Ab Initio Description of Open-Shell Nuclei: Merging In ...crunch.ikp.physik.tu-darmstadt.de/tnp/pres/2016_msu_gebrerufael.pdf · Ab Initio Description of Open-Shell Nuclei: Merging

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

12C

Motivation Why should we merge IM-SRG and NCSM?

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

SRG






IM- NCSM






convergence

SRG

- not exact method

- only for ground state

- spectroscopy not straight forward

- currently limited to even nuclei






IM- NCSM






convergence

SRG

- 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









flow parameter









flow parameter









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

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




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




Nmax=2 Nmax=4

Nm

ax=4

N

max

=2

Nm

ax=0




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

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






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 %)





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

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

HF

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

HO

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

HF






HF






HO






HO






HO

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.


7 8

6

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

7 8

6




7 8

6

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




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




Nmax=0 eigenstates




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

Nmax=0 eigenstates





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

NC

SM

normal ordering

IM-SRG NCSM

IM-NCSM Procedure


free-space SRG

NC

SM

normal ordering

IM-SRG NCSM

IM-NCSM Procedure


free-space SRG

NC

SM

normal ordering

IM-SRG NCSM

IM-NCSM Procedure

IM-SRG to get rid of spurious states:


free-space SRG

NC

SM

normal ordering

IM-SRG

CI

NCSM

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


Documents

Ab Initio Description of Open-Shell Nuclei: Merging In ...crunch.ikp.physik.tu-darmstadt.de/tnp/pres/2016_msu_gebrerufael.pdf · Ab Initio Description of Open-Shell Nuclei: Merging