ab initio method: in-medium no-core shell...

Ab Initio Method: In-Medium No-Core Shell Model

E. Gebrerufael1 K. Vobig1 H. Hergert2 R. Roth1

1 Institut für Kernphysik, TU Darmstadt 2 NSCL/FRIB Laboratory and Department of Physics & Astronomy, MSU

Eskendr Gebrerufael - TU Darmstadt - Apr. 2017 1

12C

accepted on PRL, arXiv:1610.05254

Overview


No-Core Shell Model (NCSM)

In-Medium Similarity Renormalization Group (IM-SRG)

In-Medium No-Core Shell Model

Results

o Evolution of Ground-State Energy

o Evolution of Excitation Energies

o Spectra

Summary and Outlook

No-Core Shell Model Basics

construct matrix representation of Hamiltonian using basis of HO/HF Slater determinants truncated w.r.t. excitation quanta Nmax

solve large-scale eigenvalue problem for a few smallest eigenvalues

range of applicability limited by factorial growth of basis with Nmax & A

adaptive importance-truncation extends the range of NCSM

… is one of the most powerful exact ab initio methods

for the p- and lower sd-shell

Barrett, Vary, Navratil, ...


In-Medium Similarity Renormalization Group Basics

flow equation for Hamiltonian:

… uses flow equation for normal-ordered Hamiltonian to decouple

the reference state from its excitations

Tsukiyama, Bogner, Schwenk, Hergert,..

flow parameter generator designed to decouple the reference state


In-Medium Similarity Renormalization Group Basics

flow equation for Hamiltonian:

… uses flow equation for normal-ordered Hamiltonian to decouple

the reference state from its excitations

H in multi-reference normal-ordered form w.r.t. [Kutzelnigg,Mukherjee]

Tsukiyama, Bogner, Schwenk, Hergert,..

flow parameter generator designed to decouple the reference state


In-Medium No-Core Shell Model Why should we merge…

+ easy access to heavy nuclei

+ soft computational scaling with A

+ decoupling in A-body space

IM- NCSM

- limited to light nuclei

- factorial growth of model space

- difficult to obtain model-space

convergence

SRG

- not exact method

- only for ground state

- spectroscopy not straight forward

+ exact method

+ easy access to excited states

+ spectroscopy for free


In-Medium No-Core Shell Model Why should we merge…

+ easy access to heavy nuclei

+ soft computational scaling with A

+ decoupling in A-body space

IM- NCSM

- limited to light nuclei

- factorial growth of model space

- difficult to obtain model-space

convergence

- not exact method

- only for ground state

- spectroscopy not straight forward

+ exact method

+ easy access to excited states

+ spectroscopy for free


diagonalize Hamiltonian in small model space

ground state defines reference state

In-Medium No-Core Shell Model How should we merge…

evolve Hamiltonian and other operators

pre-diagonalization in A-body space

NCSM define

reference state

IM-SRG evolve

operators

NCSM extract

observables

diagonalize IM-SRG evolved Hamiltonian

obtain eigenstates and extract observables


In-Medium No-Core Shell Model IM-NCSM is different from …

IM-NCSM is different from IM-SRG for valence-space interactions:

• build on explicit multi-reference formulation

• full no-core approach

• all model-space truncations are converged

NCSM define

reference state

IM-SRG evolve

operators

NCSM extract

observables


Nmax=0 Slater determinants

In-Medium No-Core Shell Model Hamiltonian Matrix in A-Body Basis: 12C

Slater determinants

Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0



Nmax=0 eigenstates Slater determinants

Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0



for sufficiently large flow parameter eigenvalues in Nmax=0, 2 and 4 equal


Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0





IM-SRG decouples reference state from Nmax =2 and 4 spaces

Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0





IM-SRG decouples reference state from Nmax =2 and 4 spaces

Why do E(s) and Nmax=0 eigenvalue differ? Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0


