Symposium on`Quarks to Universe in Computational Science’(QUCS2012)

No-core Monte Carlo shell model
towards ab initio nuclear structure
A02: Nuclear Physics

Takashi Abe

Nara Prefectural New Public Hall
December 13-16, 2012Collaborators• U of Tokyo– Takaharu Otsuka (Dept of Phys & CNS) : A02– Noritaka Shimizu (CNS): A02– Tooru Yoshida (CNS)• JAEA– Yutaka Utsuno• Iowa State U– James P. Vary
– Pieter Maris2

Outline• Motivation• No-Core Monte Carlo Shell Model (MCSM)• Benchmark in p-shell nuclei• Density profile from MCSM wave functions• Summary & perspective3

Current status of ab inito approaches• Major challenge of the nuclear physics- Understand the nuclear structure from ab-initio calculations innon-relativistic quantum many-body system w/ realistic nuclear forces- ab-initio approaches: GFMC, NCSM (up to A ~ 12-14), CC (closed shell +/- 1,2),
SCGF theory, IM-SRG, Lattice EFT, …
• demand for extensive computational resources ab-initio(-like) SM approaches (which attempt to go) beyond standard methods- IT-NCSM, IT-CI: R. Roth (TU Darmstadt), P. Navratil (TRIUMF), …
- SA-NCSM: T. Dytrych, K.D. Sviratcheva, J.P. Draayer, C. Bahri, J.P. Vary, …
(Louisiana State U, Iowa State U)- No-Core Monte Carlo Shell Model (MCSM)
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“Ab initio” in nuclear physics• Solve non-relativistic Schroeding eq. and obtain the eigenvalues
and eigenvectors.

– Ab initio: All nucleons are active, and use realistic NN (+ 3N) interactions.

• Two sources of errors:– Nuclear forces (interactions btw/among nucleons),
in principle, they should be obtained by QCD.
– Finite # of basis space,
we have to extrapolate to infinite basis dimensions

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Core & no-core shell models• Conventional (core) shell model vs. No-core shell model (NCSM)
CIab initio
Conventional Shell model
No-core shell model
Shell gap
sd-shell
Shell gap
p-shells-shell

Effective interactions

Realistic nuclear interactions

Talk by Y. Tsunoda

This talkNuclear shell model• Eigenvalue problem of large sparse Hamiltonian matirx

Large sparse matrix (in m-scheme)7M-scheme dimension of p-shell nuclei
DM16O (0+) M=012C (0+) M=0
10B (3+) M=3

Current FCI limitupper p-shell

8Be (0+) M=0
7Li(3/2-) M = 3/2
6He (0+) M=0, 6Li(1+) M=1

4He (0+) M=0

lower p-shell.
.
.

Moore’s law:#transistors doubles every two years. (p = 2n/2)
x 5.7 after 5 yrs, x 32 after 10 yrs(= emax + 1).
.
.

Nshell=5
Nshell=4
Nshell=3
Nshell=2
Nshell=1 8

Review: T. Otsuka , M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001)Power of the MCSM
• MCSM w/ an assumed inert core is one of the powerful shell model algorithms.

M-scheme dim.

Monte Carlo Shell Model

Conventional Shell Model

Publication Year

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UNEDF SciDAC Collaboration: http://unedf.org/MCSM
No-Core MCSM

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Review: T. Otsuka , M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001)

Monte Carlo shell model (MCSM)• Importance truncation
Standard shell model

H=Diagonalization

All Slater determinantsd > O(1010)

Monte Carlo shell model

H~

Diagonalization

Important bases stochastically selected

dMCSM < O(100)11

SM Hamiltonian & MCSM many-body w.f.
• 2nd-quantized non-rel. Hamiltonian (up to 2-body term, so far)

• Eigenvalue problem• MCSM many-body wave function & basis function
These coeff. are obtained by the diagonalization.
• Deformed SDs
This coeff. is obtained by a stochastic sampling.
( cα† … spherical HO basis)

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Sampling of basis functions in the MCSM
• Deformed Slater determinant basis( cα† … HO basis)• Stochastic sampling of deformed SDsHFHFc.f.) Imaginary-time evolution & Hubbard-Stratonovich transf.

