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Short-range and tensor correlations in light
nuclei studied with antisymmetrized molecular dynamics (TOAMD) Mengjiao LYU (RCNP) Masahiro ISAKA (Hosei) Hiroshi TOKI (RCNP) Hisashi HORIUCHI (RCNP) Kiyomi IKEDA (RIKEN) Tadahiro SUHARA (Matsue) Taiichi YAMADA (Kanto Gakuin) Takayuki MYO QNP , Tsukuba
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Outline Variational methods for finite nuclei with bare force
Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) ฮฆ TOAMD =(1+๐น+ ๐น 2 +โฏ)ร ฮฆ AMD Variational correlation function ๐น= ๐<๐ ๐ ๐๐ Power series expansion using ๐น High-Momentum AMD (HM-AMD) High-momentum nucleon pairs as 2p-2h excitations Combine TOAMD & HM-AMD F
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Deuteron properties & tensor force
Energy -2.24 MeV Kinetic 19.88 Central -4.46 Tensor -16.64 LS -1.02 P(L=2) 5.77% Radius 1.96 fm S S D ฮฆ(๐) D SD DD r [fm] S S D 1.22 ฮฆ (๐) D d-wave is โspatially compactโ (high-momentum)
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Prog. Theor. Exp. Phys. 2015, 073D02 (38 pages)
DOI: /ptep/ptv087 Tensor-optimized antisymmetrized molecular dynamics in nuclear physics (TOAMD) Takayuki Myo, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi, and Tadahiro Suhara Describe finite nuclei using VNN , in particular, for clustering Multiply pair-type correlation function "๐ญ= ๐<๐ ๐จ ๐ ๐๐ " to AMD w.f. ๐ญ ๐ซ for D-wave transition with S12 from tensor force ๐ญ ๐บ for Short-range repulsion in central force A. Sugie, P. E. Hodgson and H. H. Robertson, Proc. Phys. Soc. 70A (1957) 1 S. Nagata, T. Sasakawa, T. Sawada, R. Tamagaki, PTP22 (1959) 274
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General formulation of TOAMD
| ฮฆ TOAMD = 1 +๐น ๐ท + ๐น ๐ + ๐น ๐ท ๐น ๐ + ๐น ๐ ๐น ๐ + ๐น ๐ท ๐น ๐ท +โฏ | ฮฆ AMD tensor short-range Power series expansion, but, all F are independent tensor ๏ด short-range Variational principle ๐ฟ๐ธ=0 for ๐ธ= ฮฆ TOAMD ๐ป ฮฆ TOAMD ฮฆ TOAMD ฮฆ TOAMD Variational parameters AMD : n, Zi (i=1,โฆ, A) ๐น ๐ท : ๐ 12 ๐ ๐ ๐บ ๐ถ ๐ ๐ โ ๐ ๐ ( ๐ ๐ โ ๐ ๐ ) 2 ๐น ๐ : ๐ ๐ ๐บ ๐ถ ๐ โฒ ๐ โ ๐ ๐ โฒ ( ๐ ๐ โ ๐ ๐ ) 2 | ฮฆ AMD =det ๐ 1 โฏ ๐ ๐ด nucleon w.f. ๐ ๐ (๐)โ ๐ โ๐ (๐โ๐) 2 ๐ ๐ ๐ ๐ Gaussian wave packet spin-isospin dependent ๐ ๐ป ๐,๐ โ๐ธ ๐ ๐,๐ ๐ถ ๐ =0 Eigenvalue problem
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Matrix elements of correlated operator
Correlated Hamiltonian ฮฆ TOAMD ๐ป ฮฆ TOAMD ฮฆ TOAMD ฮฆ TOAMD = ฮฆ AMD ๐ป+ ๐น โ ๐ป+๐ป๐น+ ๐น โ ๐ป๐น+โฏ ฮฆ AMD ฮฆ AMD 1+ ๐น โ +๐น+ ๐น โ ๐น+โฏ ฮฆ AMD Correlated Norm Classify the connections of F, H into many-body operators using cluster expansion method 3-body 4-body bra Fโ V F ๐น โ ๐น= {2-body} +โฏ+ {4-body} ket i j k i j k l (2-body)2 ๐ ๐ Fourier transformation of ๐น, ๐ ๐ โ๐ ( ๐ ๐ โ ๐ ๐ ) 2 โ ๐ โ ๐ ๐ /4๐ โ ๐ ๐๐โ
๐ ๐ โ ๐ โ๐๐โ
๐ ๐ ๐น โ ๐๐น= {2-body} +โฏ+ {6-body} (2-body)3 relative single particle
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PLB 769 (2017) 213 PRC 95 (2017) PRC 96 (2017) PTEP (2017) 073D01 PTEP (2017) 111D01 Results 3H, 4He, (6He, 6Li) VNN : bare AV8โ with central, LS, tensor terms 7 Gaussians for ๐น ๐ท , ๐น ๐ to converge the solutions. Successively increase the correlation terms. | ฮฆ TOAMD = 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD Single F Double F AMD F F F 4p-4h 2p-2h 3p-3h
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TOAMD with Double ๐น 3H AV8โ
F TOAMD with Double ๐น AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD F are independent AMD 3H Add terms successively Reproduce the GFMC energy Small curvature as ๐น 2 terms increase ๏ฎ good convergence ๐ 2 =1.75 fm 1+๐น ๐ Correlation functions
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TOAMD with Double ๐น 3H AV8โ
