Monte-Carlo Shell Model for ab initio calculationbrown/EFES-2010/pdf/shimizu.pdf · Monte-Carlo Shell Model for ab initio calculation N. Shimizu, T. Abe, L. Liu,Y. Utsuno, T. Otsuka,
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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.
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