New Horizons for the No-Core Shell Model · No-Core Shell Model & Friends §complementarity of...

New Horizons for the No-Core Shell Model

Robert Roth

Robert Roth - TU Darmstadt - February 2018

No-Core Shell Model & Friends

2

No-Core Shell Model

In-Medium Similarity Renormalization Group

Many-Body Perturbation Theory

• solution of matrix eigenvalue problem in truncated many-body model space

• universality: all nuclei and all bound-state observables on the same footing

• but: limited by model-space convergence

• power-series expansion of energies and states

• simplicity: low-order contributions can be evaluated very easily and efficiently

• but: order-by-order convergence problematic

• decoupling ground-state from excitations through unitary transformation via flow equation

• efficiency: favorable scaling gives access to medium-mass nuclei

• but: limited to ground-state observables


No-Core Shell Model & Friends

§ complementarity of advantages and limitations of the different methods

§ combine NCSM with other methods to overcome limitations

§ expand reach in terms of observables, particle number or model-space size

§ target: spectroscopy of fully open-shell medium-mass nuclei

3

No-Core Shell Model

In-Medium Similarity Renormalization Group

Many-Body Perturbation Theory

Robert Roth - TU Darmstadt - February 2018 4

Hybrid NCSM Methods

IM-SRG decoupling

MBPTbasis optimization

NCSMmany-body solution

NCSMreference state

MBPTconvergence booster



NAT-NCSM IM-NCSMNCSM-PTNCSM

ground state

NCSMstrength distribution

SF-NCSM

Natural-Orbital NCSM


J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.

6

Natural-Orbital NCSM



• construct HF basis in large single-particle space

• compute perturbative corrections to one-body density matrix up to second order

• determine natural orbitals from one-body density matrix and transform matrix elements

• NCSM calculation with natural-orbital basis

• use importance truncation for large spaces and heavier nuclei (optional)

• use normal-order two-body approximation to include 3N interactions (optional)

cf. work of Ch. Constantinou, M. A. Caprio, J. P. Vary, P. Maris on construction of natural-orbital basis from NCSM solutions


Natural Orbitals from MBPT

§ perform constrained spherical Hartree-Fock calculation to obtain unperturbed single-particle basis and ground state

§ compute MBPT corrections to HF ground state up to second order

7

§ write density-matrix corrections in terms of single-particle summations, evaluation only takes minutes…

§ solve eigenvalue problem of one-body density matrix, eigenvectors define expansion coefficients of natural-orbital single-particle states

§ transform all input matrix elements to natural-orbital basis


§ evaluate one-body density matrix with perturbed state up to second order

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ℏΩ [MeV]

E[MeV

]

14 16 18 20 22 24 26 28

-170

-165

-160

-155

-150

-145

-140

Natural Orbitals

NN

-onl

y, α

=0.

08 fm

4 , e

max

=12

ℏΩ [MeV]

E[MeV

]

14 16 18 20 22 24 26 28

-170

-165

-160

-155

-150

-145

-140



8

NCSM Convergence: Energies

§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM

Harmonic Oscillator

16ONmax=2

12

4

6

Hartree-Fock

ℏΩ [MeV]

E[MeV

]

14 16 18 20 22 24 26 28

-170

-165

-160

-155

-150

-145

-140

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90



9


Natural Orbitals

NN

+3N

(400

), α

=0.

08 fm

4 , e

max

=12


Harmonic Oscillator

Nmax=2

10

4

6

Hartree-Fock

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90

8

16O

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90



10


Natural Orbitals

NN

+3N

(400

), α

=0.

08 fm

4 , e

max

=12


Harmonic Oscillator

Nmax=2

10

4

6

Hartree-Fock

ℏΩ [MeV]

E[MeV

]

12 14 16 18 20 22 24 26 28

-130

-120

-110

-100

-90

8

16O ●◆ ... explicit 3N○◇△... NO2B

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6



11

NCSM Convergence: Radii

Natural Orbitals

NN

+3N

(400

), α

=0.

08 fm

4 , e

max

=12


Harmonic Oscillator

Nmax=2

10

4

6

Hartree-Fock

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

16O ●◆ ... explicit 3N○◇△... NO2B

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6



12

NCSM Convergence: Radii

Natural Orbitals

NN

+3N

(400

), α

=0.

08 fm

4 , e

max

=12


Harmonic Oscillator

Nmax=2

10

4

6

Hartree-Fock

ℏΩ [MeV]

Rp,rms

[fm]

12 14 16 18 20 22 24 26 28

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

16O ●◆ ... explicit 3N○◇△... NO2B

Nmax

E x(1+)

[MeV

]

2 4 6 8 100

2

4

6

8

10

Nmax

E[MeV

]

2 4 6 8 10

-100

-95

-90

-85

-80



13

NCSM Convergence: Spectroscopy

NN+3N(500), α=0.08 fm4, emax=12

Nmax

E x(2+)

[MeV

]

2 4 6 8 100

2

4

6

8

10

Nmax

Q(2+)

[efm2]

2 4 6 8 100

1

2

3

4

5

6

Nmax

B(E2,2+

→0+

)[e2fm4]

2 4 6 8 100

1

2

3

4

5

6

Nmax

B(M1,1+

→0+

)[μN2]

2 4 6 8 100

0.01

0.02

12C

ground-state energy

2+ excitation energy 1+ excitation

energy

2+ quadrupolemoment

E2 transitionstrength

M1 transitionstrength

ħΩ

⦁ 16 MeV◆ 20 MeV

24 MeV



14

Oxygen Isotopes

Nmax

E[MeV

]

2 4 6 8 10 12 14-170

-160

-150

-140

-130

-120

-110

-100

-90

-80

14O

15O

16O

...