F TOAMD with Double ๐น AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD PLB769 (2017) 213 3H Kinetic/2 F are independent Reproduce the Hamiltonian components of 3H Central Tensor Correlation functions
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TOAMD with Double ๐น 4He AV8โ
F TOAMD with Double ๐น AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD F are independent AMD 4He Good energy with ๐น 2 Small curvature as ๐น 2 terms increase ๏ฎ good convergence ๐ 2 =1.50 fm Next order : ๐น 3 is ongoing 1+๐น ๐ Correlation functions
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Correlation functions ๐น ๐ , ๐น ๐ท in 3H
short-range ๐น ๐ (๐) tensor ๐น ๐ท (๐) FS 3E r2รFD 3E Amplitude intermediate long FS 1E Negative sign in ๐น ๐ to avoid short-range repulsion in VNN . Ranges of ๐น ๐ท , ๐น ๐ are NOT short, contributing to many-body terms.
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Many-body terms of ๐น โ ๐ป๐น
Correlated Hamiltonian ๐ป = ๐ป 2โbody + ๐ป 3โbody +โฏ+ ๐ป ๐ดโbody 4-body 3-body 3H 3-body 4He Energy Energy 2-body wide ๐ low compact ๐ high 2-body Many-body terms play a decisive role for energy saturation NO energy saturation within 2-body term PTEP (2017) 073D01
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High-Momentum Antisymmetrized Molecular Dynamics (HM-AMD)
TM et al PTEP (2017) 111D Tensor correlation vs. TOSM Lyu et al PTEP (2018) 011D HM-AMD + TOAMD with AV8โ TM PTEP (2018) 031D Short-range correlation with AV4โ Lyu et al arXiv: Submitted to RPC Zhao et al. arXiv: Submitted to RPC
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TOAMD for p-shell nuclei
In TOAMD, single configuration of AMD for s-shell nuclei ๏ extend to multi-configuration of AMD (AMD+GCM) Multi-AMD bases to represent the NN correlations. High-Momentum AMD (HM-AMD) Kimura (2014, priv. comm.), Itagaki-Tohsaki (PRC97(2018) ) ๐ : Centroid of Gaussian wave packet ๐ = Re ๐ Spatial position ๏ฎ clustering, halo ๐ =2โ๐ Im ๐ Momentum component ๏ฎ high-momentum High-momentum in nucleon-pair is shown to describe short-range & tensor correlations as full 2p-2h excitations. ๐ ๐ โ ๐ โ๐ (๐โ๐) 2 ๐
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High-momentum AMD (HM-AMD)
Utilize imaginary positions in Gaussian wave packets. ๐=๐๐ซ with ๐ซ =0, 1, 2,โฏ,10 fm ๐ =2โ๐ Im ๐ =2โ๐๐ซ pair ansatz : ๐ p =๐๐ซ, ๐ n =โ๐๐ซ Opposite momentum direction (Itagaki) Large relative-momentum leading to 2p-2h effect Two directions of ๐ซ : spin-โฅ (z), spin-โฅ (๐ฅ) ฮจ HMโAMD = ๐ ๐ถ ๐ ฮฆ AMD ( ๐ซ ๐ ) Extendable to multi HM-pairs ๐ โ๐ (๐โ๐๐ซ) 2 ~ 5 fm โ1 ๐๐ซ ๐ โ๐ ๐ 2 cf. TOAMD ๐=๐ โ๐๐ซ ๐ โ๐ (๐+๐๐ซ) 2 PTEP (2017) 111D01, (2018) 011D01
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Lyuโs work: HM-AMD + TOAMD
(HM-TOAMD) (RCNP, Osaka) Hybridization of HM-AMD and TOAMD TOAMD : Single correlation function ๐น ๐ , ๐น ๐ท HM-AMD : One high-momentum pair VNN: AV8โ bare potential | ฮฆ hybrid (๐ซ) = 1+ ๐น ๐ +๐น ๐ท | ฮฆ HMโAMD ๐๐ซ F ๐ โ๐ (๐ยฑ๐๐ซ) 2 ๐ =ยฑ2โ๐๐ซ | ฮฆ HMโTOAMD = ๐ ๐ถ ๐ | ฮฆ hybrid ( ๐ซ ๐ ) โ๐๐ซ
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Energy, radius and Hamiltonian components
4He with AV8โ bare interaction Superpose the basis states with different Dโs for pairs. Lyu et al. PTEP 2018 (2018) 011D01, PRC submitted 4He Add bases successively
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Energy, radius in 4He with HM-TOAMD
4He with AV8โ bare interaction Superpose the basis states with different Dโs for pairs. HM-TOAMD TOAMD-F2 GFMC Energy -24.74 -25.93 Kinetic 95.17 97.06 102.3 Central -52.33 -53.12 -55.05 Tensor -63.80 -64.84 -68.05 LS -3.77 -3.83 -4.75 Radius 1.51 1.50 1.49 [MeV] [fm] Lyu et al. PTEP 2018 (2018) 011D01, PRC p-shell nuclei with HM-TOAMD is in progress.
