrs ¯ q¯     J3 J3    J J λ   3 4 (16) Spherical tensor product: [Aλ1 Bλ2 ]λµ = ∑ µ1 µ2 A.2 λ λ λ λ λ Cµ11 µ22 µ Aµ11 Bµ22 (17) Spherical tensor operators The definition of a spherical tensor TMJ is: [Jz , TMJ ] = M¯hTMJ , [J± , TMJ ] = h¯ (18) (J ± M + 1)(J ∓ M)TJ,M±1 , (19) ∑ m pa†pm p a pm p , (20) ∑ m∓p a†pm p a pm p∓1 , (21) ( j p ± m p + 1)( j p ∓ m p ) . (22) where Jz = h¯ pm p J± = h¯ pm p and m± p = One can very quickly show that a particle creation operator transforms as a tensor: [ jz , a†jm ] = ∑ m [a† a jm , a†jm ] = ma†jm , jm (23) m [ j± , a†jm ] = ∑ m∓ [a† a jm ∓1 , a†jm ] = m± a†jm±1 . jm (24) m Attempts to show this for the particle annihilation operator fail: [ jz , a jm ] = ∑ m [a† a jm , a jm ] = −ma jm , jm (25) m [ j± , a jm ] = ∑ m∓ [a† a jm ∓1 , a jm ] = −m∓ a jm∓1 . jm m 127 (26) An operator which does in fact transform as a tensor is a˜ jm = (−1)( j+m) a j,−m , (27) as is demonstrated here: [ jz , a˜ jm ] = ∑ m [a† a jm , a j,−m ](−1)( j+m) = ma j,−m (−1)( j+m) = ma˜ jm , jm (28) m [ j± , a˜ jm ] = ∑ m∓ [a† a jm ∓1 , a j,−m ](−1)( j+m) = −m± (−1)( j+m) a j,−m∓1 = m± a˜ j,(m±1) . jm m (29) jq jp Consider now the tensor product of two spherical tensor operators Am p and Bmq coupled to total angular momentum J: [A j p B jq ]JM = ∑ m p ,mq j p jq J jq jp (30) Cm p mq M Am p Bmq It can be shown that this product is a spherical tensor of rank J: [Jz , [A j p B jq ]JM ] = ∑ (m p + mq )Cm p mq M Am p Bmq = M[A j p B jq ]JM m p ,mq [J± , [A j p B jq ]JM ] = = j p jq J jq jp j p jq J jp [( j p )± + ( jq )± ,Cm p mq M Am p Bmq ] ∑ ± (m± p Cm p mq M Am p ±1 Bmq + mq Cm p mq M Am p Bmq ±1 ) ∑ ∓ (m∓ a Cm p ∓1mq M + mb Cm p mq ∓1M )Am p Bmq m p ,mq j p jq J jp j p jq J m p ,mq j p jq J jq j p jq J j p jq J jp (31) jq ∑ m p ,mq = jq jp [( j p )z + ( jq )z ,Cm p mq M Am p Bmq ] m p ,mq = j p jq J ∑ jp jq = M ±Cm p mq M±1 Am p Bmq = M ± [A j p B jq ]JM±1 jp jq jq (32) where the identity used to arrive at the third line comes from acting with J± on both M-scheme and J-scheme versions of a two body ket and equating the two expressions. 128 A.2.1 J-scheme operators in second quantization We can use the Wigner-Eckart theorem λ jq λ j p p j p m p |Tλ µ |q jq mq = jˆ−1 p TpqCmq µm p (33) to write a one-body operator in J-scheme: ∑ pq −1 λ jq λ j p † p j p m p |Tλ µ |q jq mq a†pm p aqmq = ∑ jˆp Tpq Cmq µm p a pm p aqmq pq j p jq λ λC ( jq −mq ) a† a = ∑ λˆ −1 Tpq pm p qmq m p −mq µ (−1) pq j p j¯q λ λC † = ∑ λˆ −1 Tpq m p m¯ q µ a pm p aqmq pq λ [a† a˜ ]λ = ∑ λˆ −1 Tpq p q µ (34) pq Consider now two particle excitations on the fermi-vacuum: [a†p a†q ]JM |Φ0 = N pq ∑ m p mq j p jq J Cm p mq M a†pm p a†qmq |Φ0 (35) The excitation operator transforms as a spherical tensor. In order for the de-excitation operator to transform in this way, we take the Hermitian conjugate (N pq [a†p a†q ]JM |Φ0 )† = N pq = N pq j p jq J ∑ Cm p mq M Φ0 |aqmq a pm p ∑ C−m p −mq −M (−1)( j p + jq −J) Φ0 |aqmq a pm p m p mq m p mq = −N pq ∑ m p mq j p jq J j p jq J Cm p mq −M (−1)(J−M) Φ0 |a˜ pm p a˜qmq = −N pq Φ0 |[a˜ p a˜q ]J−M (−1)(J−M) (36) The normalization constant is obtained by calculating the overlap of the two wavefunctions. We tend to use un-normalized matrix elements to avoid this issue however. We can from here construct 129 a two-body operator: 1 ∑ pm p, qmq|Vµλ |rmr , sms a†pm p a†qmq asms armr 4 pqrs = 1 −1 N −1C j p jq J1 C jr js J2 (pq)J M |V λ |(rs)J M a† a† a N pq ∑ ∑ 1 1 µ 2 2 pm p qmq sms armr rs m p mq M1 mr ms M2 4 J J pqrs 1 2 M1 M2 = 1 j p jq J1 1 j j J J λ J1 ˜ (J1 J2 )λ † Cm p mq M Cmrr ms s2M CM2 µM Vpqrs a pm p a†qmq asms armr ∑ ∑ 1 2 2 1 4 J J pqrs Jˆ1 1 2 M1 M2 = 1 j p jq J1 1 j j J J J2 λ (J1 J2 )λ † Cm p mq M Cmrr ms s2M CM1 −M (−1)(J2 −M2 )V˜ pqrs a pm p a†qmq asms armr ∑ ∑ µ ˆ 1 2 1 2 4 J J pqrs λ 1 2 M1 M2 =− 1 jr js J2 1 J J2 λ (J1 J2 )λ † † J1 C−mr −ms M CM1 −M (−1)(J2 −M2 )V˜ pqrs [a p aq ]M ar−mr as−ms ∑ ∑ µ ˆ 2 1 2 1 4 J J pqrs λ 1 2 M1 M2 =− 1 jr js J2 1 (J1 J2 )λ † † J1 J J2 λ (−1)( jr + js +mr +ms )V˜ pqrs [a p aq ]M ar−mr as−ms Cmr ms −M CM1 −M ∑ ∑ µ 1 2 1 2 4 J J pqrs λˆ 1 2 M1 M2 =− 1 J1 J2 λ 1 J2 (J1 J2 )λ † † J1 CM −M µ V˜ pqrs [a p aq ]M [a˜r a˜s ]−M ∑ ∑ 2 1 2 1 4 J J pqrs λˆ 1 2 M1 M2 =− A.2.2 1 1 ˜ (J1 J2 )λ † † J Vpqrs [[a p aq ] 1 [a˜r a˜s ]J2 ]]λµ ∑ ∑ 4 J J pqrs λˆ 1 2 (37) Normal-ordered spherical tensor operators If we normal order these operators, we find: ∑ ∑ pq m p mq pm p |Tλ µ |qmq a†pm p aqmq λ λ {a† a˜ }λ + = ∑ λˆ −1 Tpq p q µ ∑ Tii ni pq imi (−1)( ji −mi ) ji ji λ Cm −m µ i i λˆ λ {a† a˜ }λ + ˆ λ = ∑ λˆ −1 Tpq j p jq µ ∑ ji Tii ni δλ 0 pq (38) i We find a similar zero body piece when we normal order the two-body operator: 1 (JJ)λ JˆV˜i ji j ni n j δλ 0 ∑ ∑ 2 ij J 130 (39) The in-medium one-body piece from the free-space two-body operator is:    J ji j p   Jˆ1 Jˆ2 ˜ (J1 J2 )λ 1 ( j + j +J +λ ) p i 2 V n (−1) {a†p a˜q }λµ i ∑ ∑ λˆ piqi    jq λ J2  J1 J2 pqi (40) For scalar operators, this reduces to Jˆ2 ˜ J ∑ ∑ jˆ2 Vpibini{a†paq} p (41) J pqi In summary, a spherical tensor operator Aλµ can be expressed in normal ordered form: 1 (JJ)0 {Vµλ } = δλ 0 δµ0 [∑ jˆiVii0 ni + ∑ ∑ JˆV˜i ji j ni n j ] 2 ij J i      J1 J2 λ  1 λJ J † λ λ + ∑{a p a˜q }µ (Vpq + ∑ ∑ Jˆ1 Jˆ2 (−1)(λ +J2 + j p + ji )V˜ piqi1 2 nk ) ˆλ pq    jq j p ji  i J1 J2 − 1 λJ J {[a†p a†q ]J1 [a˜r a˜s ]J2 }λµ V˜ pqrs1 2 ∑ ∑ ˆ 4λ pqrs J1 J2 (42) Typically, we use non-reduced matrix elements for the scalar operators, i.e.: p|X|q = 1 p||X 0 ||q . jˆp (43) This is possible because the non-reduced matrix elements are independent of M just as the reduced ones are. So in a typical IMSRG code you might calculate the zero body piece of a scalar operator as: 1 E0 = ∑ jˆi2Vii ni + ∑ ∑ Jˆ2V˜iJji j ni n j 2 ij J i A.3 (44) Jucys diagrams In the following sections it becomes necessary to use a diagrammatic strategy to execute the required angular momentum coupling. Without immense digression into the theory of Jucys angular momentum diagrams, we introduce some relevant definitions and identities for this work. We implore the reader to learn more about these fantastic diagrams in Ref. [182]. 131 A.3.1 Basic Components and Symmetries First we introduce the basic diagrams for Clebsch-Gordan coefficients, j1 j j j Cm11 m223m3 = jˆ3 (−1)2 j2 + j2 , (45) j3 j1 j j¯ j j j2 j3 Cm1 m2¯ 3m ≡ Cm1 −m (−1) j2 −m2 = jˆ3 1 2 m3 1 2 3 + j2 . (46) j3 Figure A.1 Diagrammatic representation of Clebsch-Gordan coefficients The diagrams consist of three oriented lines and an oriented vertex. The diagrams are based on 3 jsymbols, hence the phase and scale factors that pop out when you equate them to Clebsch-Gordan coefficients. The orientation of the lines indicates whether or not the specific angular momenta is time-reversed, and the orientation of the vertex indicates the direction you would read off a 3 j-symbol from these diagrams, with + indicating clockwise. Some diagrammatic represenations of the symmetries of the 3 j-symbol: j2 j2 + j1 = (−1) j1 + j2 + j3 j3 − j1 = j3 j2 − j2 , (47) j3 j2 + j1 j3 j1 = (−1)2 j3 + j1 , j3 Figure A.2 Symmetries of Jucys diagrams. 132 (48) Figure A.2 (cont’d) j2 j3 j2 + j1 = j3 + j1 . (49) Here we see diagrammatic rules for permutation of columns and time reversal of all angular momenta. A.3.2 Diagram manipulation The utility of Jucys diagrams comes from contracting the lines, which corresponds to summing out an angular-momentum substate, ∑ m1 j j j j j j Cm11 m223m3 Cm44 m551m1 j2 j 2( j + j ) ˆ ˆ = j1 j3 (−1) 2 5 + 3 j4 j5 + (50) j1 Figure A.3 Jucys contraction. Now we have arrived at the concept of external and internal lines. For computation of J-scheme expressions (where all m-dependence has been summed out of the problem), our goal is to construct diagrams consisting only of internal lines. We need to be able to change the orientation of a line. For internal lines labeled j, this is as simple as multiplying by (−1)2 j . You get a sign change if you flip the orientation of a half-integer angular momenta. For external lines, time reversal still has an explicit effect, as the line still has m-dependence. Thus for an external line labeled j, you must multiply by (−1) j−m in order to flip the line orientation. 133 Once all lines are connected, we must be able to factorize large diagrams. There are only two rules that are important for our purposes, j2 j2 j2 = ∑ jˆ32 j3 j1 − + j1 j3 j3 j2 = j1 j1 , j3 (51) j1 j3 j2 − + j2 . j1 (52) Figure A.4 Jucys factorization rules. With these two rules, we can factorize pretty much everything we encounter into 6 j and 9 j symbols, + j1 j4 j2 j5 j3 j6 j1 j5 = j6 , + j3 j2 + + j4 j8 j2 j5 j8  j3  j6 =  j9 − j1 + − j9 j2   j1 j  4 j7 j3 j7 j5 − + + j6 j4 Figure A.5 Jucys diagrams for 6 j- and 9 j-symbols. 