Monte-Carlo Shell Model for ab initio calculation

N. Shimizu, T. Abe, L. Liu, Y. Utsuno, T. Otsuka, P. Maris, and J. Vary

Univ. of Tokyo, JAEA, RIKEN, CNS, NSCL MSU, Iowa State Univ.

Outline

• Introduction of the Monte-Carlo Shell Model method

• NN-interaction: UCOM and JISP16• MCSM results:

4He binding energies, center of mass motion,rms radius, Coulomb int.other nuclei : 7Li, 10B

• Development of the new MCSM code

J-scheme dimension of Full Configuration Interaction

Current FCI limit

Monte-Carlo Shell ModelStochastic “importance truncation” scheme

Stochastically

selected basis

φσφ σβ )0()()( ⋅∏= ⋅∆e h

)(ˆ σh one-body Hamiltonian

σ... auxiliary field

random numbers.)(1

,∑=

Π=ΦMCSMN

ii

Ji Pc σφ

MCSM basis, deformed Slater det.

Solving quantum many-body problem with harmonic oscillator basis

Ref. K.W.Schmidt, PPNP 52, 565 (2004)

6p6h trun.

MCSM(1998)

MCSM(2001)exact

56Ni 8π8ν in pf-shell, 109dim

43keV

• Monte-Carlo Shell Model• Full Configuration Interaction• n-particle n-hole trunaction• No-Core Shell Model• Importance-Truncated Shell Model• Projected Configuration

Interaction• VAMPIR• Coupled Cluster approach• In-medium SRG• SMMC, GCM, RPA, …

Magnetic moment of Xe isotopes

Neutron Number

g fa

ctor

s

B(E2) of Sn,Te,Xe,Ba,Ce isotopes(Blue: exp. Red: MCSM results)

z-axisAxial symm. rotorTriaxial def.

Neutron Number

B(E2

; 0+ -

>2+ )

[e2 b

2 ]

MCSM in medium-heavy regionMCSM is quite sucessful in sd-, pf-shell nuclei, such as “island of inversion”, and ...

Features of MCSM

• low-lying eigenstates in huge Hilbert space are described as linear combinations of 40 projected Slater determinants

• obey variational principle ... upper limit• wave function can be obtained directly• No sign problem like SMMC• weak computation-time dependency on number of particles

shortcomings of the original MCSM code for no-core calc.• assume isospin symmetry• assume 1- and 2-body interactions, not 3-body force

Truncation Scheme for ab-initio shell-model calculation

No-Core Shell Model FCI, MCSM

x o

x x x o

ω0

ω1

ω2

ωmaxNExcitation energy up to

ω3

ω4

fermi surface

x

x

ω2

ω4

ωωωω max642 N≤=+

x o

o o x o

ω0

ω1

ω2

ω3

ω4

fermi surface

x

x x x

Single particle energies up to ωeMax

Exact decoupling with CoM motion

Many particles can excite in single Slater det.

Approximate decoupling with CoM motionLawson’s prescription

ω54shell =N

)3( =eMax

compare FCI and MCSM results FCI calc. performed by MSHELL (by T. Mizusaki)

Towards no-core MCSM calculation: NN-force

MCSM cannot treat 3-body force now

Unitary Correlation Operator Method (UCOM) using AV18 bare NN force

• soften correlations of short-

range repulsive core and tensor force

• parameter of the tensor correlator

is tuned to minimize contribution

of 3-body force

Ref. R.Roth et al. NPA788, 12 (2007)Thanks to R. Roth for UCOM int.

Towards no-core MCSM calculation: NN-force

JISP16 J-matrix Inverse Scattering Potential tuned B.E.s up to 16O with phase-shift-equivalent unitary transformation

“ab exitu approach” based on J-matrix inverse scattering approach

High quality description of NN potential thru. p-shell nuclei-> Reproduce the phase shift, deuteron properties, and

B.E.s of some light nuclei-> In this sense, JISP16 is the “bare” interaction

JISP16 NN int. is tuned to minimize the contribution of 3N (many-body) int.

JISP16: A. M. Shirokov, J.P. Vary, A. I. Mazur, T.A. Weber, Phys. Lett. B644, 33 (2007)NCFC calc of light nuclei w/ JISP16: P. Maris, J.P. Vary, A.M. Shirokov, Phys. Rev. C 79, 014308 (2009)

Hope: 3-body force

transformation U is determined so that the contributions of genuine 3-body force and induced 3-body force cancel each other

...)body3(eff

)body2(eff

)body2( ++= −−− HHUHU†

...)body3( +−H

Benchmark of no-core MCSMEg.s. of 4He, JISP16 and UCOM

exact

exact

Nshell=2

Nshell=3

UCOM + MCSM

JISP16 + MCSM

MCSM basis dimension

Number of projected Slater determinants

CoM with Lawson prescription: 4He UCOM

Nshell=2

JISP16 and UCOM : 4He case

UCOM and AV18

JISP16

Nshell=2

Nshell=3

Nshell=4

Nshell=6Nshell=5

FCI calc.