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Rough image of the search steps
Hamiltoniankernel• Basis search– HF solution is taken as the 1st basis

H(F,F’)=(n-1)*(n-1)matrix

fixed– Fix the n-1 basis states already takenn-th(to be optimized)– Requirement for the new basis: atopt the basis which makes the energy(of a many-body state) as low as possible by a stochastic sampling ( )   n e h ( ) n
HFh( n )  hHF     nO

HF14L. Liu, T. Otsuka, N. Shimizu, Y. Utsuno, R. Roth, Phys. Rev C86, 014302 (2012)

Feasibility study of MCSM for no-core calculations

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Recent developments in the MCSM
• Energy minimization by the CG method– N. Shimizu, Y. Utsuno, T. Mizusaki, M. Honma, Y. Tsunoda & T. Otsuka,
Phys. Rev. C85, 054301 (2012)• Efficient computation of TBMEs– Y. Utsuno, N. Shimizu, T. Otsuka & T. Abe,
Compt. Phys. Comm. 184, 102 (2013)• Energy variance extrapolation– N. Shimizu, Y. Utsuno, T. Mizusaki, T. Otsuka, T. Abe & M. Honma,
Phys. Rev. C82, 061305 (2010)• Summary of recent MCSM techniques– N. Shimizu, T. Abe, T. Tsunoda, Y. Utsuno, T. Yoshida, T. Mizusaki,
M. Honma, T. Otsuka, Prog. Theor. Exp. Phys. (2012)16

Energy minimization by Conjugate Gradient method
MCSM Step 1 (old)64Ge in pfg9-shell, 1014dim

E

Stochastic samplingstart
rejectacceptMCSM+CG Step 2 (new)E

Stochastic samplingstart

Step 1Step 2

rejectacceptConjugate gradient

Stochastic sampling before conjugate gradient
to minimize the expectation value energy

Reduction of the number of basis function roughly 30%

Y. Utsuno, N. Shimizu, T. Otsuka, and T. Abe, Comp. Phys. Comm., in press, arXiv:1202.2957 [nucl-th]

Efficient computation of the TBMEs• hot spot: Computation of the TBMEs(w/o projections, for simplicity)c.f.) Indirect-index method(list-vector method)• Utilization of the symmetry

sparsedense18

Y. Utsuno, N. Shimizu, T. Otsuka, and T. Abe, arXiv:1202.2957 [nucl-th] (Comp. Phys. Comm. In press)

Schematic illustration of the computation of TBMEs• Matrix-matrix method

Δm =-1-100+1x 0+1BLAS Level 3x019

Y. Utsuno, N. Shimizu, T. Otsuka, and T. Abe, Comp. Phys. Comm., in press, arXiv:1202.2957 [nucl-th]Tuning of the density matrix productNshell = 5
The performance reaches 80% ofthe theoretical peak at hotspot.
SPARC64 requires large Nbunch incomparison to Xeon
Matrix product e.g.
(390 x 390) x (390 x 2Nbunch)

Nbunch controllable tuning parameterchunk sizeStandard technique20

Extrapolations in the MCSM
• Two steps of the extrapolation
1. Extrapolation of our MCSM (approx.) results to the FCI(exact) results in fixed model space
Energy-variance extrapolation
N. Shimizu, Y. Utsuno, T.Mizusaki, T. Otsuka, T. Abe, & M. Honma, Phys. Rev. C82, 061305(R) (2010)

2. Extrapolation into the infinite model space MCSM
Exponential fit w.r.t. Nmax in the NCFC
1. Extrp into FCIFCINot applied in the MCSM, so far…