134 (53) (54) A.4 Particle-hole picture For certain derivations, it is convenient to formulate angular momentum coupling in terms of particle-hole creation and annihilation operators. The hole creation operator, which creates a time reversed particle, is given by a¯†m p = a˜m p = (−1) p+m p a−m p , (55) and the corresponding annihilation operator by a¯m p = a˜†m p = (−1) p−m p a†−m p . (56) Again we must alter the form of the annihilation operator to ensure that it is a spherical tensor. a˜¯m p = −a†m p . (57) With these definitions, we can construct particle-hole creators p |Φq (JM) = Cm p mq M a†pm p a¯†qmq |Φ0 , ∑ Cm p m¯ q M a†pm p aqmq |Φ0 , m p mq ≡ j p jq J ∑ m p mq j p j¯q J = [a†p a¯†q ]JM |Φ0 , (58) and corresponding annihilators p Φq (JM)| = ∑ jq j p J mq m p p Φq (JM)| = Cmq m p −M (−1)( jq + j p +M) Φ0 |a˜¯qmq a˜ pm p j¯p jq J¯ Cm¯ m M¯ Φ0 |a pm p a†qmq p q mq m p ∑ = (−1)J−M Φ0 |[a˜ p a˜¯q ]J−M . (59) Now we may construct a two-body operator in the particle-hole picture. − 1 (J1 J2 )λ † † J V˜ pqrs [[a p aq ] 1 [a˜r a˜s ]J2 ]λµ ∑ 4λˆ pqrs J1 J2 = 1 j p jq J (J1 J2 )λ j j J J J¯2 λ † † V˜ pqrs Cm p mq1M Cmrr ms s2M CM1 M ∑ ∑ ¯ µ am p amq ams amr ˆ 1 2 1 2 4λ pqrs m p mq mr ms J1 J2 M1 M2 135 (60) = 1 (J1 J2 )λ j js (61) V˜ pqrs a†m p a†mq ams amr (−1)(J1 +J2 +λ ) Jˆ1 Jˆ2 p jq λ jr ∑ ∑ 4 pqrs m p mq mr ms + + + J1 J2 J1 J2 Figure A.6 Jucys diagram for a two-body operator. The sum over the Clebsch-Gordan coefficients, represented as a Jucys diagram, can be recoupled using Jucys factorization rules, jp j = (−1)(J1 −m p −mq ) q js jq λ jr + + + J1 J2 jq − = ∑ (Jˆ3Jˆ4 )2 jr J4 λ jr j p + J3 J4 jp js − J3 (−1)(J1 −m p −mq ) + jq js + + J1 = ∑ + J2 (−1)(J2 +λ −m p −mq ) (Jˆ3 Jˆ4 )2 J3 J4 js j p λ jr + + + J1 J2 J3 + J4 jp jr − jq +   j p js (J −m −m ) 2 p q ˆ ˆ jq jr = ∑ (−1) 1 (J3 J4 )  J3 J4 J1 J2   jp (J +J +λ + j − j ) 2 q s ˆ ˆ 1 4 jq = ∑ (−1) (J3 J4 )  J3 J4 J1 − js + jq λ J2 js jr λ j p + + − J4 J3 − J1  J3  js J4 jq j λj  − r − p − λ  J4 J3 js J3  js jr J4 jr jq λ j p  + + + J2 λ J4 J3    j p js J3  jr j¯q J j p j¯s J J λ J3 = ∑ (−1)(J1 +J4 +λ + jq + js ) Jˆ4 jq jr J4 Cmr m¯q4M Cms m¯p3M CM4 µM . ∑ 4 3 4 3 M M  J3 J4 J1 J2 λ 3 4 Figure A.7 Diagrammatic derivation of the Pandya transform. 136 (62) Thus, − 1 (J1 J2 )λ † † J V˜ pqrs [[a p aq ] 1 [a˜r a˜s ]J2 ]λµ ∑ ˆ 4λ pqrs J1 J2      j p js J3        1 (J J )λ (J +J + j + j ) 1 2 q s ˜ ˆ ˆ ˆ 2 4 V (−1) J J J = ∑ pqrs 1 2 4 jq jr J4   4 pqrs      J1 J2 J3 J4 J J λ   1 2 j p j¯s J jr j¯q J J λJ ∑ Cm pm¯s3M3Cmr m¯q4M4CM44 µM3 3 a†m p ams amr a†mq m m m × mp q r s M3 M4      j p js J3        1 (J J )λ (J +J + j + j ) 1 2 q s ˜ ˆ ˆ ˆ 2 4 V (−1) J J J = ∑ pqrs 1 2 4 jq jr J4   4 pqrs      J1 J2 J3 J4 J J λ   1 2 × ˆ j p j¯s J3 † a C j¯r jq J¯4 a a† J3 CJ3 J¯4 λ C a m m ∑ m p m¯s M3 m p s m¯ r mq M¯ 4 r mq ˆ M3 M¯ 4 µ λ m p mq mr ms M3 M4      j p js J3        1 (J J )λ (J +J + j + j ) † †J J λ 1 2 q s ˜ ˆ ˆ ˆ ˆ 2 4 = V (−1) J J J J pqrs ∑ 1 2 3 4 jq jr J4 [[a p a¯s ] 3 [a˜r a˜¯q ] 4 ]µ ˆ   4λ pqrs      J1 J2 J3 J4 J J λ   1 2 (63) The particle-hole formalism two-body operator has been “Pandya transformed”, − 1 1 (J3 J4 )λ † † J (J1 J2 )λ † † J V˜ pqrs [[a p aq ] 1 [a˜r a˜s ]J2 ]λµ ≡ − V¯ psr [[a p a¯s ] 3 [a˜r a˜¯q ]J4 ]λµ ∑ ∑ ¯ q¯ ˆ ˆ 4λ pqrs 4λ pqr ¯ s¯ J1 J2 where (64) J3 J4      j p js J3        (J3 J4 )λ (J1 J2 )λ (J +J + j + j ) q s ¯ ˆ ˆ ˆ ˆ 2 4 Vpsr = − ∑ (−1) J1 J2 J3 J4 jq jr J4 V˜ pqrs ¯ q¯     J1 J2    J J λ   1 2 (65) Pandya transformed matrix elements are very useful for computing terms with multiple Wick contractions acting on particle and hole states. 137 A.5 Derivation of the general tensor product In order to calculate moments and certain transitions in the EOM-IMSRG(2,2), the spherical tensor product [Aλ1 Bλ2 ]λµ = ∑ µ1 µ2 µ λ λ λ λ λ Cµ11 µ22 µ Aµ11 Bµ22 (66) must be computed. We will include up to two-body operators in accordance with the truncations of both the EOM and IMSRG calculations. The tensor product of A and B is named Z, for which we may compute reduced matrix-elements XJX ||Z λ ||Y JY = 1 λ λ λ λ λ J λ JX CMY µM Cµ11 µ22 µ XJX MX |Aµ11 Bµ22 |Y JY MY , ∑ ∑ ˆ Y X JX µM M µ1 µ2 X Y (67) 1 λˆ (68) or equivalently XJX ||Z λ ||Y JY = J J¯ λ λ λ λ λ λ CMX MY¯ µ Cµ11 µ22 µ XJX MX |Aµ11 Bµ22 |Y JY MY . x Y µ µ µMX MY 1 2 ∑ ∑ This calculation proceeds in two steps: 1. Expand A and B in m-scheme and perform all possible Wick contractions. 2. Return full expression to J-scheme by summing out remaining angular momentum substates. A.5.1 Step one: Wick Contractions We write out the relevant multiplications of two normal-ordered operator strings, {a†p aq }{a†r as } = δqr δ ps n p n¯ q (69) = δqr n¯ q {a†p as } − δ ps n p {a†q ar } (70) = {a†p a†r as aq } (71) = (1 − Ppq )(1 − Prs )δ pu δrt n p n¯ r {a†q as } (72) = (1 − Prs )δtr n¯ r {a†p a†q as au } + (1 − Ppq )δ pu n p {a†q at† as ar } (73) 0b {a†p aq }{a†r as } 1b {a†p aq }{a†r as } 2b {a†p a†q as ar }{at† au } {a†p a†q as ar }{at† au } 1b 2b 138 {a†p a†q as ar }{at† a†u aw av } = (1 − Prs )(1 − Ppq )δ pv δqw δsu δrt n p nq n¯ s n¯ r (74) 0b {a†p a†q as ar }{at† a†u aw av } = (1 − Ppq )(1 − Prs )(1 − Pwv )δqw δsu δrt nq n¯ r n¯ s {a†p av } 1b − (1 − Ppq )(1 − Prs )(1 − Ptu )δ pv δqw δsu n p nq n¯ s {at† ar } {a†p a†q as ar }{at† a†u aw av } (75) = (1 − Prs )δrt δsu n¯ r n¯ s {a†p a†q aw av } 2b + (1 − Ppq )δ pv δqw n p nq {at† a†u as ar } − (1 − Ppq )(1 − Prs )(1 − Ptu )(1 − Pvw )δrt δqw nq n¯ q {a†p at† as av } . (76) With these expressions, we can perform step one for all terms up to and including two-body operators. We will do this for one example only. For simplicity, we denote the R-body product of an N and M body operator as Z(NM → R). Z λpq (11 → 1) ∼ 1 λˆ λˆ j j¯ λ j j¯ λ j p j¯q λ ∑ ∑ Ars1 Btu2Cmrr ms¯ s 1µ1Cmtt mu¯ u 2µ2Cµ11 µ22 µ Cm pm¯ q µ λ λ λ λ λ {a†r as }{at† au } 1 2 {m} rstu = 1 ˆλ λˆ j j¯ λ j j¯ λ j p j¯q λ ∑ ∑ Ars1 Btu2Cmrr ms¯ s 1µ1Cmtt mu¯ u 2µ2Cµ11 µ22 µ Cm pm¯ q µ λ λ λ λ λ 1b δst n¯ s {a†r au } − δru nr {at† as } . 1 2 {m} rstu (77) Z λpq (11 → 1) = 1 ˆλ λˆ j p j¯r λ λ λ js j¯q λ j p j¯q λ λ λ λ ∑ ∑ A pr1 Bsq2Cm pm¯ r 1µ1Cmsm¯ q 2µ2Cµ11 µ22 µ Cm pm¯ q µ n¯r δrs 1 2 {m} rs − 1 λˆ λˆ λ λ j p j¯r λ js j¯q λ λ λ λ j p j¯q λ ∑ ∑ B pr2 Asq1Cm pm¯ r 2µ2Cmsm¯ q 1µ1Cµ11 µ22 µ Cm pm¯ q µ nr δrs 1 2 {m} rs = 1 λˆ λˆ λ 1 2 r − j p j¯r λ λ jr j¯q λ λ λ λ j p j¯q λ ∑ A pr1 Brq2 ∑ Cm pm¯ r 1µ1Cmr m¯ q 2µ2Cµ11 µ22 µ Cm pm¯ q µ n¯r 1 λˆ λˆ {m} λ j p j¯r λ λ jr j¯q λ λ λ λ j p j¯q λ ∑ B pr2 Arq1 ∑ Cm pm¯ r 2µ2Cmr m¯ q 1µ1Cµ11 µ22 µ Cm pm¯ q µ nr 1 2 r (78) {m} From here, we may proceed with step two and sum out all of the m-dependence remaining in the expression. While we will explicitly demonstrate step two for each term in the multiplication, we will skip the demonstration of step one for all except Z(11 → 1), and immediately jump to step 139 two. A.5.2 Step two: angular momentum coupling To verify the final expressions, we have derived them using both the angular-momentum relations of [183] and the Jucys diagrammatic formalism summarized in [182]. As the latter method is more transparent (after an admittedly steep learning curve has been surmounted), we will use this strategy here. A.5.2.1 Zero-body terms Z0 (11 → 0) = 1 ˆλ λˆ j p j¯q λ λ λ λ jq j¯p λ λ λ ∑ Cµ11 µ22 µ ∑ Cm pm¯ q 1µ1Cmqm¯ p 2µ2 A pq1 Bqp2 nqn¯ p 1 2 {m} pq 1 λ λ λ λ λ Cµ11 µ22 µ ∑ δλ λ δµ1 −µ2 A pq1 Bqp2 (−1)( j p − jq −µ1 ) nq n¯ p ∑ 1 2 λˆ 1 λˆ 2 µ1 µ2 µ pq 1 λ λ = δλ 0 δλ λ ∑ A pq1 Bqp2 (−1)( j p − jq +λ1 ) nq n¯ p 1 2 ˆλ pq = (79) 1 Analogously, Z0 (22 → 0) = 1 (J1 J2 )λ1 (J2 J1 )λ2 δ δ Brspq (−1)(J1 −J2 +λ1 ) n p nq n¯ r n¯ s ∑ A pqrs ˆλ λ 0 λ1 λ2 pqrs 1 A.5.2.2 (80) J1 J2 One-body terms Z λpq (11 → 1) = × 1 jq λ j p λ λ λ Cmq µm p Cµ11 µ22 µ ˆ ˆ jˆp λ1 λ2 ∑ r{m} j p j¯r λ jr j¯q λ λ λ j p j¯r λ jr j¯q λ λ λ n¯ rCm p m¯ r 1µ Cmr m¯ q 2µ A pr1 Brq2 − nrCm p m¯ r 2µ Cmr m¯ q 1µ B pr2 Arq1 . 2 1 1 2 We will look at only the first term, as the second follows analogously. 140 (81) 1 jq λ j p λ λ λ j p j¯r λ jr j¯q λ Cmq µm p Cµ11 µ22 µ Cm p m¯ r 1µ Cmr m¯ q 2µ ∑ 1 2 jˆp λˆ 1 λˆ 2 {m} jq = ∑ λˆ jp λ1 + + jr jq jr + + (82) λ2 λ jp λ λ1 λ2 Figure A.8 Expression of the first term of eq. 81 with Jucys Diagrams. Here we see that each external line can be contracted if we reverse time for the two leftmost diagrams and permute the indices of all but the first. + λ2 jq 1 jq λ j p λ λ λ j p j¯r λ jr j¯q λ Cmq µm p Cµ11 µ22 µ Cm p m¯ r 1µ Cmr m¯ q 2µ = λˆ ∑ 1 2 jˆp λˆ 1 λˆ 2 {m} jr (−1)( j p + jq +λ ) , + λ λ1 + + jp λ1 λ2 λ = λˆ (−1)( j p + jq +λ ) . jq j p jr (83) Figure A.9 Contraction of all Jucys diagrams in eq. 82. The second term is the same up to an index exchange and a phase. Hence, Z λpq (11 → 1) = λˆ (−1)( j p + jq +λ ) ∑ n¯ r jr   λ 1 λ2   λ   jq jp  jr  λ λ A pr1 Brq2 − nr   λ 1 λ2   λ   jp jq  jr  λ λ (−1)(λ1 +λ2 +λ ) B pr2 Arq1 . (84) For the next term, we start from the Pandya two-body representation, which is more straightforward 141 for sums over particle and hole states, Z λpq (21 → 1) = = = 1 jˆp λˆ 1 λˆ 2 ∑ rs jq λ j p λ λ λ j j¯ λ j p j¯q J j j¯ J J J¯2 λ1 ¯ (J1 J2 )λ1 λ2 Cmq µm p Cµ11 µ22 µ Cmrr ms¯ s 2µ Cm p m¯ q1M Cmrr ms¯ s2M CM1 M A pqr Brs ns n¯ r , ¯ ¯ s¯ 2 1 2 1 2 µ1 J1 J2 {m} 1 jq λ j p λ λ λ j p j¯q J J λ¯ λ (J1 λ2 )λ1 λ2 Cmq µm p Cµ11 µ22 µ Cm p m¯ q1M CM1 µ¯2 µ1 A¯ pqr Brs ns n¯ r , ∑ ¯ s¯ 1 1 2 1 jˆp λˆ 1 λˆ 2 rs{m}J 1 ∑ jˆp λˆ λˆ 2 rs{m} 1 jq λ j p j p j¯q λ (λ λ2 )λ1 λ2 Cmq µm p Cm p m¯ q µ A¯ pqr Brs ns n¯ r (−1)(λ1 +λ2 +λ ) , ¯ s¯ 1 (λ λ )λ λ ∑ A¯ pqr¯ s2¯ 1 Brs2 nsn¯r (−1)(λ1+λ2+λ ) , ˆλ 2 λˆ rsµ 2 1 (λ λ2 )λ1 λ2 Brs ns n¯ r (−1)(λ1 +λ2 +λ ) . = A¯ pqr ¯ s¯ ˆλ ∑ rs = (85) 2 Likewise, 1 λ (λ1 λ )λ2 Ars1 B¯ rsp nr n¯ s . ∑ ¯ q¯ λˆ 1 rs The final contribution to Z pq is from two two-body operators, Z λpq (12 → 1) = Z λpq (22 → 1) = (86) 1 1 ∑ 2 jˆp λˆ 1 λˆ 2 rst{m} J1 J2 J3 ¯ ¯ jq jr J3 J2 J3 λ2 J J2 λ1 (J1 J2 )λ1 ˜ (J2 J3 )λ2 jq λ j p λ1 λ2 λ j p jr J1 nr n¯ s n¯t A˜ prst Bstqr Cmq µm p Cµ1 µ2 µ Cm p mr M CM1 M ¯ µ Cmq mr M3 CM M¯ µ 1 2 3 2 1 2 1 ¯ ¯ jq jr J J J3 λ1 (J1 J2 )λ2 ˜ (J2 J3 )λ1 jq λ j p λ1 λ2 λ j p jr J1 J J2 λ2 − n¯ r ns nt B˜ prst Astqr Cmq µm p Cµ1 µ2 µ Cm p mr M CM1 M Cmq mr3M CM2 M ¯ ¯ µ 1 3 1 2 2 2 3 µ1 The angular momentum coupling is achieved using Jucys diagrams, 1 jq λ j p λ λ λ j p jr J jq jr J3 J2 J¯3 λ2 J J¯2 λ1 Cmq µm p Cµ11 µ22 µ Cm p mr1M CM1 M C C ∑ ¯ ¯ 1 1 2 µ1 mq mr M3 M2 M3 µ2 jˆp λˆ 1 λˆ 2 {m} jq = λˆ Jˆ1 Jˆ2 ∑ + + + λ {m} jp λ1 jr λ2 jp λ J1 Figure A.10 Derivation of Z λpq (22 → 1) with Jucys diagrams. 