FCI and MCSM results

• Comparison of MCSM w/ FCI @ Nshell = 2 (sp), & 3 (spsd)

Removal of spurious CM motion effect for 4He G.S.E, JISP16

MCSM

FCI

hw = 30 MeVNshell = 2

MCSM

FCI

Lawson method w/ H = Hint + β Hcm

w/o Coulomb force

MCSM

FCI

hw = 30 MeVNshell = 3

w/o Coulomb force

~ 30 keV deviations

• Comparison of MCSM w/ FCI @ Nshell = 2 (sp), & 3 (spsd)

Removal of spurious CM motion effect for 4He G.S.E

MCSM

FCI

MCSM

FCI

Lawson method w/ H = Hint + β Hcm

MCSMFCI

MCSMFCINshell = 2

Nshell = 3

hw = 30 MeVw/o Coulomb force

cf. decoupling of center of mass motion

G. Hagen, T. Papenbrock, D. J. Dean, Phys. Rev. Lett. 103, 062503 (2009)

4He calc w/ Coulomb interaction

hw = 25 MeV

Nshell <Hint+Hq> (MeV) by MCSM

<Hint+Hq> (MeV) by FCI

<Hint> (MeV)by MCSM

<Hint> (MeV)by FCI

2 -24.869 -24.869 -25.759 -25.759

3 -26.813 -26.838 -27.687 -27.715

4 -27.681 -27.859 -28.552 -28.733

Occupation Number

4HeNshell = 2Lawson’s beta = 0

0s1/2

0p1/2

0p3/2

Occupation Number

4HeNshell = 3Lawson’s beta = 0

Occupation Number

4HeNshell = 4Lawson’s beta = 0

Closed Core of 4HeOccupation number of single-particle orbits

Overlap probability with (0s1/2)4

Point-proton rms radius of 4He

exp. 1.4fm

• Comparison of MCSM (solid symbols) w/ FCI (open symbols w/ curves) calcs of G.S.E.s for some p-shell nuclei

G.S.E.s of the other p-shell nuclei

6Li (1+)6He (0+)

8Be (0+) 12C (0+)

β = 0

H = Hint + β Hcm

2H (1+) 16O (0+)

EXPERIMENT:16O G.S.E. ~ -128 MeV w/ Coulomb force

Caution that the soft 2N int’s tend to give the overbinding for heavier nuclei !!

level-ordering in Li-isotopesS.C. Pieper, Enrico Fermi Course CLXIX (2008)

Incorrect level ordering for GFMC calc w/ 2N only

GFMC Calc w/ & w/o 3N int

1/2- & 3/2- 1/2- & 3/2-

LS-splitting btw 3/2- & 1/2- states in 7Li

7Li(1/2-)7Li(3/2-)

MCSM/FCI calc , hw = 20 MeV, Nshell = 3

-24.230 / -26.0625 MeV (3/2-) -22.441 / -23.4872 MeV (1/2- )

1.789 / 2.5753 MeV

MCSM/FCI calc, hw = 20 MeV, Nshell = 2

-16.767 / -17.2324 MeV (3/2-) -14.458 / -14.4587 MeV (1/2- )

2.309 / 2.7737 MeV

Correct level-ordering & relatively large LS-splitting already for NN int. only

exp. 0.472MeV

Level-ordering S.C. Pieper, Enrico Fermi Course CLXIX (2008)

Q < 0

Q > 0Q < 0

Q > 0Q > 0 ?

Q < 0 ?

Level-ordering of 1st & 2nd

2+ states in 10Be

Level-ordering of 1+ & 3+ states in 10B

GFMC Calc w/ & w/o 3N int

NCSM Calc w/ & w/o 3N int

Level-ordering of 3+ & 1+ states in 10B

10B(1+)10B(3+)

MCSM/FCI calc , hw = 25 MeV, Nshell = 3

- 42.241/ -46.6018 MeV (3+) - 39.075/ -42.3384 MeV (1+)

MCSM/FCI calc, hw = 25 MeV, Nshell = 2

- 34.170/ -34.2208 MeV (3+) - 29.749/ -29.7547 MeV (1+)

Correct level-ordering already for NN int. only exp. 0.718MeV

New code for MCSMOriginal MCSM code

since 1995 for conventionallarge-scale shell-model calc.

Useful even nowadays, but …

• Fortran 77 • PVM (less popular now)• Isospin-conserving interaction

only• Developed on Alpha chip

New MCSM code since 2010written from scratch

• not only for shell-model calc. with core, but also for no-core calc.

• Fortran 95 • MPI (+OpenMP hybrid parallel)• Isospin-breaking interaction• Developed on Intel chip • New algorithm to evaluate

Hamiltonian matrix elements in Slater det. Basisdifficult to run it on

state-of-the-art supercomputerswith minor modification 80% complete and

Benchmark test nowRef. T. Otsuka, T. Mizusaki, and M. Honma: Phys. Rev. Lett. 75 1284 (1995)

Benchmark for new MCSM codeScalability benchmark for parallel computationBenchmark test for codes

(56Ni in pf-shell)

Fortran 77 => 9530% improve

New algorithm50% improve

Totally, 100% improve

4He in Nshell=5

Estimation : Light nuclei with Nshell=5 is feasible on T2K supercomputerat Univ. of Tokyo.

perf

orm

ance

Computer developmentsAlpheet-1 since 1999

62GFlops @RIKENmainly for MCSM calc.

Alpheet-2 since 2002~200GFlops

T2K supercomputer @ Univ. of TokyoSince 2008 83TFlops

~5TFlops/job

Japanese next-generationsupercomputerSince 2012, 10PFlops(?)

New code

4shell =N

5shell =N

(?)7,6shell =N

J-scheme dimension of Full Configuration Interaction

Current FCI limit

Summary

• MCSM succeeds in reproducing the results of no-core FCI results of 4He and some p-shell nuclei with both JISP16 and UCOM interaction.

• Feasibility of the treatment of spurious center-of-mass motion, excited states, root-mean-square radius is also tested in the MCSM calc.

• Development of new MCSM code is now in progress. Its benchmark result provides us with a promising perspective of the study of p-shell nuclei up to Nshell=5 and more.