Short-range and tensor correlations in light nuclei

ELPH 研究会 C024 「ハドロン構造における多粒子相関」, 東北大学ELPH, Oct.16-17, 2019

Myo, Takayuki

大阪工業大学

明 孝之

Outline

• Properties of tensor force from p-exchange

• Variational method for nuclei with nuclear force

• Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD)

– ΦTOAMD = (1 + 𝐹 + 𝐹2 +⋯) × ΦAMD

– Variational correlation function 𝐹 = σ𝑖<𝑗𝐴 𝑓𝑖𝑗

• Apply TOAMD to s-shell & p-shell nuclei

– AV8’ , Bare nucleon-nucleon interaction

– AV4’ , Central interaction with short-range repulsion

– Comparison with Jastrow method

F

3

Pion exchange interaction & Vtensor

d interaction

Yukawa interaction

involve largemomentum

2 2 2

1 2 1 2 122 2 2 2 2 2

2 2 2 2

1 2 122 2 2 2 2 2

ˆ ˆ3( )( ) ( )

( )

q q qS

m q m q m q

m q m qS

m q m q m q

σ q σ q σ σ

σ σ

12 1 2 1 2ˆ ˆ3( )( ) ( )S σ q σ q σ σ

Tensor operator

• Vtensor produces the high-momentum component

AV18 NN potential , Triplet-Even (S=1, T=0)

4

𝑆12 ො𝒓 = 3 𝝈1 ∙ ො𝒓 𝝈2 ∙ ො𝒓 − 𝝈1 ∙ 𝝈2 = ቊ−1 (𝒓 ⊥ 𝝈)2 (𝒓 ∥ 𝝈)

𝒓,スピンの向きが全て揃うと強い引力

(テンソル力)

斥力芯は共通に存在する (中心力)

R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C51, 38 (1995)R. Roth, T. Neff and H. Feldmeier, Prog. Part. Nucl. Phys. 65 (2010) 50.

AV8’ bare potential

5

R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89,182501 (2002)

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

Deuteron properties & tensor force

D-wave is “spatially compact” (high-momentum)

S 11.31D 8.57

SD -18.93DD 2.29

S

D

S

D

r [fm]

S 2.00D 1.22

Φ(𝒓)

෩Φ(𝒌)

7

• Describe finite nuclei using VNN , in particular, for clustering

• Multiply pair-type correlation function "𝑭 = σ𝒊<𝒋𝑨 𝒇𝒊𝒋 " to AMD w.f.

𝑭𝑫 for D-wave transition with S12 in tensor force

𝑭𝑺 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

Prog. Theor. Exp. Phys. 2015, 073D02 (38 pages)

DOI: 10.1093/ptep/ptv087

Takayuki Myo, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi,

and Tadahiro Suhara

Tensor-optimized antisymmetrized molecular

dynamics in nuclear physics (TOAMD)

• Variational principle

– 𝛿𝐸 = 0 for 𝐸 =ΦTOAMD 𝐻 ΦTOAMD

ΦTOAMD ΦTOAMD

• Variational parameters

– AMD : n, Zi (i=1,…, A)

– 𝐹𝐷 ∶ 𝑆12 σ𝑛𝑁𝐺 𝐶𝑛𝑒

−𝑎𝑛(𝒓𝑖−𝒓𝑗)2

– 𝐹𝑆 ∶ σ𝑛𝑁𝐺 𝐶𝑛

′ 𝑒−𝑎𝑛′ (𝒓𝑖−𝒓𝑗)

2

| ۧΦTOAMD = 1 + 𝐹𝑆 + 𝐹𝐷 + 𝐹𝐷𝐹𝑆 + 𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝐷 +⋯ | ۧΦAMD

General formulation of TOAMD

𝜑𝒁(𝒓) ∝ 𝑒−𝜈(𝒓−𝒁)2𝜒𝜎𝜒𝜏

Gaussian wave packet with range n

nucleon w.f.

Eigenvalue problemσ𝑛 𝐻𝑚,𝑛 − 𝐸 𝑁𝑚,𝑛 𝐶𝑛 = 0

short-range

tensor short-range

spin-isospin dependent

Power series expansion, but, all 𝐹 are independent

| ۧΦAMD = det 𝜑1⋯𝜑𝐴

𝐹 are scalar function

tensor

En’yo, Horiuchi, ...