142 (87) Figure A.10 (cont’d) × jq J1 + + J2 J2 + J3 jr J3 λ1 jq λ2 + + jr J3 = λˆ Jˆ1 Jˆ2 (−1)( jq − jr +J3 +λ +λ1 +λ2 ) jp λ + λ2 + + J1 J2 + λ1 + J1 = λˆ Jˆ1 Jˆ2 (−1)( j p + jr +J1 ) + J2 λ J3 + J J λ = λˆ Jˆ1 Jˆ2 (−1)( j p + jr +J1 ) 1 2 1 λ2 λ J3 J3 λ1 jq jr λ - λ2 J3 jr jq j p λ J1 J1 jp + . (88) Applying this expression to eq. 87, we arrive at          j p jq λ    1ˆ (J1 J2 )λ2 ˜ (J2 J3 )λ1 λ1 λ2 λ λ ˆ ˆ ˜ n¯ r ns nt Brpst Astqr Z pq (22 → 1) = λ ∑ J1 J2   2  J3 J1 jr    J1 J3 J2   rst J1 J2 J3       (J1 J2 )λ1 ˜ (J2 J3 )λ2 λ1 λ2 λ ˜ − nr n¯ s n¯t Arpst Bstqr (89)   J3 J1 J2   A.5.2.3 Two-body terms For two-body operators, we start with the disconnected term, 143 (J J )λ 1 2 (11 → 2) = Z pqrs × 1 λ λ (1 − Ppq (J1 ))(1 − Prs (J2 ))A pr1 Bqs2 ˆλ λˆ λˆ 1 2 j p jq J J J¯ λ j j J jq j¯s λ j p j¯r λ ∑ Cm pmq1M1Cmrr ms s2M2CM11M2¯ 2 µ Cµ11 µ22 µ Cm pm¯ r 1µ1Cmqm¯ s 2µ2 λ λ λ {m} λ λ = (1 − Ppq (J1 ))(1 − Prs (J2 ))A pr1 Bqs2 jp jr × ∑ λˆ Jˆ1Jˆ2 + {m} + + jq λ1 js λ2 J1 J2 λ jp jq J1 + jr + js λ2 λ1 + J2 (90) jp λ + jr λ λ = (1 − Ppq (J1 ))(1 − Prs (J2 ))A pr1 Bqs2 + λ1 J1 ∑ λˆ Jˆ1Jˆ2 + j + q J 2 {m} js λ + +    j p jr λ1 λ2 λ λ = (1 − Ppq (J1 ))(1 − Prs (J2 ))A pr1 Bqs2 λˆ Jˆ1 Jˆ2 jq js λ2 . (91)   J1 J2 λ (J J )λ 1 2 (11 → 2) using Jucys diagrams. Figure A.11 Derivation of Z pqrs Next, we couple the (21 → 2) and (12 → 2) multiplications. There are four unique terms here, which we will express in m-scheme first for clarity. Z pqrs (21 → 2) = ∑ (1 − Prs )A pqts Btr n¯t − (1 − Ppq )B pt Atqrs nt (92) Z pqrs (12 → 2) = ∑ (1 − Ppq )A pt Btqrs n¯t − (1 − Prs )B pqts Atr nt (93) tmt tmt Now we will express the first term of eq. 92 in J-scheme, 144 1 (J1 J3 )λ1 λ2 (1 − Prs (J2 ))A pqts Btr ˆλ λˆ λˆ ∑ 1 2 tJ3 (J J )λ 1 2 (21 → 2a) = Z pqrs × J J¯ λ j j J J J¯ λ j j J j j¯ λ ∑ CM11M2¯ 2 µ Cmrr ms s2M2Cµ11 µ22 µ Cmtt ms s3M1CM11M3¯ 31µ1Cmtt mr¯ r 2µ2 λ λ λ (94) {m} where 1 λˆ λˆ λˆ J J¯ λ j j J J J¯ λ j j J j j¯ λ ∑ CM11M2¯ 2 µ Cmrr ms s2M2Cµ11 µ22 µ Cmtt ms s3M1CM11M3¯ 31µ1Cmtt mr¯ r 2µ2 λ λ λ 1 2 {m} jr = ∑ λˆ Jˆ2Jˆ3 jt + js J2 J1 × λ + + js {m} + λ1 J3 λ J1 J2 λ2 + jt J3 jr + λ2 λ1 jr + λ2 = λˆ Jˆ2 Jˆ3 (−1)(J2 +J3 + jt − jr ) + jt js J2 + λ + + J3 λ1 + J1 + + λ1 = λˆ Jˆ2 Jˆ3 (−1)( jt + js +J1 +λ2 +λ ) J2 - jt - jr + = λˆ Jˆ2 Jˆ3 (−1)( jt + js +J1 +λ2 +λ ) λ λ1 λ2 J3 J2 J1 (J J )λ J2 - J3 J1 J3 js λ2 λ + js jt J3 λ2 J2 jr . 1 2 (21 → 1) using Jucys diagrams. Figure A.12 Derivation of Z pqrs 145 λ2 (95) From this, and analagous derivations, we arrive at (J J )λ 1 2 (12 → 2) Z pqrs = (1 − Ppq (J1 ))λˆ Jˆ1 ∑ ∑ Jˆ3 (−1)( j p + jq +J1 +J2 +J3 +λ1 +λ ) t J  3       j p jq J  λ λ λ   1 1 2 λ (J3 J2 )λ2 × A pt1 B˜tqrs n¯t   J3 λ1 ja    J2 J1 J3        jr js J   λ λ λ   2 1 2 λ (J1 J3 )λ2 (J +J +λ ) ˆ ˆ ˆ 1 3 2 +(1 − Prs (J2 ))λ J2 ∑ ∑ J3 (−1) Atr1 B˜ pqst nt (96)  J3 λ1 ji    J1 J2 J3   t J3 and (J J )λ 1 2 (21 → 2) Z pqrs    jr      js J2  λ1 λ2 λ  λ (J1 J3 )λ1 (J +J +λ +λ ) ˆ = −(1 − Prs (J2 ))λ Jˆ2 ∑ ∑ Jˆ3 (−1) 1 3 2 Btr2 A˜ pqst n¯t     J3 λ2 ja   J2 J1 J3  t J3 −(1 − Ppq (J1 ))λˆ Jˆ1 ∑ ∑ Jˆ3 (−1)( j p + jq +J1 +J2 +J3 +λ1 ) t J  3       j p jq J  λ λ λ   1 1 2 λ (J3 J2 )λ2 B pt2 A˜ tqrs nt . ×      J3 λ2 ji   J1 J2 J3  (97) The (22 → 2) term has three unique types of contractions. As demonstrated in eq. 76, we have pp and hh ladder diagrams, and the ph chain diagrams. We will start with the ladder diagrams, which are quite simple compared with the chain. (J J )λ 1 2 (22 → 2(pp, hh)) = Z pqrs 1 (J1 J3 )λ1 (J3 J2 )λ2 A˜ pqtu B˜turs n¯t n¯ u ∑ ˆ ˆ ˆ 2λ λ1 λ2 tuJ3 × J J¯ λ λ λ λ J J¯ λ J J¯ λ ∑ CM11M2¯ 2 µ Cµ11 µ22 µ CM11M3¯ 31µ1CM33M2¯ 22µ2 {m} + 1 (J1 J3 )λ2 ˜ (J3 J2 )λ1 B˜ pqtu Aturs nt nu ∑ ˆ ˆ ˆ 2λ λ1 λ2 tuJ3 × J J¯ λ λ λ λ J J¯ λ J J¯ λ ∑ CM11M2¯ 2 µ Cµ11 µ22 µ CM11M3¯ 32µ2CM33M2¯ 21µ1 {m} 146 (98) where 1 λˆ λˆ λˆ λ λ λ J J¯ λ J J¯ λ J J¯ λ ∑ CM11M2¯ 2 µ Cµ11 µ22 µ CM11M3¯ 31µ1CM33M2¯ 22µ2 1 2 {m} = ∑ λˆ λ1 J1 + + J3 J1 J2 + J3 + J2 λ2 {m} λ λ λ1 λ2 + λ1 λ λ2 = λˆ (−1)(J1 +J2 +λ ) J1 J3 J2 + λ λ λ = λˆ (−1)(J1 +J2 +λ ) 1 2 J2 J1 J3 , (99) (J J )λ 1 2 (22 → 2(pp, hh)) using Jucys diagrams. Figure A.13 Derivation of Z pqrs hence, (J J )λ 1 2 (22 → 