In-Medium No-Core Shell Model Hamilton Matrix in A-Body Basis: 12C

Nmax=0 states couple to

reference state

E(s) and Nmax=0 eigenvalue

not identical

diagonalization of evolved Hamiltonian

necessary

first basis state = reference state

Nmax=0 eigenstates


Results Evolution of Ground-State Energy




arXiv:1610.05254


drastically enhanced model-space convergence for IM-NCSM


arXiv:1610.05254



NCSM with explicit 3N


arXiv:1610.05254



NO2B approximation + induced many-body contribution = 4.0 MeV (≈ 5 %)



arXiv:1610.05254



NO2B approximation + induced many-body contribution = 4.0 MeV (≈ 5 %)

for s > 0.3 MeV-1 induced many-body contribution becomes significant



arXiv:1610.05254


E(s) more robust than in 12C case

NO2B approximation + induced many-body contribution = 2.3 MeV (< 2 %)



Results NCSM vs. IM-NCSM vs. MR-IM-SRG

NCSM @ Nmax extrap. IM-NCSM @ Nmax=4 MR-IM-SRG (HFB, White) Experiment


10% <1% 3% 5% 4%

10% largest deviation in 12C

best case 14C

very good agreement

largest deviation in 26O

<2%

arXiv:1610.05254

Results Evolution of Excitation Energies

E* of 2+ increases abruptly at the end due to kink in ground-state energy




Nmax convergence from above in decoupled regime variational principle




Nmax convergence from above in decoupled regime variational principle

first excited 0 + behaves differently and drops by ≈5 MeV Hoyle state?


Results Signatures of Hoyle State in 12C


trends are compatible with Hoyle-state interpretation

experiment

exp. value for Hoyle state 0.4 fm larger than the ground state [Phys. Rev. C, 80:054603]

need better control of induced many-body terms for quantitative statements

exp. value 5.5 e fm2 [Nuclear Physics A, 506(1)]


large dependence on s in Nmax=0

dependence on s reduces with increasing Nmax



large dependence on s in Nmax=0


E* converges monotonically from above for evolved Hamiltonian



analyze E* as function of Nmax large dependence on s in Nmax=0


E* converges monotonically from above for evolved Hamiltonian


Results Spectra

IM-NCSM


Results Spectra

IM-NCSM

uncertainty band due to flow-parameter variation between smax/2 and smax


Results Spectra

NCSM IM-NCSM



Results Spectra

NCSM IM-NCSM


2+ and 1+ in IM-NCSM and NCSM in good agreement


Results Spectra

NCSM IM-NCSM


2+ and 1+ in IM-NCSM and NCSM in good agreement

second 0+ in NCSM (Hoyle?) slow convergence, IM-NCSM closer to experiment


Results Spectra

first 2+ and 4+ robust and well converged in IM-NCSM

higher-lying states show small flow-parameter dependence

1+ not yet observed experimentally theoretical prediction

NCSM IM-NCSM


Summary

established a many-body technique IM-NCSM = IM-SRG + NCSM

IM-SRG decouples reference state from higher Nmax

extremely enhanced Nmax convergence for subsequent NCSM

Nmax≤4 sufficient to extract converged ground-state energies

variational principle becomes valid for excitation energies

since ground-state energy converged

preliminary ab initio studies regarding Hoyle state in 12C


Outlook

o more detailed analysis of the Hoyle state in 12C

o study of exotic nuclei: island-of-inversion physics, …

o evolve vector operator, for instance E1 transition operators

o extend applicability of IM-NCSM to odd nuclei

using particle-attached particle-removed formalism

o …


Thanks to my group & collaborator

• S. Alexa, T. Hüther, L. Mertes, R. Roth, S. Schulz,

H. Spielvogel, C. Stumpf, A. Tichai, K. Vobig, R. Wirth

Institut für Kernphysik, TU Darmstadt

• H. Hergert

NSCL/FRIB, Michigan State University

Thank You For Your Attention

JURECA LOEWE-CSC LICHTENBERG

COMPUTING TIME


ab initio method: in-medium no-core shell...

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