Tensor-optimized antisymmetrized molecular dynamics (TOAMD) with bare forces for light nuclei Takayuki MYO Mengjiao LYU (RCNP) Masahiro ISAKA (RCNP) Hiroshi TOKI (RCNP) Hisashi HORIUCHI (RCNP) Kiyomi IKEDA (RIKEN) Tadahiro SUHARA (Matsue) Taiichi YAMADA (Kanto Gakuin) SOTANCP4 in Galveston, Texas 2018.5
Outline Variational methods for finite nuclei with bare force Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) Single correlation function : Φ AMD Double correlation function : 2 Φ AMD Comparison with Jastrow method High-Momentum AMD (HM-AMD) High-momentum nucleon pairs in nuclei as 2p-2h excitation Combine TOAMD & HM-AMD
Deuteron properties & tensor force Energy -2.24 MeV S D Kinetic 19.88 Central -4.46 Tensor -16.64 LS -1.02 P(L=2) 5.77% S 11.31 D 8.57 SD -18.93 DD 2.29 AV8 Radius 1.96 fm V central V tensor r R m (s) = 2.00 fm R m (d) = 1.22 fm d-wave is spatially compact (high momentum)
Prog. Theor. Exp. Phys. 2015, 073D02 (38 pages) DOI: 10.1093/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 V NN, in particular, for clustering Multiply pair-type correlation function " = σ A i<j f ij " to AMD w.f. 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 D for D-wave transition from tensor force S for Short-range repulsion in central force cf. UCOM (Neff, eldmeier) 4
ormulation of TOAMD Deuteron wave function Deuteron S-wave D-wave K. Ikeda, TM, K. Kato, H. Toki Lecture Notes in Physics 818 (2010) R D wave~0.6 R S wave Involve high-k component induced by V tensor Tensor-optimized AMD (TOAMD) 1 TOAMD D AMD isospin 1 A N ( ) G t D D ri rj τi τ j, D( r) 12 n t 0 i j n f f S C e spatially compact Pair excitation via tensor operator with D-wave transition Optimize relative motion with Gaussian expansion General formulation with respect to mass number A t a n r 2
General formulation of TOAMD Φ TOAMD = 1 + D + S + D S + S S + D D + Φ AMD tensor tensor short-range Variational principle δe = 0 for E = Φ TOAMD H Φ TOAMD Φ TOAMD Φ TOAMD Variational parameters short-range AMD : n, Z i (i=1,, A) D S 12 σ n N G C n e a n(r i r j ) 2 S σ n N G C n e a n (r i r j ) 2 nucleon w.f. Power series expansion, but, all are independent Φ AMD = det φ 1 φ A φ Z (r) e ν(r Z)2 χ σ χ τ Gaussian wave packet spin-isospin dependent σ n H m,n E N m,n C n = 0 Eigenvalue problem
Matrix elements of correlated operator Correlated Hamiltonian Φ TOAMD H Φ TOAMD Φ TOAMD Φ TOAMD = Φ AMD H + H + H + H + Φ AMD Φ AMD 1 + + + + Φ AMD Correlated Norm Classify the connections of, H into many-body operators using cluster expansion = {2-body} + + {4-body} (2-body) 2 3-body V A i j k bra ket 4-body A i j k l V = {2-body} + + {6-body} ourier transformation of, V (2-body) 3 e a(r i r j ) 2 e k2 /4a e ik r i e ik r j relative single particle
Diagrams of cluster expansion - V NN - bra 2-body 3-body 4-body V V V ket 5-body 6-body
Diagrams of cluster expansion, Kinetic energy 1-body bra 2-body 3-body 4-body T T T ket uncorrelated kinetic energy 5-body
Matrix elements with ourier trans. ourier transformation of the interaction V & D, S. Y. Goto and H. Horiuchi, Prog. Theor. Phys., 62 (1979) 662 wave packet Gaussian expansion of V, D, S for relative motion. Multi-body operators are represented in the separable form for particle coordinates with opposite momentum direction. Three-body interaction can be treated in the same manner. e a(r i r j ) 2 = 1 relative 4πa r 2 ij S 12 r e a r 2 i r j 1 = 4πa 3/2 dk e k2 4a e ik r ie ik r j overlap 3 2 i 2a Z e ik r Z = Z Z exp( Quadratic + Linear terms of k single particle (separable) 2 dk e k2 4a e ik r ie ik r j k 2 S 12 (k) k 2 Τ8ν + ik Z + Z Τ2) tensor-type 10
3 H, 4 He, ( 6 He, 6 Li) Results V NN : bare AV8 with central, LS, tensor terms 7 Gaussians for D, S to converge the solutions. Successively increase the correlation terms. PLB 769 (2017) 213 PRC 95 (2017) 044314 PRC 96 (2017) 034309 PTEP (2017) 073D01 PTEP (2017) 111D01 Φ TOAMD = 1 + S + D + S S + S D + D S + D D Φ AMD AMD Single Double 2p-2h 3p-3h 4p-4h
TOAMD with single 12
TOAMD with single 1 + S + D Φ AMD Kinetic 1 2 3 H 4 He Energy LS Tensor Central wide ρ low φ e ν(r Z)2 compact ρ high ν = 1 2b 2, ħ 2 m ν = ħω 2 Large cancelation of T & V makes the small total energy. Z i = 0, s-wave configuration of AMD w.f. 13
Many-body terms of H Correlated Hamiltonian H = H 2 body + H 3 body + + H A body 3 H 3-body 4 He 3-body 4-body Energy 2-body Energy wide ρ low compact ρ high 2-body Many-body terms play a decisive role for energy saturation NO energy saturation within 2-body term PTEP (2017) 073D01 14
Amplitude Correlation functions S, D in 3 H short-range S (r) tensor D (r) S 3 E r 2 D 3 E intermediate long S 1 E Negative sign in S to avoid short-range repulsion in V NN. Ranges of D, S are NOT short, contributing to many-body terms.