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Summary Tensor-Optimized AMD (TOAMD) High-momentum AMD (HM-AMD)
Successive variational method for nuclei to treat VNN directly. Correlation functions : FD (tensor) , FS (short-range). F are independently optimized, better than Jastrow method. High-momentum AMD (HM-AMD) High-momentum pairs using imaginary centroids in Gaussian wave packets Comparison with shell model : โOne high-momentum pairโ = 2p-2h correlation Hybrid : HM-TOAMD using VNN PLB 769 (2017) 213 PRC 96 (2017) PTEP (2017) 111D01 PTEP (2018) 011D01 19
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Backup
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TOAMD with Double ๐น 4He AV8โ
F TOAMD with Double ๐น AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD 4He F are independent Kinetic/2 Good energy with ๐น 2 Next order is ๐น such as ๐น ๐ ๐น ๐ท ๐น ๐ท . Central Tensor Correlation functions
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TOAMD & Jastrow method TOAMD : Describe finite nuclei using VNN
Power series expansion using the correlation function ๐ญ ฮฆ TOAMD = 1+๐น+ ๐น 2 + ๐น 3 +โฏ ร ฮฆ AMD ๐ญ ๐ซ : D-wave transition from tensor force ๐ญ ๐บ : Short-range repulsion in central force Jastrow method ฮฆ Jastrow = ๐<๐ ๐ด 1+ ๐ ๐๐ ร ฮฆ 0 (reference state) Product form : common correlation ๐ ๐๐ among all pairs. Variational Monte Carlo (VMC) is utilized to calculate ๐ป . Widely used in various fields (cond. matter, atomic, molecular,...) F are independent ๐น ๐ =๐นโ๐นโฒโ๐นโฒโฒโฏ R. Jastrow, Phys. Rev. 98, 1479 (1955) f
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TOAMD & VMC with Jastrow
VNN : AV6 for central & tensor forces omit LS, L2, (LS)2 from AV14 Common ๐น Energy (MeV) TOAMD (power series) gives better energy than VMC (Jastrow) from variational point of view 3H Few-body Independent optimization of all F 4He PRC 96 (2017) Correlation functions ๐น ๐ท ๐น ๐ท JPS magazine (2017) Dec. 867
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Variation of multi-๐น in TOAMD
AV6 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD Few-body same F Kinetic/2 Central Common ๐น 3H Tensor Free F Correlation functions
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Diagrams of cluster expansion - VNN -
2-body 3-body 4-body bra F F V V V F ket 5-body 6-body
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Diagrams of cluster expansion, Kinetic energy
1-body 2-body 3-body 4-body bra F F T T T F ket uncorrelated kinetic energy 5-body
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Add bases successively
Convergence of HM-AMD VNN : Central : Volkov No.2 with M= LS & Tensor : G3RS bare force ๐๐ซ โ๐๐ซ 4He HM-AMD with 1 pair Tensor-optimized shell model (TOSM), complete 2p-2h. โCriterionโ (0s)4 Add bases successively
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Energy surface with the shift Dz
VNN : Central Volkov No.2 with M=0.6, Coulomb, LS & Tensor : G3RS bare force TM, K. Kato, K. Ikeda PTP113 (2005) 763 ๐=0.25 fm โ2 (0s)4 2 bases : (0s)4 + pn pair with Dz Converge ๐ =2๐ ๐ท ๐ง =2.5 fm โ1 > ๐ ๐น 4He Superpose by increasing Dz ๐ = 5 fm โ1 low-k high-k
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Hamiltonian Component in HM-AMD
4He ๐ ~ 5 fm โ1 ๐ = โ๐ =2โ๐ Im ๐ ๐=0.25 fm โ2