Matrix elements of correlated operator

Correlated Norm

Classify the connections of F, Hinto many-body operatorsusing cluster expansion method

(2-body)3

(2-body)2

ΦTOAMD 𝐻 ΦTOAMD

ΦTOAMD ΦTOAMD

=ΦAMD 𝐻 + 𝐹†𝐻 + 𝐻𝐹 + 𝐹†𝐻𝐹 +⋯ ΦAMD

ΦAMD 1 + 𝐹† + 𝐹 + 𝐹†𝐹 +⋯ ΦAMD

Correlated Hamiltonian

𝐹†𝐹 = {2-body} +⋯+ {4-body}

𝐹†𝑉𝐹 = {2-body} +⋯+ {6-body}

VF†

F

3-body

i j k𝒜

Fourier transformation of 𝐹, 𝑉

𝑒−𝑎(𝒓𝑖−𝒓𝑗)2⇒ 𝑒−𝒌

𝟐/4𝑎 ∙ 𝑒𝑖𝒌⋅𝒓𝑖 ∙ 𝑒−𝑖𝒌⋅𝒓𝑗

relative single particle

i j k l𝒜

4-bodybra

ket

Diagrams of cluster expansion - VNN -

4-body3-body

F

2-body

V

F

F

5-body

V V

6-body

bra

ket

Diagrams of cluster expansion, Kinetic energy

1-body 4-body

T

3-body

F

2-body

T

F

T

F

5-body

bra

uncorrelated kinetic energy

ket

Results

• 3H, 4He, (6He, 6Li)

• VNN : bare AV8’ with central, LS, tensor terms

• 7 Gaussians for 𝐹𝐷, 𝐹𝑆 to converge the solutions.

• Successively increase the correlation terms.

PLB 769 (2017) 213 PRC 95 (2017) 044314PRC 96 (2017) 034309

PTEP (2017) 073D01PTEP (2017) 111D01

| ۧΦTOAMD = 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 +⋯ | ۧΦAMD

Single F Double F

F

F

F

F

F

2p-2h 4p-4h3p-3h

AMD

• Add terms successively

• Reproduce the GFMC energy

• Small curvature as 𝐹2 terms increase good convergence

• 𝑟2 = 1.75 fm

Correlation functions

3H

AMD

TOAMD with Double 𝐹

F are independent

1 + 𝐹𝑆

F

F

AV8’ 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D SS DDS DS

Correlation functions

3H F are independent

TOAMD with Double 𝐹

Kinetic/2

Tensor

Central

F

F

PLB769 (2017) 213

• Reproduce the Hamiltonian components of 3H

AV8’ 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D SS DDS DS

• Good energy with 𝐹2

• Small curvature as 𝐹2 terms increase good convergence

• 𝑟2 = 1.50 fm

• Next order : 𝐹3 is ongoing

Correlation functions

AMD

4He

F are independent

TOAMD with Double 𝐹

1 + 𝐹𝑆

F

F

AV8’ 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D SS DDS DS

Correlation functions

Kinetic/2

4He

• Good energy with 𝐹2

• Next order is 𝐹3

such as 𝐹𝑆𝐹𝐷𝐹𝐷.

TensorCentral

TOAMD with Double 𝐹 F

F

AV8’ 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D SS DDS DS

F are independent

Correlation functions 𝐹𝑆 , 𝐹𝐷 in 3H

r2×FD 3E

FS 1E

FS 3E

• Negative sign in 𝐹𝑆 to avoid short-range repulsion in VNN .

• Short-range repulsion : 1E > 3E in AV8’.

• Ranges of 𝐹𝐷, 𝐹𝑆 are NOT short, contributing to many-body terms.

tensor 𝐹𝐷(𝑟)

Am

pli

tude

short-range 𝐹𝑆(𝑟)

longintermediate

𝐹(𝑟) = σ𝑛𝑁𝐺 𝐶𝑛𝑒

−𝑎𝑛𝑟2

Importance of many-body terms in 𝐹†𝐻𝐹

• Include all of many-body terms in the correlated operators.

• ෩𝑯𝟐−𝐛𝐨𝐝𝐲 is two-body

term only, which violates the variational principle.

