Tensor-Optimized Antisymmetrized Molecular Dynamics

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Explore the innovative Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) method for light nuclei with direct treatment of VNN interactions. Gain insights into the variational approach for nuclei, deuteron properties, and the formulation of TOAMD. Discover how TOAMD describes finite nuclei using VNN and pair-type correlation functions, optimizing relative motions and excitations through detailed analyses.

  • Tensor-Optimized
  • Molecular Dynamics
  • Nuclei
  • Variational Method
  • Clustering
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  1. Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) for light nuclei with bare nuclear interactions TOAMD ... New variational method for nuclei to treat VNNdirectly Takayuki MYO Hiroshi TOKI (RCNP) Kiyomi IKEDA (RIKEN) Hisashi HORIUCHI (RCNP) Tadahiro SUHARA (Matsue) The Seventh Asia-Pacific Conference on Few-Body Problems in Physics (APFB 2017), 2017.8.25-30

  2. Deuteron properties & tensor force Energy -2.24 MeV S 11.31 D 8.57 S Kinetic 19.88 Central -4.46 D SD -18.93 DD 2.29 Tensor -16.64 LS -1.02 P(L=2) 5.77% Radius 1.96 fm AV8 Vcentral Rm(s) = 2.00 fm Rm(d) = 1.22 fm d-wave is spatially compact (high momentum) r Vtensor

  3. 12C in Antisymmetrized Molecular Dynamics = det Veff: Effective central force + LS force NO tensor force AMD 1 A nucleon w.f. ? ? ?( ? ?)2???? Gaussian wave packet 0+(Hoyle) triple- config. 12C -particle ... (0s)4 AMD 0+(GS) shell-model-like + T. Suhara. and Y. Kanada-En yo, Phys. Rev. C 82, 044301 (2010) neutron proton

  4. Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) TM, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi, and Tadahiro Suhara Describe finite nuclei using VNN, in particular, for clustering Multiply pair-type correlation function ? 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 ??for tensor force with D-wave transition ??for Short-range repulsion in central force 4

  5. Formulation of TOAMD K. Ikeda, TM, K. Kato, H. Toki Lecture Notes in Physics 818 (2010) Deuteron wave function = + Deuteron -wave -wave S D ?? wave~0.6 ?? wave spatially compact Involve high-k component induced by Vtensor Tensor-optimized AMD (TOAMD) ( TOAMD 1 F = + ) isospin AMD D F t N ( ) 1 A G 2 a r = = t ( ) , ( ) D f F f r r r S C e n i j i j 12 D D n = i j 0 t n Pair excitation via tensor operator with D-wave transition Optimize relative motion with Gaussian expansion General formulation with respect to mass number A

  6. General formulation of TOAMD = + + + + + + (1 ) F F F F F F F F TOAMD AMD D S D S D D S S tensor short-range tensor short-range F are independent Variational principle ?? = 0 for ? = TOAMD? TOAMD TOAMD TOAMD Variational parameters AMD : , Zi(i=1, , A) ?? ?12 ?=1 ?? ?=1 = det AMD 1 A ????? ???2 ???? ?(?) ? ?( ? ?)2???? nucleon w.f. ?2 ? ?? Gaussian wave packet spin-isospin dependent ??,? ? ??,???= 0 Eigenvalue problem ?

  7. Matrix elements of correlated operator Correlated Hamiltonian AMD? + ? ? + ?? + ? ?? + AMD AMD1 + ? + ? + ? ? + AMD TOAMD? TOAMD TOAMD TOAMD = Correlated Norm Classify the connections of F, H into many-body operators using cluster expansion method 3-body 4-body bra F V F ? ? = {2-body} + + {4-body} ket i j k i j k l (2-body)2 ? ? ? ?? = {2-body} + + {6-body} Fourier transformation of ?,? ? ?( ?? ??)2 ? ?2/4? ??? ?? ? ?? ?? relative (2-body)3 single particle

  8. Diagrams of cluster expansion - VNN- 4-body 2-body 3-body bra F V F V V F ket 5-body 6-body

  9. Results PLB 769 (2017) 213 PRC 95 (2017) 044314 PTEP (2017) 073D01 PRC (2017) in press 3H, 4He, (6He, 6Li) VNN: AV8 (central, LS, tensor) 7 Gaussians for ??,??to converge the solutions. Full treatment of many-body operators (all diagrams) to retain the variational principle. Successively increase the correlation terms. TOAMD = 1 + ??+??+????+ ????+ ????+ ????| Single F | AMD Double F F F 2p-2h F F F 4p-4h 3p-3h

  10. TOAMD with single ? F 1 + ??+??| AMD 1 Kinetic 2 3H 4He Energy LS Central Tensor wide ?low compact ?high 2 ?? = ? 1 ? = 2?2, ? ? ?( ? ?)2 2 Large cancelation of Kinetic & V makes the small total energy. ?? = 0, s-wave configuration of AMD w.f. 10

  11. Many-body operators of ??? 3-body 4-body 3-body 4He 3H Energy Energy 2-body 2-body wide ?low compact ?high NO energy saturation within 2-body 2 ?? = ? 1 ?? = 0 ? = 2?2, PTEP (2017) 073D01 2 Many-body correlation plays a decisive role for energy saturation. Similar to Bethe-Brueckner-Goldstone approach with G-matrix (Baldo) 11

  12. F Double ? effect in TOAMD F 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS F are independent AMD 3H Add terms successively Reproduce the GFMC energy Small curvature as ?2terms increase good convergence 1 + ?? ?2= 1.75 fm Correlation functions

  13. F Double ? effect in TOAMD F 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS PLB769 (2017) 213 3H F are independent Kinetic/2 Reproduce the Hamiltonian components of 3H Central Tensor Correlation functions