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HM-AMD & TOSM for 4He Tensor-Optimized Shell Model HM-AMD TOSM Energy
Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 Tensor-Optimized Shell Model | TOSM = ๐ 0 ๐ถ ๐ 0 |0p0h, ๐ ๐ 1 ๐ถ ๐ 1 |1p1h, ๐ ๐ 2 ๐ถ ๐ 2 |2p2h, ๐ 2 (MeV) HM-AMD TOSM Energy โ64.7 โ65.2 Kinetic 83.6 84.8 Central โ83.6 โ84.1 Tensor โ66.3 โ67.7 LS 0.8 0.9 Coulomb NO truncation of particle states in TOSM using Gaussian expansion with high-L more than 10โ HM-AMD = TOSM (1 pair) (full 2p-2h) ๐ โ๐ (๐ยฑ๐๐ซ) 2 Next, bare VNN (Lyu, Isaka)
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Tensor-optimized Shell Model for 4He
| TOSM = ๐ 0 ๐ถ ๐ 0 |0p0h, ๐ ๐ 1 ๐ถ ๐ 1 |1p1h, ๐ ๐ 2 ๐ถ ๐ 2 |2p2h, ๐ 2 NO truncation of particle states using Gaussian expansion 4He Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 9 Gaussian bases for each orbit with L โCriterionโ to examine 2p-2h effect (0s)4 (single particle orbit)
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Double ๐น effect in TOAMD
AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD Single SS SD DS DD ๐โ ๐ โ๐ (๐โ๐) 2 3H ๐= 1 2 ๐ 2 โ 2 ๐ ๐= โ๐ 2 Single F are independent +SS Getting close to the GFMC energy. n-dependence is small due to the flexibility of F. +SD+DS +DD wide, ๐ low compact, ๐ high
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TOAMD with single ๐น 3H 4He ร 1 2 ๐โ ๐ โ๐ (๐โ๐) 2 1+ ๐น ๐ท +๐น ๐ | ฮฆ AMD
F 1+ ๐น ๐ท +๐น ๐ | ฮฆ AMD ร 1 2 Kinetic 3H 4He Energy LS Central Tensor wide ๐ low compact ๐ high ๐= 1 2 ๐ 2 , ใ โ 2 ๐ ๐= โ๐ 2 ๐โ ๐ โ๐ (๐โ๐) 2 Large cancelation of T & V makes the small total energy. ๐ ๐ =๐ , s-wave configuration of AMD w.f.
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Many-body terms of ๐น โ ๐ป๐น in 3H
uncorrelated kinetic โ 2 ๐ ๐= โ๐ 2 Kinetic Full T+ 2-body same trend in central & tensor 2-body > 3-body not negligible 3-body Central Tensor wide, ๐ low comprtact, ๐ high
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Many-body Hamiltonian terms in 4He
PTEP (2017) 073D01 Kinetic 1+2 body Full 3-body Higher-body term tends to give smaller scale, but, not ignored. 4-body Central Tensor wide, ๐ ๐๐๐ค compact, ๐ โ๐๐โ
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Variation of multi-๐น in TOAMD
AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD PTEP (2017) 073D01 cf. Jastrow ansatz ๐น ๐ท ๐ , ๐น ๐ (๐) are fixed at single level, and used in ๐น 2 terms 3H DE=1.4 MeV F are independent as a full variation Correlation functions
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Variation of multi-๐น in TOAMD
AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD same Kinetic/2 F 3H Fixed ๐น Central Free F Few-body Tensor Correlation functions
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Variation of multi-๐น in TOAMD
AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD PTEP (2017) 073D01 cf. Jastrow ansatz ๐น ๐ท ๐ , ๐น ๐ (๐) are fixed at single level, and used in ๐น 2 terms 4He DE=2.3 MeV F are independent as a full variation Correlation functions
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Variation of multi-๐น in TOAMD
AV8โ 1+ ๐น ๐ +๐น ๐ท + ๐น ๐ ๐น ๐ + ๐น ๐ ๐น ๐ท + ๐น ๐ท ๐น ๐ + ๐น ๐ท ๐น ๐ท | ฮฆ AMD S D SS SD DS DD same Kinetic/2 F 4He Fixed ๐น Central Free F Tensor Independent optimization of all F Correlation functions
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