Many-Body Perturbation Theory Calculations (Baishan Hu) Supervisor: - PowerPoint PPT Presentation

FRIB-Theory Alliance workshop Many-Body Perturbation Theory Calculations 胡柏山 (Baishan Hu) Supervisor: 许甫荣 (Furong Xu) 教授 E-mail: hubsh@pku.edu.cn 北京大学物理学院（ Peking University, Beijing, China ） June 21, 2018 @ East Lansing, MI

Outline I. Introduction II. Theoretical framework and calculations a) MBPT for closed-shell nuclei b) MBPT ( Q -box + folded diagrams) for open-shell nuclei As Prof. F.R. Xu’s talk c) MBPT for nuclear reaction (RGM) III. Summary and Outlook 2

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 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 3

History of MBPT Key tool from 1950’s to 1970’s  Rayleigh-Schr Ö dinger perturbation theory  G -matrix, Brueckner-Hartree-Fock method  Valence-space shell-model interaction ( Q -box + folded diagram) Great depression in 1980’s  Depending on a starting energy parameter of G -matrix  Poor convergence of the intermediate-state summations (tensor part)  Intruder states Today MBPT is coming back ...  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) B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 4

MBPT for closed-shell nuclei Ra Rayleigh-Sch Schr Ö din dinger ger pert rturb urbatio ation n theory ry B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 5

Advantages in HF basis, compared with HO basis Harmoni nic Oscillat ator or Faster convergence Some perturbation diagrams are cancelled out Hartree e Fock In HO basis, calculations 16 O 16 could be ħω dependent, while much less in HF basis A. Tichai et al., PLB 756, 283 (2016) 6 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

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 (3) -E (1) E (2) Ψ = Φ HF + ψ (1) + ψ (2) + … ψ (1) ψ (2) B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 7

HF-RSPT calculations for 16 O with N 3 LO-SRG, N shell =13, ħω =35 MeV Binding energy B.S. Hu , F.R. Xu , et al., PRC 94, 014303 (2017) 3NF important ! Point-proton rms radius B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 8

Inclusion of 3NF 𝑰 HF Normal ordering with HF reference state Discard residual 3B part: NO2B approximation B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 9

ASG dia iagram am expansi sion on when 3NF is is in inclu luded ed E (3) : 56 terms (Derived for the first time) … B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 10

ASG diagram am expansi sion on wh when 3NF NF is included ed NO2B B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 11

NO2B approximation: B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 12

NN -only N 3 LO-SRG + Bare JISP16 N shell =7, ħω =30 MeV B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 13

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 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 14

Nucleus as an open quantum system J. Phys. G: Nucl. Part. Phys. 36(2009) 013101 Input 6 Li NN +3NFs, • Neutron/Proton-rich Many-body operators isotopes dynamics • Clustering in nuclei • New magic numbers • Intrinsic resonance • New collective modes Open channels • Halo Bound, resonant and scattering states may be strongly coupled. Need nuclear theory including the continuum. G. Hupin, S. Quaglioni, P. Navratil, PRL114, 212502 (2015) 15

MBPT for open-shell nuclei Realistic Gamow shell model Workflow As Prof. F.R. Xu’s talk Bare forces: 𝑾 𝒎𝒑𝒙−𝒍 , Strong repulsion ， SRG, tensor force , pp,nn: OLS, … slow convergence pn: Up to third ird order rder HO/HF; Gamow-Berggren basis (a,b): WS/GHF Extended Kuo-Krenciglowa Valence-space effective method (EKK) interactions Complex shell model code (Jacobi-Davidson, N. Michel) B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 16

Advantages in HF basis, compared with HO basis 18 O 18 Faster convergence 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, 061304(R) (2012) B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 17

Diagrammatic expansion S -box (1-body) Q -box (2-body) 2nd 2nd 3rd 3rd 18 The pictures are taken from M. Hjorth-Jensen, et al ., Physics Reports 261 (1995) 125.

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 19 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

Results of GHF sp energies MeV sp resonance B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 20

Order-by-order convergence in real-energy space SPEs Bare NNLO opt ħω =20 MeV N shell =12 22 O core, sd -shell Q -box Q -box 21 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

…… …… 1d 5/2 1d 5/2 d 3/2 channel Q-space 40 40 Continuum 0f 5/2 0f 5/2 Q-box 0f 7/2 0f 7/2 1d 3/2 , 2d 3/2 … folded 1p 1/2 1p 1/2 diagrams 1p 3/2 1p 3/2 20 20 0d 3/2 Model 0d 3/2 0d 3/2 Resonance 1s 1/2 Space 1s 1/2 1s 1/2 0d 5/2 14 0d 5/2 8 22 O 0p 1/2 Q-space 8 0p 1/2 core 0p 3/2 0p 3/2 2 2 0s 1/2 0s 1/2 Proton Neutron 12

Oxygen isotopes from RGSM Kondo, et al , (2016): ~0.110 MeVLunderberg, et al , (2012): ~0.005 MeV Kohley et al , (2013): 4.5 +11 -1.5 (stat) + 3 (sys) ps 23 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

Fluorine isotopes from RGSM + isomer, T 1/2 =2.2(1) ms, Long-lived 4 1 Lepailleur, et al ., PRL110, 082502 (2013) B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018 24

MBPT for nuclear reaction RG RGSM-RGM RGM COSM 25

MBPT for nuclear reaction RG RGSM-RGM RGM 26 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

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) 27 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

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 , … 28 B. S. Hu - FRIB Theory Alliance Workshop: From Bound States to the Continuum, Jun 21, 2018

北京大学攻读博士学位研究生选题报告 Thank you for Thank you for your atte your attention ntion ！ 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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