18

3H

Correlated Hamiltonian ෩𝐻 = ෩𝐻2−body + ෩𝐻3−body +⋯+ ෩𝐻𝐴−body

compact, 𝜌high

𝜑 ∝ 𝑒−𝜈(𝒓−𝒁)2

nucleon w.f. common n for all nucleons

wide, 𝜌low

ℏ2

𝑚𝜈 =

ℏ𝜔

2

19

• Many-body terms play a decisive role for energy saturation

• NO energy saturation within 2-body term

3-body

2-body

Energy

Many-body terms of 𝐹†𝐻𝐹

4-body

3H 4He

Energy

3-body

2-body

PTEP (2017) 073D01

Correlated Hamiltonian ෩𝐻 = ෩𝐻2−body + ෩𝐻3−body +⋯+ ෩𝐻𝐴−body

wide𝜌low

compact𝜌high

Τℏ2 𝑚 ⋅ 𝜈 = Τℏ𝜔 2

TOAMD & Jastrow method

F are independent

𝐹𝑛 = 𝐹 ∙ 𝐹′ ∙ 𝐹′′⋯

R. Jastrow, Phys. Rev. 98, 1479 (1955)

f

f

• TOAMD : Describe finite nuclei using VNN

– Power series expansion using the correlation function 𝑭

– ΦTOAMD = 1 + 𝐹 + 𝐹2 + 𝐹3 +⋯ ×ΦAMD

– 𝑭𝑫 : D-wave transition from tensor force

– 𝑭𝑺 : Short-range repulsion in central force

• Jastrow method

– ΦJastrow = ς𝑖<𝑗𝐴 1 + 𝑓𝑖𝑗 ×Φ0(reference state)

– Product form : common correlation 𝒇𝒊𝒋 among all pairs.

– Variational Monte Carlo (VMC) is utilized to calculate 𝐻 .

– Widely used in various fields (cond. matter, atomic, molecular,...)

TOAMD & VMC with Jastrow

Correlation functions

Few-body

VNN : AV6 for central & tensor forces

omit LS, L2, (LS)2 from AV14

4He

3H

Energy (MeV)

TOAMD (power series)gives lower energies than VMC (Jastrow) from variational point of view

Common 𝐹

PRC 96 (2017) 034309

𝐹𝐷𝐹𝐷

Independent optimization of all F

JPS magazine (2017) Dec. 867

3H

Variation of multi-𝐹 in TOAMD

Correlation functions

AV6 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D DDSSS DS

F

F

Tensor

Common 𝐹

Free F

Kinetic/2

Few-body same

Central

P-shell nuclei in TOAMD

23

• AV8’ Bare nucleon-nucleon interaction

• AV4’ Central interaction with short-range repulsion.

• Add terms successively• w/o Projection• FDFD is ongoing

Correlation functions

5He with Double 𝐹

F are independent

1 + 𝐹𝑆

AV8’ 1 + 𝐹𝑆 +𝐹𝐷 +𝐹𝑆𝐹𝑆 + 𝐹𝐷𝐹𝑆 + 𝐹𝐷𝐹𝐷 | ۧΦAMD

D SS DDS DS

AMD

Correlation functions

a2 fm

n

5He

Tensor

Central

Kinetic/2

𝑁G = 4 in 𝐹

E

6He & 6Li with single 𝐹

25

Central

Kinetic

Tensor

× 12

6Li > 6He

6He > 6Li

a

a1 fm

n

p

nn

compact, 𝜌highwide, 𝜌low

Energy

CoolingAV8’

• Larger tensor contribution in 6Li due to last pn pair

• 6He : 𝑬𝒙 𝟐+ = 1.6 MeV (exp : 1.8 MeV)

• Next, +F2, GCM (multi configuration of AMD)

ℏ2

𝑚𝜈 =

ℏ𝜔

2

6He, 6Li with Double 𝐹

26

• Add terms successively• w/o Projection• Converge to the GFMC energy 𝐹𝑆 are independent

a n

pa

1 fm

nn

1 + 𝐹𝑆 + 𝐹𝑆𝐹𝑆 | ۧΦAMD

Correlation functions Correlation functions

AV4’ (central)

Kinetic Kinetic

6He 6Li

7Li, 8Be in TOAMD (single 𝐹)

27

a aa t

AV4’

TOAMDAMD 1 + 𝐹𝑆 | ۧΦAMD

7Li 8Be

AMD AMD

Basic configuration

𝐹𝑆 effect

Component of AMD = 91-92%

Correlation function 𝐹𝑆 in 8Be

Even channel Odd channel

28

• EVEN channel : Similar shape to s-shell nuclei (universality).• ODD channel : Obtained in the solution of eigenvalue problem.

cf. In VMC with Jastrow method, Form of F is assumed.

intermediate

decrease

short

Am

pli

tude

long

aa: 1 fm

Singlet

Triplet

Singlet

Triplet

Summary

• Tensor-Optimized AMD (TOAMD)

– Successive variational method for nuclei to treat VNN directly.

– ΦTOAMD = (1 + 𝐹 + 𝐹2 +⋯) × ΦAMD with 𝐹 = σ𝑖<𝑗𝐴 𝑓𝑖𝑗

– Correlation functions, 𝐹𝐷 (tensor), 𝐹𝑆 (short-range).

– 𝐹 are independently optimized, better than Jastrow method

– Successful results for S-shell nuclei (A≤4) with double F (F2)

– P-shell nuclei (A≥5) are ongoing.

– Triple F (F3) is ongoing.

𝐹𝐷𝐹𝐷𝐹𝑆 in 3H : Δ𝐸(𝐹2) = 80 keV → 30 keV w.r.t. 𝐸GFMC.