Many-Body Perturbation Theory Calculations

Transcription

1 TALENT Course 2018 Many-Body Perturbation Theory Calculations 胡柏山 (Baishan Hu) 北京大学物理学院 (Peking University, Beijing, China) July 31, Xinxiang

2 Outline I. Introduction II. Theoretical framework and calculations a) MBPT for closed-shell nuclei b) MBPT (Q-box + folded diagrams) for open-shell nuclei III. Summary and Outlook 2

3 Perturbation theory takes us from a simple, exactly solvable (unperturbed) problem to a corresponding real (perturbed) problem Bai Shan The key formulas in perturbation theory are In MBPT the ambition is to include in H 0 as much physics as possible, so that V represents a small perturbation 3

4 Key tool from 1950 s to 1970 s Great depression in 1980 s Today MBPT is coming back... History of MBPT Rayleigh-SchrÖdinger perturbation theory G-matrix, Brueckner-Hartree-Fock method Valence-space shell-model interaction (Q-box + folded diagram) Depending on a starting energy parameter of G-matrix Poor convergence of the intermediate-state summations (tensor part) Intruder states RSPT with soft potential (V low-k, SRG, UCOM, OLS) Bogoliubov MBPT Auxiliary method (importance truncation, natural orbital basis) Realistic Gamow shell model (Extended Kuo-Krenciglowa method) 4

5 MBPT for closed-shell nuclei Rayleigh-SchrÖdinger perturbation theory 5

6 Advantages in HF basis, compared with HO basis Faster convergence Harmonic Oscillator Some perturbation diagrams are cancelled out In HO basis, calculations could be ħω dependent, while much less in HF basis Hartree Fock 16 O A. Tichai et al., PLB 756, 283 (2016) 6

7 RSPT for closed-shell nuclei Perform a Hartree-Fock calculation HF state is chosen as a reference state H 0 In the HF basis, make RSPT corrections Energy up to 3 rd order Wave function up to 2 nd order (One-body density) E=E (0) +E (1) +E (2) +E (3) + -E (1) E (3) E (2) Ψ =Φ HF + ψ (1) + ψ (2) + ψ (1) ψ (2) 7

8 HF-RSPT calculations for 16 O with N 3 LO-SRG, N shell =13, ħω=35 MeV Binding energy 3NF important! B.S. Hu, F.R. Xu, et al., PRC 94, (2017) Point-proton rms radius 8

9 Inclusion of 3NF Normal ordering with HF reference state H HF Discard residual 3B part: NO2B approximation 9

10 ASG diagram expansion when 3NF is included E (3) : 56 terms (Derived for the first time) 10

11 ASG diagram expansion when 3NF is included NO2B 11

12 NO2B approximation: 12

13 NN-only N 3 LO-SRG Bare JISP16 + N shell =7, ħω=30 MeV 13

14 HF-RSPT calculations with NO2B N 2 LO sat, N shell =13, ħω=22 MeV Binding energy B.S. Hu, T. Li, F.R. Xu, in preparation (2018) Charge radius 14

15 MBPT for open-shell nuclei Realistic Interaction Hartree-Fock Hartree-Fock Hartree-Fock- Bogoliubov Single-Reference Normal Ordering Q-box + Folded Diagram Multi-Reference Normal Ordering HF-MBPT SM Interaction BMBPT closed-shell nuclei open-shell nuclei 15

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17 MBPT for open-shell nuclei Kuo-Krenciglowa method (Folded-Diagram method) Factorization theorem: the core is separated out and only quantities relative to the core are concerned Q-box is made up of non-folded diagrams which are irreducible and valence linked 17

18 Bloch Horowitz MBPT for open-shell nuclei BWPT Kuo-Krenciglowa degenerate model space Extended Kuo-Krenciglowa 18

19 Advantages in HF basis, compared with HO basis Faster convergence 18 O Some perturbation diagrams are cancelled out In HO basis, calculations could be ħω dependent, while much less in HF basis K. Tsukiyama, et al., PRC 85, (R) (2012) 19

20 Diagrammatic expansion S-box (1-body) 2nd 2nd Q-box (2-body) 3rd 3rd 20 The pictures are taken from M. Hjorth-Jensen, et al., Physics Reports 261 (1995) 125.