2(pp, hh) Z pqrs   λ   λ 1 1 λ2 (J1 J3 )λ1 (J3 J2 )λ2 = λˆ ∑ (−1)(J1 +J2 +λ ) A˜ pqtu B˜turs n¯t n¯ u   2 tuJ   J J J 3 2 1 3    λ λ λ   1ˆ 1 2 (J1 J3 )λ2 ˜ (J3 J2 )λ1 (J +J +λ +λ ) B˜ pqtu Aturs nt nu + λ ∑ (−1) 1 2 1 2   2 tuJ   J1 J2 J3 3 For the final term, the ph chain, we use Pandya-transformed matrix elements, 1 (1 + Ppq (J1 ))(1 + Ppq (J2 )) ˆλ λˆ λˆ 1 2     j p js J3         (J3 J5 )λ1 (J5 J4 )λ2 ( j + j +J +J ) q s ˆ ˆ ˆ ˆ 2 4 × ∑ J1 J2 J3 J4 (−1) B¯t ur n¯t nu jq jr J4 A¯ pst¯ u¯ ¯ q¯     tu    J3 J4 J5 J J λ   1 2 (J1 J2 )λ Z˜ pqrs (22 → 2(ph)) = 147 (100) × λ λ λ J J¯ λ J J¯ λ J J¯ λ ∑ CM33M4¯ 4 µ Cµ11 µ22 µ CM33M5¯ 51µ1CM55M4¯ 42µ2 (101) {m} Here we have applied the inverse pandya transform and anti-symmetrization operators to construct (J J )λ 1 2 . The angular momentum coupling is exactly the same as in the last case, Z pqrs (J1 J2 )λ Z˜ pqrs (22 → 2(ph)) = − λˆ (1 + Ppq (J1 ))(1 + Ppq (J2 ))        j p js J3       λ λ λ   (J J )λ (J5 J4 )λ2 1 2 ( j + j +J +J +λ ) q s ˆ ˆ ˆ ˆ 2 3 × ∑ J1 J2 J3 J4 (−1) A¯ pst¯3u¯5 1 B¯t ur n¯t nu jq jr J4 ¯ q¯         tu J J J   4 3 5  J3 J4 J5 J J λ   1 2 (102) A.6 Compilation of the full tensor product Here we compile the results of the previous section for use in the construction of transition matrix elements in the EOM-IMSRG(2,2). For nuclear physics applications, EOM-IMSRG(2,2) ladder operators are linear combinations of one- and two-body excitation operators coupled to desired spin J Π Xν† (J Π M) = ∑ pq [a†p a˜q ]JM 1 J X pq (ν) √ n¯ p nq + 2J + 1 ∑ ∑ 4 pqrs J J (J1 J2 )J X˜ pqrs (ν) 1 2 [[a†p a†q ]J1 [a˜r a˜s ]J2 ]JM √ . 2J + 1 (103) We suppress parity and isospin quantum numbers, as their selection rules do not affect the Jscheme expressions. However, ladder operators X † do have definite parity and charge exchange characteristic, so commensurate selection rules apply. Note that we break from the convention that a, b, c, . . . denote particle states here. We will instead use p, q, r, . . . for non-contracted indices (external lines), and a, b, c, . . . for contracted or summed indices. A.6.1 Transitions to the ground state For transitions to the ground state, we compute ¯ ν = [O¯ λ X¯ν† (Jν )]0 . M0ν = Φ0 ||O¯ λ ||Ψ 148 (104) Adding eqs. 79 and 80, we arrive at: [Aλ1 Bλ2 ]0 = 1 λ λ δ A 1 B 2 (−1)( ja − jb +λ ) n¯ a nb ˆλ λ1 λ2 ∑ ba ab 2 1 + δλ λ 1 2 λˆ 2 A.6.2 ab (J J )λ (J2 J1 )λ2 Babcd (−1)(J1 −J2 +λ ) n¯ a n¯ b nc nd . 1 2 ∑ Acdab (105) abcd J1 J2 Transitions between excited states and moments For transitions between states constructed via a ladder operator acting on the reference, or moments involving these states, we must compute the full tensor-product J ≡ [O ¯ λ X¯ν† (Jν )]J , ZM M (106) Mµν = [X(Jµ )Z J ]0 . (107) and then the zero-body product Here, we combine eqs. 84 through 89 to reveal    λ λ λ   1 2 λ1 λ2 λ λ ( j + j ) ˆ p q (−1) n¯ a Z pq = λ (−1) B A ∑ pa aq   jq j p ja   a    λ λ λ   1 2 λ2 λ1 (λ +λ ) 1 2 na − B pa Aaq (−1)    j p jq ja  1 λ1 ¯ (λ λ1 )λ2 (−1)(λ1 +λ2 +λ ) ¯ (λ λ2 )λ1 λ2 A B ¯ a¯ + A ¯ Bab )nb n¯ a pqa ¯ b ˆ ba pqb ˆ λ λ 1 2 ab            j p jq λ λ1 λ2 λ  (J J )λ (J2 J3 )λ2 1 1 2 1 B˜ ˆ ˆ ˆ A˜ cpab n¯ a n¯ b nc − ∑ ∑ λ J1 J3 abqc     2 abc J J J     J J j J J J c 1 2 3 3 1 3 1 2     λ λ λ  2 1 (J1 J2 )λ2 ˜ (J2 J3 )λ1 (λ +λ +λ ) − (−1) 1 2 B˜ cpab Aabqc na nb n¯ c    J3 J1 J2  + ∑( 149 (108) and eqs. 91 through 102 to arrive at      j p jr λ1        (J J )λ λ λ 1 2 1 2 ˆ Z˜ pqrs = (1 − Ppq (J1 ))(1 − Prs (J2 ))A pr Bqs λ Jˆ1 Jˆ2 jq js λ2        J J λ   1 2 +(1 − Ppq (J1 ))λˆ Jˆ1 ∑ ∑ Jˆ3 (−1)( j p + jq +J1 +J2 +J3 +λ1 +λ ) a J3    jp        jq J1 λ1 λ2 λ  λ (J3 J2 )λ2 A pa1 B˜ aqrs n¯ a      J3 λ1 ja   J2 J1 J3  +(1 − Prs (J2 ))λˆ Jˆ2 ∑ ∑ Jˆ3 (−1)(J1 +J3 +λ2 ) a J3    jr      js J2  λ1 λ2 λ  λ (J1 J3 )λ2 Aar1 B˜ pqsa na     J3 λ1 ja   J1 J2 J3  −(1 − Prs (J2 ))λˆ Jˆ2 ∑ ∑ Jˆ3 (−1)(J1 +J3 +λ2 +λ ) a J3    jr      js J2  λ1 λ2 λ  λ (J1 J3 )λ1 Bar2 A˜ pqsa n¯ a     J3 λ2 ja   J2 J1 J3  −(1 − Ppq (J1 ))λˆ Jˆ1 ∑ ∑ Jˆ3 (−1)( j p + jq +J1 +J2 +J3 +λ1 ) a J3    jp   J3   λ 1 1 + λˆ ∑ ∑(−1)(J1 +J2 +λ )  2 ab J  J2 3   λ 1ˆ 1 (J +J +λ +λ ) + λ ∑ ∑(−1) 1 2 1 2  2 ab J  J1 3      jq J1  λ1 λ2 λ  λ (J3 J2 )λ2 na B pa2 A˜ aqrs       λ2 ja J1 J2 J3   λ2 λ  (J1 J3 )λ1 (J3 J2 )λ2 n¯ a n¯ b A˜ pqab B˜ abrs  J1 J3    λ2 λ  (J1 J3 )λ2 (J3 J2 )λ1 B˜ pqab A˜ abrs na nb   J2 J3 +(1 − Ppq (J1 ))(1 − Prs (J2 ))λˆ Jˆ1 Jˆ2 ∑ J3 J4 J5 150 Jˆ3 Jˆ5 (−1)( js − jq +J3 +λ )       j p jr J3         λ λ λ   1 2 (J J )λ (J J )λ × jq js J5 A¯ 3 ¯4 1 B¯ ¯4 5 2 n¯ a nb ∑ absq¯       J5 J3 J4   ab p¯rab     J J λ  1 2 A.7 (109) Tensor-scalar Commutators The most frequently used result of the derivation of Sec. A.5 are the