  14. F Double ? effect in TOAMD F 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS F are independent AMD Good energy with ?2 Small curvature as ?2terms increase good convergence ?2= 1.50 fm Next order is ?3 4He 1 + ?? Correlation functions

  15. F Double ? effect in TOAMD F 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS 4He F are independent Kinetic/2 Good energy with ?2 Next order is ?3 such as ??????. Central Tensor Correlation functions

  16. Summary Tensor-Optimized AMD (TOAMD). Successive variational method for nuclei to treat VNNdirectly. Correlation functions : FD(tensor) , FS (short-range). F is independently optimized, better than Jastrow method. PRC (2017) in press At F2 level, good reproduction of s-shell nuclei. Next, we shall apply TOAMD to p-shell nuclei. Multi-configuration of AMD as TOAMD+GCM. Triple-F (??) is ongoing. Include VNNNsuch as Fujita-Miyazawa type in the same manner of many-body operators.

  17. Backup 17

  18. Short-range correlation in TOAMD power series expansion | TOAMD = 1 + ??1+ ??2 ??3| AMD single double PRC95 (2017) 044314 VNN: Malfliet-Tjon V (central short-range) f Nucl. Phys. A127, 161 (1969). ? ? = 1458.05? 3.11? f 578.09? 1.55? (MeV) ? ? TOAMD vs. Variational Monte Carlo with Jastrow-type ? (reference state) Jastrow= ?<? ??? 0 same 1 + ??? =1 + ?<? ???+ ?<?<? ??? ???+ ? ? ? ? ?<? ???= ?<? ~ ? ? (3-body term) ? (2-body term)

  19. TOAMD vs. VMC with Jastrow VNN: Malfliet-Tjon V (central short-range) PRC95 (2017) 044314 TOAMD (double) 8.24 VMC (Jastrow) 8.22(2) Few-body 8.25 3H (MeV) 31.36 31.28 31.19(5) 4He Carlson, Pandharipande, Nucl. Phys. A371, 301 (1981). FY, SVM, GFMC TOAMD (power series) provides the better energy than VMC (Jastrow) from variational point of view. TOAMD is extendable by increasing the power n of correlation functions ??in the power series expansion.

  20. TOAMD vs. VMC with Jastrow VNN: AV6 with central & tensor forces (omit LS, L2, (LS)2from AV14) Common ? (MeV) F 3H F Few-body TOAMD (power series) gives better energy than VMC (Jastrow) from variational point of view 4He Correlation functions

  21. Variation of multi-? in TOAMD 1 + ??+??+????+ ????+ ????+ ????| AMD AV6 SD DS S DD D SS same Few-body F Kinetic/2 F Central Fixed ? 3H Tensor Free F Correlation functions

  22. Correlation functions ??, ??in 3H short-range ??(?) tensor ??(?) FS 3E r2 FD 3E Amplitude intermediate long FS 1E Negative sign in ??to avoid short-range repulsion in VNN Ranges of ??,??are NOT short. Range b of ?? AMD 0.6 bAMD spatially compact, high-k (= 1/ 2? + ?) (1/ ?) Tensor-optimized shell model TM, K. Kato, K. Ikeda PTP113 (2005) 763

  23. Many-body terms of ??? in 3H 2 ?? = ? uncorrelated kinetic Kinetic Full 2 T+ 2-body same trend in central & tensor 2-body > 3-body (not negligible) 3-body Central Tensor 23 compact, ?high wide, ?low

  24. Many-body Hamiltonian terms in 4He PTEP (2017) 073D01 Kinetic Full 1+2 body 3-body Higher-body term tends to give smaller scale, but, not ignored. 4-body Central Tensor 24 wide, ???? compact, ? ??

  25. Variation of multi-? in TOAMD 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS PTEP (2017) 073D01 cf. Jastrow ansatz ??? ,??(?) are fixed at single level, and used in ?2terms 3H E=1.4 MeV F are independent as a full variation Correlation functions

  26. Variation of multi-? in TOAMD 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS same Kinetic/2 F 3H F Fixed ? Central Free F Few-body Tensor Correlation functions

  27. Variation of multi-? in TOAMD 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS PTEP (2017) 073D01 cf. Jastrow ansatz ??? ,??(?) are fixed at single level, and used in ?2terms 4He E=2.3 MeV F are independent as a full variation Correlation functions

  28. Variation of multi-? in TOAMD 1 + ??+??+????+ ????+ ????+ ????| AMD AV8 SD DS S DD D SS same Kinetic/2 F 4He F Fixed ? Central Free F Tensor Independent optimization of all F Correlation functions

  29. Hamiltonian components in 6He & 6Li n n 1 fm 1 Kinetic p 2 n Up to 6-body terms Energy 6He > 6Li Central 6Li >6He Tensor compact, ?high wide, ?low Larger tensor contribution in 6Li due to last pn pair Next, TOAMD+GCM (multi configuration of AMD) 29

  30. 6He with +n2 n n 1 fm Kinetic 1+2 body Full 4-body 3-body 5-body small 6-body small Central Tensor compact, ? ?? wide, ????

  31. Pion exchange interaction & Vtensor 2 2 2 q q q = + q q 3( )( ) ( ) S 1 2 1 2 12 + + + 2 2 2 2 2 2 m q m q m q + + 2 2 2 2 m m q q m q = + ( ) S 1 2 12 + + 2 2 2 2 2 2 m q m q involve large momentum interaction Yukawa interaction Tensor operator = ) ( q q 3( )( ) S 12 - Vtensorproduces the high momentum component. 1 2 1 2 31

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