2. Extrp into inf.
basis dimNCFC21Energy-variance extrapolation• Originally proposed in condensed matter physics
Path Integral Renormalization Group method
M. Imada and T. Kashima, J. Phys. Soc. Jpn 69, 2723 (2000)• Imported to nuclear physics
Lanczos diagonalization with particle-hole truncation
T. Mizusaki and M. Imada Phys. Rev. C65 064319 (2002)
T. Mizusaki and M. Imada Phys. Rev. C68 041301 (2003)

single deformed Slater determinant
T. Mizusaki, Phys. Rev. C70 044316 (2004)

Apply to the MCSM (multi deformed SDs)
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N. Shimizu, Y. Utsuno, T. Mizusaki, T. Otsuka, T. Abe & M. Honma, Phys. Rev. C82, 061305 (2010)

Energy-variance Extrapolation of 12C 0+ g.s. Energy

Exact value is unknown
DM ~ 6 x 1011
E = -90.030 MeV (MCSM) [81 dim]
Variational upper boundDJ = 11,384,214,614 ~ 1.1 x 1010
Estimated error ~ 144 keV
12C (0+)E = -92.18(14) MeV (quadratic)
E = -92.58 MeV (linear)
Eeffective lower boundNshell = 4 (spsdpf)
hw = 30 MeV
w/o Coulomb force?
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Benchmarks in p-shell nuclei

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Helium-4 & carbon-12 gs energies
Nshell = 24He(0+;gs)

w/ optimum hw
w/o Coulomb force
w/o spurious CoM treatment

Nshell = 3

Nshell = 4
Nshell = 5

Nshell = 212C(0+;gs)

Nshell = 3

Exact result is unknown

Nshell = 425

T. Abe, P. Maris, T. Otsuka, N. Shimizu, Y. Utsuno, J. P. Vary, Phys Rev C86, 054301 (2012)

Energies of the Light Nuclei

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Density Profiles from MCSM wave functions

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Density Profile from ab initio calc.

VMC

8Be (0+; gs)• Green’s function Monte Carlo (GFMC)“Intrinsic” density is constructedby aligning the moment of inertia among
samples
Intrinsic density

R. B. Wiringa, S. C. Pieper, J. Carlson & V. R.
Pandharipande, Phys. Rev. C62, 014001 (2000)

8Li (2+; gs)• No-core full configuration (NCFC)
Translationally-invariant density is obtainedby deconvoluting the intrinsic & CM w.f.
C. Cockrell J. P. Vary & P. Maris,Phys. Rev. C86, 034325 (2012)

Translationallyinvariant
neutron density

Space-fixed
neutron density28

Density profile in MCSM
NB

Φ=｜NBC1 ( D)   ci P J ,  ( D ( n ) )+c2+ c3＋・・・+ C98・・・+ c99

n 1+ c100〉

 c  (D )n 1(n)iAngular-momentum projection

Laboratory frame

Alignment by Q-moment“Intrinsic” (body-fixed) frame

N. Shimizu, T. Abe, Y. Tsunoda, Y. Utsuno, T. Yoshida, T. Mizusaki, M. Honma, T. Otsuka,
Progress in Theoretical and Experimental Physics, in print (2012)

Density profile of 8Be 0+ g.s. state from MCSM w.f.
“Intrinsic” density

before alignment

after alignment

ρ/2100 bases

10 bases

1 basis8fm
Yaxis8fm

X = 0 fm

X = 1 fm

8fm
Yaxis8fm

X = 0 fm

X = 1 fm

Nshell = 4

Summary• MCSM can be applied to the no-core calculations of p-shell nuclei.
- Benchmarks for the p-shell nuclei have been performed and gave goodagreements w/ FCI results.
- Density profiles from MCSM many-body w.f. are investigated and thecluster-like distributions are reproduced.

Perspective• MCSM algorithm- Access to larger model spaces (Nshell = 5, 6, …)- Inclusion of the 3-body force by effective 2-body force.
• Physics- Cluster(-like) states (Be isotopes, 12C Hoyle state, …)
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END

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