commutator expressions for a scalar operator (expressed in non-reduced matrix elements) and an arbitrary rank spherical tensor operator (expressed in reduced matrix elements). There are two uses for these expressions. One use is the solution of the EOM-IMSRG(2,2) equations. Analogous to the m-scheme case, (J J (ν) and X˜ ab ladder the amplitudes Xai abi j Ji j )J (ν), as well as excitation energies, are obtained through application of the Lanczos algorithm to solve the eigenvalue problem in eq. 4.13. The commutator expression [H, Xν† ] requires the commutator of a scalar operator H, and the ladder operator, which is a spherical tensor. A more general application is to compute effective operators in the IMSRG. For all IMSRG calculations, observables must be consistently evolved using eq. 3.8 in the flow-equation approach or eq. 3.39 in the Magnus formalism. If the operator is of tensor character, tensor-scalar commutators are required. Thus we must compute Zµλ ≡ ABλµ − Aλµ B , (110) which can be derived as a limiting case of the general tensor-product in eqs. 105 through A.6.2, A corresponds to the scalar operator, where λ1 → 0 and the Wigner-Eckart theorem has been used to revert to non-reduced matrix elements. B is a tensor operator of rank λ2 = λ . The resultant 151 operator Z is also of rank λ . (λ 0)λ Z λpq = ∑ A pa Bλaq − Bλpa Aaq − ∑(na − nb ) A¯ λpqa Bλ − jˆa B¯ ¯ Aab ¯ b¯ ab pqa ¯ b a + ab 1 ∑ (nanbn¯c + n¯an¯bnc)Jˆ1Jˆ2(−1) j p+ jc+J1+λ 2 abc J1 J2    J J λ   1 2 J1 ˜ (J1 J2 )λ (J1 J2 )λ ˜ J2 A˜ cpab Babcq − B˜ cpab Aabcq ×   jq j p jc   (111) and (J1 J2 )λ (J1 J2 )λ (J1 J2 )λ (J1 J2 )λ (J1 J2 )λ Z˜ pqrs = ∑ A pa B˜ aqrs + Aqa B˜ pars − B˜ pqas Aar − B˜ pqra Aas a     J J λ  2 1 J2 λ j + j +J p q ˆ ˆ 2 − J1 J2 (−1) ∑ (1 − Ppq (J1 ))(−1) Bλpa A˜ aqrs    j p ja jq  a    J J λ   1 2 J1 j + j −J r s 1 − (1 − Prs (J2 ))(−1) A˜ pqra Bλas    js ja jr  1 J1 ˜ (J1 J2 )λ (J1 J2 )λ ˜ J2 (1 − na − nb )(A˜ pqab Babrs − B˜ pqab Aabrs ) + ∑ Jˆ1 Jˆ2 Jˆ3 Jˆ4 (na − nb ) ∑ 2 ab abJ3 J4      j p js J3        J3 (J J )λ j + j +J +J q s 2 4 × (1 − Ppq (J1 ))(1 − Prs (J2 ))(−1) B¯ ¯3 4 . jq jr J4 A¯ psa ¯ ¯ b abrq¯        J J λ   1 2 (112) + Here we have employed the scalar Pandya-transformed matrix elements defined by     j p jq J   1 J J2 1 ˆ A¯ pqr = − J A˜ psrq ∑ 2 ¯ s¯   jr js J2  J2 152 (113) A.8 Scalar Commutators Finally, taking λ → 0 for both operators reveals the scalar limit, with non-reduced matrix elements. Z0 = ∑(na − nb ) jˆa2 Aab Bba + ab 1 ∑ (nanbn¯cn¯d − ncnd n¯an¯b)Jˆ2A˜ Jabcd B˜Jcdab , 4 abcd (114) J Z pq = ∑(A pa Baq − B pa Aaq ) + δ j p jq a Jˆ2 (n − nb )(Aab B˜ Jbpaq − Bab A˜ Jbpaq ) ˆj2p a abJ ∑ Jˆ2 1 + δ j p jq ∑ 2 (na nb n¯ c + n¯ a n¯ b nc )(A˜ Jpcab B˜ Jabqc − B˜ Jpcab A˜ Jabqc ) , ˆ 2 abJ j p (115) Z˜ Jpqrs = ∑(A pa B˜ Jaqrs + Aqa B˜ Jpars − B˜ Jpqas Aar − B˜ Jaqra Aas ) a − ∑(B pa A˜ Jaqrs + Bqa A˜ Jpars − A˜ Jpqas Bar − A˜ Jaqra Bas ) a 1 (1 − na − nb )(A˜ Jpqab B˜ Jabrs − B˜ Jpqab A˜ Jabrs ) 4∑ ab     j p js J   1 J J 2 + (1 − Ppq (J))(1 − Prs (J)) ∑ Jˆ1 A¯ 1 ¯ B¯ 1¯ (na − nb ) . psa ¯ b abrq¯   abJ1  jr jq J  + (116) These expressions are used for the evaluation of the IMSRG unitary transformation involving the scalar operators η and H. A.9 EOM-IMSRG({3},2) triples expression For the computation of the EOM-IMSRG({3},2) triples correction in m-scheme, we must compute ν = δ E{3} |Wabci jk |2 1 ∑ i jk 36 abci jk D (117) abc i jk where Dabc is the energy denominator, and Wabci jk = [H, Xν† ]abci jk = (1 − Pab − Pac )(1 − Pi j − Pik ) ∑(Γbcqi Xaq jk − Xbcqi Γaq jk ) (118) jq In J-scheme, eq. 117 becomes jˆ12 1 | [(ab)Jab c] j1 ||W Jν ||[(i j)Ji j k] j2 |2 , ∑ ∑ 2 i jk ˆ 36Jν abc j j D 1 2 abc i jk J J ab i j 153 (119) where the triples amplitudes are given by [(ab)Jab c] j1 ||W Jν ||[(i j)Ji j k] j2 = 1 Jˆν ja j J ji j j Ji j J jc j Ji j j j j j¯ J k 2 ab 1 1 2 ν ∑ Cmambbab Mab Cmi m j Mi j CMab mc m1 CMi j mk m2 Cm1 m¯ 2 Mν {m} Jν ama , bmb , cmc |WM |imi , jm j , kmk , ν (120) i jk and the Epstein-Nesbet energy denominator Dabc is approximated with up to two-body monopole matrix elements, i jk Dabc = fii + f j j + fkk − faa − fbb − fcc + abab + acac + bcbc + i ji j + ikik + jk jk − iaia − ibib − icic − ja ja − jb jb − jc jc − kaka − kbkb − kckc . (121) Where monopole matrix elements are given by pqrs = ∑ Γ˜ Jpqrs (2J + 1) . (122) J To compute the triples amplitudes, there are 18 terms to evaluate. We now derive the J-scheme expressions. We will only perform the derivation for the m-scheme term T1A = ∑ Γaqi j Xbcqk , (123) q which is expressed in J-scheme as Ji j jk j2 1 J jc j ja j J j j¯ J CMab mc1m CM m Cm1 m2¯ νMν T1A = 2 ∑ Cma mb ab m M ij k 2 1 1 2 b ab