Ab Initio Description of Open-Shell Nuclei: Merging In...

Ab Initio Description of Open-Shell Nuclei: Merging In-Medium SRG and 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 - Oct. 2016 1

Motivation Why should we merge IM-SRG and NCSM?

- limited to light nuclei

- factorial growth of model space

- computationally demanding

- difficult to obtain model-space

convergence

+ easy access to heavy nuclei

+ soft computational scaling with A

+ computationally very efficient

+ decoupling in A-body space

IM- NCSM

convergence

IM- NCSM

convergence

- not exact method

- only for ground state

- spectroscopy not straight forward

- currently limited to even nuclei

IM- NCSM

convergence

- not exact method

+ exact method

+ easy access to excited states

+ spectroscopy for free

+ no limitation to even nuclei

IM- NCSM

convergence

- not exact method

+ exact method

+ easy access to excited states

+ spectroscopy for free

+ no limitation to even nuclei

Overview

No-Core Shell Model (NCSM)

In-Medium Similarity Renormalization Group (IM-SRG)

Novel Approach: IM-NCSM

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

… is one of the most powerful exact ab initio methods

for the p- and lower sd-shell

Barrett, Vary, Navratil, ...

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 by

reducing the model space to physically relevant states

… 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,..

Eskendr Gebrerufael - TU Darmstadt - March 2016 5

flow parameter

generator designed to decouple the reference state

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

flow parameter

Novel Approach: IM-NCSM How should we merge…

NCSM define

reference state

IM-SRG evolve

operators

NCSM extract

observables

diagonalize Hamiltonian in small model space

ground state defines reference state

NCSM define

reference state

IM-SRG evolve

operators

NCSM extract

observables

evolve Hamiltonian and other operators

pre-diagonalization in A-body space

NCSM define

reference state

IM-SRG evolve

operators

NCSM extract

observables

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

Nmax=0 Slater determinants

Novel Approach: IM-NCSM Hamiltonian Matrix in A-Body Basis: 12C

Slater determinants

Nmax=2 Nmax=4

Nmax=0 eigenstates Slater determinants

Nmax=2 Nmax=4

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

Nmax=2 Nmax=4

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

Nmax=2 Nmax=4

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

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

Nmax=2 Nmax=4

Novel Approach: IM-NCSM Hamilton Matrix in A-Body Basis: 12C

Nmax=0 states couple to

reference state

first basis state = reference state

Nmax=0 eigenstates

reference state

Nmax=0 eigenstates

reference state

E(s) and Nmax=0 eigenvalue

cannot be identical

Nmax=0 eigenstates

reference state

E(s) and Nmax=0 eigenvalue

cannot be identical

diagonalization of evolved Hamiltonian

necessary

Nmax=0 eigenstates

Overview

No-Core Shell Model (NCSM)

In-Medium Similarity Renormalization Group (IM-SRG)

Novel Approach: IM-NCSM

Results

o Evolution of Ground-State Energy

o Evolution of Excitation Energies

o Spectra

Summary and Outlook

Results Evolution of Ground-State Energy

drastically enhanced model-space convergence for IM-NCSM

NCSM with explicit 3N

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

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

E(s) more robust than in 12C case

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

Results Evolution of Excitation Energies

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

E* converges monotonically from above w.r.t. Nmax for evolved Hamiltonian

variational principle valid for E* since ground-state energy converged

first excited 0 + very sensitive to flow parameter Hoyle state? Eskendr Gebrerufael - TU Darmstadt - Oct. 2016 12

large dependence on s in Nmax=0

dependence on s reduces with increasing Nmax

E* converges monotonically from above for evolved Hamiltonian

analyze E* as function of Nmax

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

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

Results Spectra

NCSM IM-NCSM

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

Results Spectra

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

NCSM IM-NCSM

Results Spectra

higher-lying states show small flow-parameter dependence

NCSM IM-NCSM

Results Spectra

1+ not yet observed experimentally theoretical prediction

NCSM IM-NCSM

Results Spectra

1+ not yet observed experimentally theoretical prediction

NCSM IM-NCSM

Summary

introduced new many-body technique IM-NCSM = IM-SRG + NCSM

exploits the advantages of both approaches

Summary

IM-SRG decouples reference state from higher Nmax

extremely enhanced Nmax convergence for subsequent NCSM

Summary

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

Outlook

o variation of several parameters: generator,

o consistent evolution radius, electromagnetic, … operators

o detailed analysis of the Hoyle state in 12C

Outlook

o variation of several parameters: generator,

o consistent evolution radius, electromagnetic, … operators

o detailed analysis of the Hoyle state in 12C

o extend applicability of IM-NCSM to odd nuclei

using particle-attached particle-removed formalism

o include three-body operators in IM-SRG

Thanks to my group & collaborator • S. Alexa, S. Dentinger, T. Hüther, L. Kreher,

L. Mertes, S. Ruppert, R. Roth, S. Schulz,

H. Spiess, H. Spielvogel, C. Stumpf, A. Tichai,

R. Trippel, 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

Appendix

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

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

Results Dependence on Single-Particle Basis

For small flow parameter Nmax convergence slow in HF basis

induced many-body in HF 4 MeV and in HO < 1 MeV

NCSM with 3N

Results Dependence on Single-Particle Basis

For small flow parameter Nmax convergence slow in HF basis

induced many-body in HF 4 MeV and in HO < 1 MeV

NCSM with 3N

Nmax-Extrapolation at s=0

Results Dependence on

quite insensitive on

E(s) equal to eigenvalue obtained in for small flow parameter s

since the reference state is an eigenstate obtained in model space

Results Spectra

excellent agreement between IM-NCSM and NCSM

NCSM IM-NCSM

Results Spectra

4+ simply not calculated in NCSM

first excited 0+ shows same behaviour as in 12C

NCSM IM-NCSM

Results IM-NCSM: Particle-Attached Particle-Removed Form.

parent

eigenvalues for small flow parameter independent on parent nucleus

eigenvalues for large flow paremter show dependence on parent nucleus

deviation at the level of 4 MeV (< 4%)

parent

Results Evolution of Excitation Energies – On Absolute Scale

2+ perfectly converged on absolute scale

induced many-body contribution different for each state

Slater det.

Novel Approach: IM-NCSM Hamilton Matrix in A-Body Basis: 16O

E(s) converges monotonically against Nmax=2 eigenvalue

IM-SRG decouples reference state

from Nmax =2 space

Nmax=0

Slater det.

from Nmax =2 space

Nmax=0

Slater det.

from Nmax =2 space

Nmax=0

Slater det.

from Nmax =2 space

Nmax=0

Nmax=0 eigenstates

for sufficient large flow parameter eigenvalues in Nmax=0 and Nmax=2 equal

Nmax=0 eigenstates

IM-SRG decouples reference state (not only)

from Nmax =2 space

Nmax=0 eigenstates

IM-SRG decouples reference state (not only)

from Nmax =2 space

-> why do E(s) and Nmax=0 eigenvalue differ?

free-space SRG

normal ordering

IM-SRG NCSM

IM-NCSM Procedure

free-space SRG

normal ordering

IM-SRG NCSM

IM-NCSM Procedure

free-space SRG

normal ordering

IM-SRG NCSM

IM-NCSM Procedure

IM-SRG to get rid of spurious states:

free-space SRG

normal ordering

IM-SRG

IM-NCSM Procedure

IM-SRG to get rid of spurious states:

IM-NCSM Current Implementation of Imaginary Time

natural orbitals: ni occupation number

… missing terms contain irreducible density matrix

Ab Initio Description of Open-Shell Nuclei: Merging In...

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