21 MBPT for open-shell nuclei Realistic Gamow shell model Workflow Bare forces: Strong repulsion, tensor force, slow convergence V low k, SRG, OLS, pp,nn: pn: HO/HF; Gamow-Berggren basis (a,b): WS/GHF Up to third order Valence-space effective interactions Complex shell model code (Jacobi-Davidson, N. Michel) Extended Kuo-Krenciglowa method (EKK) 21

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23 Nucleus as an open quantum system J. Phys. G: Nucl. Part. Phys. 36(2009) Li Many-body dynamics Input NN+3NFs, operators Open channels Neutron/Proton-rich isotopes Clustering in nuclei New magic numbers Intrinsic resonance New collective modes Halo G. Hupin, S. Quaglioni, P. Navratil, PRL114, (2015) Bound, resonant and scattering states may be strongly coupled. Need nuclear theory including the continuum. 23

24 Gamow-Berggren basis The wave function of a resonance with a peak at energy e 0 and a width γ Through the Fourier transform, we obtain the time evolution of the resonance Gamow state: Complex energy G. Gamow, Z. Phys.51 (1928) 204 Berggren basis in complex-k plane, describing bound, resonance and scattering on equal footing. T. Berggren, Nucl. Phys. A109 (1968) 265 Orthogonality and completeness bound, resonance scattering (discretized) 24

25 Gamow Hartree-Fock Step ❶: Solve the Hartree-Fock equations in HO representation using H int, Step ❷: Extract the non-local HF potential ʋ(r,r ) Step ❸: Obtain the radial wave function u(r)/r in complex-k plane Exterior complex scaling Outgoing solution at large distance 25

26 Results of GHF sp energies MeV sp resonance 26

27 Order-by-order convergence in real-energy space Bare NNLO opt SPEs ħω=20 MeV N shell =12 22 O core, sd-shell Q-box Q-box 27

28 Q-space Q-space 1d 5/2 1d 5/2 d 3/2 channel Continuum 0f 5/2 0f 7/2 1p 1/2 1p 3/2 Q-box folded diagrams 0f 5/2 0f 7/2 1p 1/2 1p 3/2 1d 3/2, 2d 3/ d 3/2 1s 1/2 0d 5/2 0p 1/2 0p 3/2 0s 1/2 8 2 Proton Model Space 22 O core 0d 3/2 1s 1/2 0d 5/2 0p 1/2 0p 3/2 0s 1/ Resonance Neutron 0d 3/2 1s 1/2 12

29 Oxygen isotopes from RGSM Kondo, et al, (2016): ~0.110 MeVLunderberg, et al, (2012): ~0.005 MeV Kohley et al, (2013): (stat) + 3 (sys) ps 29

30 Fluorine isotopes from RGSM Long-lived 4 1+ isomer, T 1/2 =2.2(1) ms, Lepailleur, et al., PRL110, (2013) 30

31 Gamow IM-SRG Hermite (HO basis / real-energy HF) Complex symmetric (Gamow-Berggren basis) Gamow IM-SRG vs RGSM Gamow IM-SRG 31

32 Summary Develop HF-RSPT including the wave-function and three-body force corrections. Develop the third-order RGSM within HF basis Describe oxygen and fluorine isotopes Outlook Derive the cross-shell effective interactions (sdpf-shell, ) Reconciling the microscopic valence-space effective interactions with the nuclear reaction theory (GSM-RGM) 32

33 Collaborators: Furong Xu (Peking University) Qiang Wu (Peking University) Tong Li (Michigan State University) Zhonghao Sun (Oak Ridge National Laboratory) Jianguo Li (Peking University) Nicolas Michel (Michigan State University) Gratitude: Junchen Pei (Peking University) Simin Wang (Michigan State University) Thomas Papenbrock (Oak Ridge National Laboratory) Gustav R. Jansen (Oak Ridge National Laboratory) Luigi Coraggio (INFN-Naples) James P. Vary (Iowa State University) Ruprecht Machleidt (University of Idaho) Marek Ploszajczak (GANIL) Our group: Yuanzhuo Ma, Bo Dai, Sijie Dai, Yifang Geng, 33

34 北京大学攻读博士学位研究生选题报告 Thank you for your attention! 第一性原理计算原子核结构 胡柏山导师 : 许甫荣教授 北京大学物理学院粒子物理与原子核物理专业 2015 年 11 月 25 日

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