Hamiltonian components a n n in 6 He & 6 Li Kinetic 1 2 a 1 fm p n Up to 6-body terms Energy Central 6 He > 6 Li Tensor 6 Li > 6 He wide, ρ low compact, ρ high Larger tensor contribution in 6 Li due to last pn pair Next, TOAMD+GCM (multi configuration of AMD) 16
TOAMD with double 17
Double effect in TOAMD AV8 1 + S + D + S S + S D + D S + D D Φ AMD Single SS SD DS DD φ e ν(r Z)2 3 H ν = 1 2b 2 ħ 2 m ν = ħω 2 Single +SS +SD+DS +DD are independent Getting close to the GMC energy. n-dependence is small due to the flexibility of. wide, ρ low compact, ρ high
TOAMD with Double AV8 1 + S + D + S S + S D + D S + D D Φ AMD S D SS SD DS DD AMD 1 + S 3 H are independent Add terms successively Reproduce the GMC energy Small curvature as 2 terms increase good convergence r 2 = 1.75 fm Correlation functions
TOAMD with Double AV8 1 + S + D + S S + S D + D S + D D Φ AMD S D SS SD 3 H are independent Kinetic/2 DS DD Central Tensor PLB769 (2017) 213 Reproduce the Hamiltonian components of 3 H Correlation functions
TOAMD with Double AV8 1 + S + D + S S + S D + D S + D D Φ AMD S D SS SD DS DD AMD 1 + S 4 He are independent Good energy with 2 Small curvature as 2 terms increase good convergence r 2 = 1.50 fm Next order is triple- Correlation functions
TOAMD with Double AV8 1 + S + D + S S + S D + D S + D D Φ AMD S D SS SD DS DD 4 He Kinetic/2 Central Tensor are independent Good energy with 2 Next order is triple- such as S D D. (on going) Correlation functions
TOAMD & Jastrow method TOAMD : Describe finite nuclei using V NN Power series expansion using the correlation function Φ TOAMD = 1 + + 2 + 3 + Φ AMD D : D-wave transition from tensor force S : Short-range repulsion in central force are independent n = Jastrow method A R. Jastrow, Phys. Rev. 98, 1479 (1955) Φ Jastrow = ς i<j 1 + f ij Φ 0 (reference state) Product form : common correlation f ij among all pairs. f f Variational Monte Carlo (VMC) is utilized to calculate H. Widely used in various fields (cond. matter, atomic, molecular,...)