ab Jˆν q{m} J1 J2 J3 ja jq J jq j J j jc J J J¯3 Jν ˜ Ji j ˜ (J2 J3 )Jν ×Cma mq1M Cmb mc2M Cmq mk 3M CM2 M ¯ Mν Γaqi j Xbcqk 1 2 b k 3 2 3 We sum the angular momentum substates out, 154 . (124) 1 Jˆν2 ja j J J j j¯ J Ji j j j jc j k 2 ab 1 1 2 ν ∑ Cmambbab Mab CMab mc m1 CMi j mk m2 Cm1 m¯ 2 Mν {m} ja jq Ji j J J¯ J jq j J j jc J 3 ν ×Cma mq M Cmb mc2M Cmq mk 3M CM2 M ¯ ij 2 b k 3 2 3 Mν jq + + ja = Jˆ2 Jˆ3 Jˆab Jˆi j jˆ1 jˆ2 jb - + Jab + + jc J2 jk J3 Ji j j2 Jν - + j1 + + Jab ja jb = Jˆ2 Jˆ3 Jˆab Jˆi j jˆ1 jˆ2 j1 - + J3 j1 - j2 ja + Ji j j2 - Jν J3 + ja × jc J2 jq J2 - (−1)( j2 + ja + jb + jc + jk + jq +J3 +Jν ) jk + = Jˆ2 Jˆ3 Jˆab Jˆi j jˆ1 jˆ2 (−1)( j2 + ja + jb + jc + jk + jq +J3 +Jν ) × ja jc jb Jab j1 J2 J2 J3 Jν j2 j1 ja ja jk jq Ji j j2 J3 , (125) Figure A.14 Derivation of induced three-body force with Jucys diagrams. and thus T1A = ∑ ∑ = Jˆ2 Jˆ3 Jˆab Jˆi j jˆ1 jˆ2 (−1)( j2 + ja + jb + jc + jk + jq +J3 +Jν ) q J2 J3 ×    ja   jc        jb Jab J2 J3 Jν   ja   j2 j1 ja    jk j1 J2  155   jq Ji j  Ji j (J2 J3 )Jν Γ˜ aqi j X˜bcqk  j2 J3  (126) The remainder of the terms may be computed with more or less the same procedure, and the full expression is given by [(ab)Jab c] j1 ||W Jν ||[(i j)Ji j k] j2 = 9 ∑ TN , (127) N=1 where ˆ T1 = jˆ1 jˆ2 Jab ∑ Jˆ2 (−1)( jc + jb + jq + j2 +Jν ) J1 J2 q         jb Jab   j2 ja J2   ja jq Ji j  Ji j (J J )Jν ( j − j +J ) 1 2 A˜ aqi j X˜bcqk × Jˆ1 Jˆi j (−1) a k 2        jc j1 J1  J1 Jν j1   jk j2 J2            ja j J   j j Ji j   ja jq J   (J J )J 2 1 b ab k 1 i j ν ˜ J2 J ˆ ˆ − J2 J1 (−1) 1 X˜aqi j Abcqk (128)  j j J   J J  j j J   j     ja c ˆ Jˆi j T2 = jˆ1 jˆ2 Jab 1 2 1 ν 1 k 1 ( jb + jc + j j + jk + jq − j2 +Jν ) ∑ Jˆ1Jˆ2Jˆ3(−1) 2    ja   jc qJ1 J2 J3   jb Jab   ji  j j J  1 3 k   j j Ji j   j2 J2    J        J3 Jν ja jq J2  J 1 (J3 J1 )Jν ( j − j +J +J ) × Jˆ2 (−1) a i 2 1 A˜ aq2 jk X˜bcqi j j j     a  ji j2 J1  1 2       J J Jν   ja j J   2 1 1 3 (J J )J J3 (J +J ) − Jˆ3 (−1) 1 2 X˜aq1jk2 ν A˜ bcqi  j j  j j J   j  1 ˆ Jˆi j T3 = jˆ1 jˆ2 Jab 2 i i q 1 ( jb + jc + ji + jq − j2 +Jν +Ji j ) ∑ Jˆ1Jˆ2Jˆ3(−1)    ja   jc qJ1 J2 J3 (129)   jb Jab   ji  j j J  1 3         J J j j J ν a q 1 3 2 J2 ˜ (J3 J1 )Jν ( j +J ) a ˆ 1 A˜ aqki Xbcq j × J2 (−1)   j1 j2 ja    j j j2 J1         J J Jν   ja jq J   1 2 1 (J1 J2 )Jν ˜ J3 ( j +J ) − Jˆ3 (−1) j 1 X˜aqki Abcq j   j2 j1 j j    j j j1 J3   2   j j Ji j   jk J2    J 156 (130) ˆ T4 = jˆ1 jˆ2 Jab ∑ Jˆ1 Jˆ2 (−1)( ja + jq + j2 +Jν +Jab +J1 +J2 ) qJ1 J2         J j J J J J ν i j 1 2 b (J1 J2 )Jν ij j ˆ k A˜ bqi j X˜caqk × Ji j (−1)     j1 c 1 2 jk J2   j2 j1 jb            (J J )J  ja j J   jq j J  J Ji j Jν  1 1 b b ab 1 i j ν ˜ J2 j X˜bqi j − Jˆ2 (−1) b Acaqk    j1 jk J2    j2 j1 jk    j1 jc J2     ja ˆ Jˆi j T5 = jˆ1 jˆ2 Jab   jb Jab   jq  j j J  (131) ( ja + jq + jk + j j − j2 +Jν +Jab +J1 +J2 +J3 ) ∑ Jˆ1Jˆ2Jˆ3(−1) qJ1 J2 J3             j j Ji j jb jq J1 J3 J2 Jν  J (J2 J3 )Jν j ˆ i A˜ bq1 jk X˜caqi × J1 (−1)     j1 c   ji j2 J3    j1 j2 jb  2 k j2 J1              ja j J   j jq J   ji j j Ji j  J J Jν   1 2 1 b ab b (J J )J J3 j − Jˆ3 (−1) b X˜bq1jk2 ν A˜ caqi          j1 jc J3   ji j1 J3   jk j2 J2   j1 j2 ji     ja   jb Jab   ji  j j J  (132) ˆ Jˆi j T6 = jˆ1 jˆ2 Jab ( ja + jq + ji + j2 +Jν +Jab +Ji j +J1 +J3 ) ∑ Jˆ1Jˆ2Jˆ3(−1) qJ1 J2 J3    ja            jb Jab   ji j j Ji j   jb jq J1  J3 J2 Jν  J J +J 1 X˜ (J2 J3 )Jν × Jˆ1 (−1) 2 1 A˜ bqki caq j   j1 jc J2    j2 jk J1    j j j2 J3    j1 j2 jb          ja j J    j jq J    ji j j Ji j   J J Jν   2 1 1 b ab b (J1 J2 )Jν ˜ J3 ( jb − j j ) ˆ X˜bqki Acaq j −J3 (−1)   j1 jc J3    j j j1 J3    j2 jk J2    j1 j2 j j   (133) 157 T7 = jˆ1 jˆ2 ∑ Jˆ1 (−1)( jq − j2 +Jν +Jab +J1 ) qJ1      jq Ji j  J1 Jab Jν  Ji j (Jab J1 )Jν ( j − j ) c ˆ k A˜ cqi j X˜abqk × [Ji j (−1)      jk j2 J1   j1 j2 jc       jc jq J   J Ji j Jν   (J J )J 1 1 1 i j ν ˜ Jab ˆ − Jab X˜cqi j Aabqk   j  j j J  j j     jc k T8 = jˆ1 jˆ2 Jˆi j ∑ 1 ab 2 1 (134) k ( j + j + j + j +J +J +J +J ) Jˆ1 Jˆ2 (−1) k q j 2 ν ab 1 2 qJ1 J2    jq         ji J2   ji j j Ji j  Jab J2 Jν  J (Jab J2 )Jν ( j − j ) c i × Jˆ1 (−1) A˜ cq1 jk X˜abqi   j2 jc J1    jk j2 J1    j2 j1 jc         jc jq J    ji j j Ji j   J J Jν   1 1 2 (J J )J Jab ˆ − Jab X˜cq1jk 2 ν A˜ abqi (135)   ji j1 Jab    jk j2 J2    j2 j1 ji   T9 = jˆ1 jˆ2 Jˆi j ∑ ( j + j + j +J +J +J ) Jˆ1 Jˆ2 (−1) q i 2 ν ab i j qJ1 J2    jc   jq J1   ji  j j J         j j Ji j Jab J2 Jν  J ( j +J ) 1 X˜ (Jab J2 )Jν × Jˆ1 (−1) c 2 A˜ cqki abq j    jj 2 2 2 jk J1   j2 j1 jc            jc jq J   ji j j Ji j  J J Jν   1 1 2 (J1 J2 )Jν ˜ Jab (J − j ) j 1 ˆ (−1) − Jab X˜cqki Aabq j   j j j1 Jab    j2 jk J2 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