Strong tensor correlation in light nucleiwith tensor-optimized antisymmetrized

molecular dynamics (TOAMD)

International symposium on “High-resolution Spectroscopy and Tensor interactions” (HST15), Osaka, 2015.11

Takayuki MYO

2

Outline

• Tensor Optimized Shell Model (TOSM)• Importance of 2p2h excitation involving high-

momentum component for tensor correlation

• Applications to He, Li, Be isotopes

• Tensor Optimized AMD (TOAMD)• Toward clustering with tensor correlation

• Formulation given in PTEP 2015, 073D02

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 qq q Sm q m q m q

m q m qSm q m q m q

12 1 2 1 2ˆ ˆ3( )( ) ( )S q q

Tensor operator

- Vtensor produces the high momentum component.

S

D

Energy -2.24 MeV

Kinetic 19.88

Central -4.46

Tensor -16.64

LS -1.02

P(L=2) 5.77%

Radius 1.96 fmVcentral

Vtensor

AV8’

Deuteron properties & tensor force

Rm(s)=2.00 fm

Rm(d)=1.22 fm d-wave is “spatially compact” (high momentum)

r

0,k

H EC

1

,A A

i G iji i j

H t T v

( ) ( )k kk

A C A

,0

lj n

H Eb

c.m. excitation is excluded by Lawson’s method

(0p0h+1p1h+2p2h)

1

' ', ,( ) ( )

N

n

n nlj lj n lj nC

r r

,1/2

2

,1 ˆ( ) exp ( ),2 lj n

llj n l j

rr Yb

r r

particle state Gaussian expansion for each orbit

Gaussian basis function' ''', ''

n nlj lj n n d

Hiyama, Kino, Kamimura PPNP51(2003)223

Shell model type configuration with mass number A

Tensor-optimized shell model (TOSM)TM, Sugimoto, Kato, Toki,

IkedaPTP117(2007)257

particle

hole

6

Tensor force matrix elements

6• Centrifugal potential (1GeV@0.5fm) pushes away D-wave.

p+r (Bonn) AV8’

VT = Vresidual

VT ≠ Vresidual

MSD(r) = r2·S(r,bS)·VT·D(r,bD)

Integrand of Tensor MEbD ~ bS

bD ~ bS0.5

1st order

0p0h-2p2h

3.8 MeV15.0 MeV

high momentum

He, Li, Be isotopes in TOSM

7

TM, A. Umeya, H. Toki, K. Ikeda PRC84 (2011) 034315TM, A. Umeya, H. Toki, K. Ikeda PRC86 (2012) 024318TM, A. Umeya, K. Horii, H. Toki, K. Ikeda PTEP (2014) 033D01TM, A. Umeya, H. Toki, K. Ikeda PTEP (2015) 073D02

T

VT

VLS

VC

E

(exact)Kamada et al.PRC64 (Jacobi)

• Gaussian expansion at most with 9 bases

• variational calculation

TM, H. Toki, K. IkedaPTP121(2009)511

4He in TOSM + central UCOM

good convergence

• High-k components bring large kinetic energy

Selectivity of the tensor coupling in 4He

9

DL = 2 DS = 2Selectivity of S12

s1 s2 s3 s41+(pn) 1+(pn)

s1 s2 s3 s4 L=2

s1 s2 s3 s4 L=2

VT

VT2𝑝2 h :(0 𝑠)10

2 [ (1𝑠 ) (𝑑3/2 ) ]10

2𝑝2 h :(0 𝑠)102 (𝑝1/2)10

2

0𝑝 0 h : (0 𝑠 )4

l1=l2=l3=l4=0l1=l2=0

l1=l2=0

l3=1l4=1

l3=0l4=2

5-8He with TOSM+UCOM• Excitation energies in MeV

• Excitation energy spectra are reproduced well

TM, A. Umeya, H. Toki, K. Ikeda PRC84 (2011) 034315

5-9Li with TOSM+UCOM• Excitation energies in MeV

• Excitation energy spectra are reproduced well

TM, A. Umeya, H. Toki, K. Ikeda PRC86(2012) 024318

1212

Radius of He & Li isotopes

I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007)252501

Expt

Halo SkinA. Dobrovolsky, NPA 766(2006)1

TOSM with AV8’

8Be spectrum

13a-a structure

• Argonne Group– Green’s function Monte Carlo

C. Pieper, R. B. Wiringa, Annu.Rev.Nucl.Part.Sci.51 (2001)

TUNL

G~ few MeV

G< 1MeV

8Be in TOSM AV8’

• VT 1.1, VLS 1.4– simulate 4He benchmark

(Kamada et al., PRC64) • correct level order (T=0,1)• tensor contribution : T=0 > T=1• a : 0p0h+2p2h with high-k

– 2a needs 4p4h.– spatial asymptotic form of 2a

14

a aclustering

TOSM

Expt. (TUNL)

TOAMD

Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD)

15

TM, H. Toki, K. Ikeda, H. Horiuchi, T. Suhara, PTEP 2015, 073D02

Toward the clustering with tensor correlation explicitly

(TOAMD)

AMD AMD

2 212

0,1

2

,

TOAMD AMD

( ) , ( ) exp( )

( ) , ( ) exp( )

D S

At

D D i j D n nt i j n

Ast

S S i j S n ns t i j n

F F

F f r r f r C a r r S

F f r r f r C a r

tensor correlation

short-range correlation

2p2h0p0h Gaussian expansion

Formulation of TOAMD

17

0 AMD AMD

AMD AMD AMD

2 212

0,1

TOAMD AMD

( ) , ( ) exp( )

D SD S

SD DD SSS D D D S S

At

D D i j D n nt i j n

GN

C F F

F F F F F F

F f r r f r r S C a r

2p2h

4p4h

0p0h

• Variational parameters– n, Zi (i=1,…, A) , spin-direction (up/down) in AMD

– C0 , Cn , an , NG in Gaussian expansion

– Solve cooling equation for

Matrix elements of multi-body operator

• generates at most 6-body matrix elements.

• Classify the connections between the operators.

' ' Aˆ... ...i j i jO

1

2

AMD1 det ,...,

!

exp

AA

n

Z

r Z r ZT T

ˆ , , , , ,

D D S S S D

D D S S

O F F F F F FF V F F V F

VT

FD

FD

2-body

VT

FD

FD

4-body

Matrix elements

particlecoordinate

centroid

range

Matrix elements with Fourier trans.• Fourier transformation of the interaction V & FD, FS.

– Y. Goto and H. Horiuchi, Prog. Theor. Phys., 62 (1979) 662– Gaussian expansion of V, FD, FS for relative motion– Multi-body operators are represented in the separable form

for particle coordinates.– Three-body interaction can be treated in the same manner.

19

3/2

3

3/2 22 2

12 123

2

2 2

2 2

( ) /4

( ) /4

1 (2 )

1( ) ( )(2 ) 2

exp / 8 ( ) / 2

ij

ikr

i j i

i j

j

ji

a r r k a

a r r k a

ikrikr

ikrikr

e dk e e ea

ir S r e dk e e e k S ka a

e k ik

pp

pp

n

Z Z Z Z Z Z

relativesingle particle

tensor-type

overlap Quadratic + Linear terms of k

Matrix elements with Fourier trans.