TOAMD & VMC with Jastrow Common Energy (MeV) 3 H ew-body V NN : AV6 for central & tensor forces omit LS, L 2, (LS) 2 from AV14 TOAMD (power series) gives better energy than VMC (Jastrow) from variational point of view (Jastrow) 4 He Correlation functions D D Independent optimization of all PRC 96 (2017) 034309 JPS magazine (2017) Dec. 867
Variation of multi- in TOAMD AV6 1 + S + D + S S + S D + D S + D D Φ AMD S D SS SD DS DD ew-body same Kinetic/2 Central 3 H Tensor Common cf. Jastrow ree Correlation functions
High-Momentum Antisymmetrized Molecular Dynamics (HM-AMD) TM et al. PTEP (2017) 111D01 Tensor correlation vs. TOSM Lyu et al. PTEP (2018) 011D01 HM-AMD + TOAMD with AV8 TM PTEP (2018) 031D01 Short-range correlation with AV4 26
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 should represent efficiently the correlations (tensor, short-range, cluster,...). High-Momentum AMD (HM-AMD) Kimura (2014, priv. comm.), Itagaki-Tohsaki (PRC97(2018) 014304) Z : Centroid of Gaussian wave packet r = Re Z p = 2ħν Im Z φ N e ν(r Z)2 Spatial position clustering, halo Momentum component high-momentum High-momentum in one nucleon-pair is shown to describe tensor correlation as full 2p-2h effect in the shell model. 27
High-momentum AMD (HM-AMD) Utilize imaginary positions in Gaussian wave packets. Z = id with D = 0, 1, 2,, 10 fm p = 2ħν Im Z = 2ħνD ~ 5 fm 1 e ν(r id)2 pair ansatz : Z p = id, Z n = id Opposite momentum direction (Itagaki) Large relative-momentum leading to 2p-2h effect Two directions of D : spin- (z), spin- (x) Ψ HM AMD = σ i C i Φ AMD (D i ) Extendable to multi HM-pairs cf. TOAMD PTEP (2017) 111D01, (2018) 011D01 e νr2 Z = 0 id id e ν(r+id)2
Energy surface with the shift D z V NN : Central Volkov No.2 with M=0.6, Coulomb, LS & Tensor : G3RS bare force (0s) 4 ν = 0.25 fm 2 TM, K. Kato, K. Ikeda PTP113 (2005) 763 2 bases : (0s) 4 + pn pair with D z 4 He Converge Superpose by increasing D z k = 2νD z = 2.5 fm low-k 1 > k high-k k = 5 fm 1
Convergence of HM-AMD V NN : Central : Volkov No.2 with M=0.6 LS & Tensor : G3RS bare force 4 He id HM-AMD with 1 pair id (0s) 4 Add bases successively Tensor-optimized shell model (TOSM), full 2p-2h Criterion
Tensor-optimized Shell Model for 4 He TOSM = σ i0 C i0 0p0h, i 0 + σ i1 C i1 1p1h, i 1 + σ i2 C i2 2p2h, i 2 4 He NO truncation of particle states using Gaussian expansion Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 9 Gaussian bases for each orbit with L Criterion to examine 2p-2h effect (0s) 4 (single particle orbit)
HM-AMD & TOSM for 4 He Tensor-Optimized Shell Model Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 TOSM = σ i0 C i0 0p0h, i 0 + σ i1 C i1 1p1h, i 1 + σ i2 C i2 2p2h, i 2 (MeV) HM-AMD TOSM Energy 64.7 65.2 Kinetic 83.6 84.8 Central 83.6 84.1 Tensor 66.3 67.7 LS 0.8 0.9 Coulomb 0.9 0.9 NO truncation of particle states in TOSM using Gaussian expansion with high-l more than 10ħ HM-AMD = TOSM (1 pair) (full 2p-2h) e ν(r±id)2 Next, bare V NN (Lyu, Isaka)
Lyu s work: HM-AMD + TOAMD (RCNP, Osaka) (HM-TOAMD) Hybridization of HM-AMD and TOAMD TOAMD : Single correlation function S, D HM-AMD : One high-momentum pair Φ hybrid (D) ൿ = 1 + S + D Φ HM AMD e ν(r±id)2 p = ±2ħνD id Φ HM TOAMD = σ i C i Φ hybrid (D i ) ൿ V NN : AV8 bare potential id
Total energy of 3 H with imaginary shift D Nucleon pair e ν r 1 Z 1 +id 2 e ν r 2 Z 2 id 2 3 pair-types Single TOAMD Tensor minimum Short range minimum p = ħk = 2ħνD ν = 0.15 fm 2 k = 3~ 4 fm 1 igure: Energy curve with single D z (z direction)
Energy, radius and Hamiltonian components 3 H with AV8 bare interaction Superpose the basis states with different D s for pairs. HM-TOAMD [MeV] [fm] Lyu et al. PTEP 2018 (2018) 011D01 4 He : reproduce the results of 2 -TOAMD. Apply HM-TOAMD to p-shell nuclei.
Summary Tensor-Optimized AMD (TOAMD) Successive variational method for nuclei to treat V NN directly. Correlation functions : D (tensor), S (short-range). are independently optimized, better than Jastrow method. PLB 769 (2017) 213 High-momentum AMD (HM-AMD) PTEP (2017) 111D01 High-momentum pairs using imaginary centroids in Gaussian wave packets Comparison with shell model and TOAMD: One high-momentum pair = 2p-2h effect Hybrid : HM-TOAMD using V NN PRC 96 (2017) 034309 PTEP (2018) 011D01