• Example : FDVT

AMD T 2 AMD 1 2

ijk, i'j'k'

1 1 2 2 ' ' ' ' ' ' ' '' '' '

1 1' ' ' ' ' '

2 21 1 2 2

1 2 1 2

/4 /4( )

3δ δ δ δ 3δ δ δ δ

det

D

i i j j

x y u v xx yy xy x y uu vv uv u vxyx yuvu v

i x u i j y v j i i j j

k a k a

ik r ik r ik r ik r

F V dk dk e e

e e e e

k k k k

B B

Ζ Z Z Z

VT

FD

2-body (2-body)(2-body) = (2-body)+(3-body)+(4-body)

k-integral

spatial part

(tensor)2

spin-isospin part antisymmetrization

𝑘2

𝑘1 a1

a2

Bij=ij

overlap

Matrix elements with Fourier trans.

• Example : FDVT

21

AMD T 3 AMD 1 2

ijk, i'j'k'

1 1 2 2 ' ' ' ' ' ' ' '' '' '

' ' ' ' '

2 21 1 2 2

1 1 2 2

/4 /4( )

3δ δ δ δ 3δ δ δ δ

det

D

i i j j k k

x y u v xx yy xy x y uu vv uv u vxyx yuvu v

i x i j y u j k v k i i

k a k a

ik r ik r ik r ik r

F V dk dk e e

e e e e

k k k k

B

Ζ Z Z Z Z Z

1 1 1' 'j j k kB B

VT

FD

3-body (2-body)(2-body) = (2-body)+(3-body)+(4-body)

k-integral

spatial part

(tensor)2

spin-isospin part antisymmetrization

𝑘2

𝑘1 a1

a2

Bij=ij

overlap

TOSM vs. TOAMD

22

TOSM TOAMD

Correlation 1p1h, 2p2h(single particle)

Relative motion in FD, FS

CM excitation Lawson method Nothing with single n

Hole states harmonic oscillator basis

Can optimize in each basis

Short-range repulsion central-UCOM

Correlation function, FS

Results

• 3H, 4He, 8Be (preliminary)• AV8’ (central+LS+tensor) • |TOAMD= |AMD+ |AMD+ |AMD• Up to 3-body term of the products of H, FD, FS

3-body2-body

VT

FD

FD

FD

FD

VT

others...

FD

FD

VT

FS

FS

T

FS

FS

T

3H energy surface

24

∑𝑛=1

𝑁𝐺

𝐶𝑛𝑒−𝑎𝑛𝑟2

• Good convergence for Gaussian numbers NG.• Obtain the saturation property with good radius.

Gaussian expansion

compactwide

Radius=1.76 fm

Three-body term in energy of 3H

25

• No saturation point with one- & two-body terms.• Three-body term has a saturation behavior.

compactwide

3-body term

1& 2-body terms

Full

Radius=1.76 fm

Energy components

EAMD ETOAMD T VC VT VLS PD

3H 16.9 -5.0 37.9 -17.4 -24.1 -1.3 6.4

4He 41.4 -13.3 61.5 -34.9 -38.3 -1.6 7.3

8Be 110.3 -27.6 134.7 -91.0 -69.1 -2.2 9.4

• PD ... D-state probability = [%] • 3H, 4He ... are (0s)3 & (0s)4 with Z=0 (spherical)• 8Be ... is a-a structure.• Need -type correlation to gain the energy more. 26

in MeVpreliminary

in progress

Correlation functions FD, FS in 3H

27

2r2FD 3E

FS 1E

FS 3E

same trendin 3H, 4He, 8Be

• Ranges of and are not short. • Range b of ~ 0.6 bAMD spatially compact

high-k, similar to TOSM

preliminary

28

Summary

• Tensor-Optimized Shell Model (TOSM) using Vbare.– Strong tensor correlation from 0p0h-2p2h involving high-k

components.

– He, Li, Be isotopes : Energy spectra, Radius of n-rich nuclei.– For 8Be, two aspects : grand state region & highly excited states,

which indicates of more configurations, such as aa states.• Tensor-Optimized AMD (TOAMD).

– Two-kinds of correlation functions FD (tensor) & FS (short-range)

– Three-body term contributes to the energy saturation. – Ranges of FD , FS are not short.

– Spatially compact behavior of FD produces high-k component.