24O

§ excellent convergence with natural-orbital basis for all oxygen isotopes

chiral NN+3N Λ3N=400 MeV α=0.08 fm4

ħΩ=20 MeV emax=12



15

Oxygen Isotopes

§ excellent convergence with natural-orbital basis for all oxygen isotopes

§ very good agreement with experimental systematics and dripline

§ NO2B instead of explicit 3N causes ~1% overbinding

A

E[MeV

]

14 16 18 20 22 24 26

-170

-160

-150

-140

-130

-120

-110

-100

-90

AO


ħΩ=20 MeV emax=12

⦁ NAT-NCSM, explicit 3N

⦁ NAT-NCSM, NO2B

Perturbatively Improved NCSM

• multi-configurational MBPT at second order for individual unperturbed states

• capture couplings in huge model-space through perturbative corrections


Perturbatively Improved NCSM

17

• eigenstates from NCSM at small Nmax as unperturbed states

• access to all open-shell nuclei and systematically improvable



Tichai, Gebrerufael, Vobig, Roth; arXiv:1703.05664



Multi-Configurational Perturbation Theory

§ prior NCSM calculation: reference or unperturbed state is superposition of Slater determinants from reference space

18

§ define partitioning and unperturbed Hamiltonian

§ evaluate second-order correction to the energy at many-body level

§ reformulation in terms of single-particle summations gives access to very large model spaces

|�refi =X

�2Mref

C� |��i

E(2) = �X

� /2Mref

|h�� |H |�refi|2

�� ref

H0 = �ref |�refih�ref| +X

� /2Mref

�� |��ih�� |



19

Oxygen Isotopes

A

E[MeV

]

14 16 18 20 22 24 26

-170

-160

-150

-140

-130

-120

-110

-100

-90

AO


ħΩ=20 MeV emax=12

HF basis Nmax=2 reference

Computational CostNAT-NCSM, expl. 3N 20000 CPU-hNAT-NCSM, NO2B 16000 CPU-hNCSM-PT 1000 CPU-h

§ excellent agreement with direct NCSM

§ all isotopes are accessible due simple m-scheme implementation

NCSM-PT


⦁ NAT-NCSM, NO2B


Tichai, et al.; in prep.

20

Oxygen Isotopes: Excited 2+ States

A

E*(2+)

[MeV

]

14 16 18 20 22 24 260

2

4

6

8

10

AO§ all methods can treat

excited states natively

§ example: first 2+ states in even oxygen isotopes

§ excellent agreement among methods except for closed (sub-)shells 22O, 24O…


ħΩ=20 MeV emax=12


NCSM-PT


⦁ NAT-NCSM, NO2B



21

Exploring sd-Shell PhenomenaE

[MeV

]

-170

-160

-150

-140

-130

-120

A

E[MeV

]

16 18 20 22 24 26 28 30 32-200-190-180-170-160-150-140-130-120

w/o chiral 3Nwith chiral 3N

§ exploring sd-shell phenomena, e.g., oxygen anomaly

§ low computational cost enables surveys with different interactions

AO

AF

NCSM-PT chiral NN+3N Λ3N=400 MeV α=0.08 fm4

ħΩ=20 MeV emax=12 HF basis Nmax=2 reference

In-Medium NCSM


In-Medium NCSM

23

• use in-medium evolved Hamiltonian for a subsequent NCSM calculation

• access to ground and excited states and full suite of observables

• IM-SRG evolution of multi-reference normal-ordered Hamiltonian (and other operators)

• decoupling of particle-hole excitations, i.e., pre-diagonalization in many-body space

• ground-state from NCSM at small Nmax as reference state for multi-reference IM-SRG

• access to all open-shell nuclei and systematically improvable

NCSMreference state

IM-SRG decoupling


talk by Klaus Vobig

on Thursday


Gebrerufael, Vobig, Hergert, Roth; PRL 118, 152503 (2017)

24

Oxygen Isotopes

A

E[MeV

]

14 16 18 20 22 24 26

-170

-160

-150

-140

-130

-120

-110

-100

-90

AO


ħΩ=20 MeV emax=12


§ excellent agreement with direct NCSM

§ IM-SRG evolution limited to J=0 reference states and thus even-mass isotopes

◆ IM-NCSM


⦁ NAT-NCSM, NO2B


Vobig, Gebrerufael, Roth; in prep.

25

Oxygen Isotopes

A

E[MeV

]

14 16 18 20 22 24 26

-170

-160

-150

-140

-130

-120

-110

-100

-90

AO§ excellent agreement with

direct NCSM

§ IM-SRG evolution limited to J=0 reference states and thus even-mass isotopes

§ odd-mass nuclei via simple particle attachment or removal in final NCSM run


ħΩ=20 MeV emax=12


IM-NCSM attached▼ IM-NCSM removed

◆ IM-NCSM


⦁ NAT-NCSM, NO2B

Strength-Function NCSM

• prepare pivot vector by applying transition operator to ground-state vector

• use simplistic Lanczos iterations to generate strength distribution



27

• regular NCSM calculation for ground state for a range of Nmax truncations

• access to all open-shell nuclei

NCSMground state


Stumpf, Wolfgruber, Roth; arXiv:1709.06840




§ perform NCSM calculation for ground state

§ prepare pivot vector with transition operator

§ perform Lanczos algorithm with Hamiltonian: obtain eigenvectors as superposition of Lanczos vectors

28

10�3

10�2

10�1

100

101100 LV

10�3

10�2

10�1

100

101200 LV

10�3

10�2

10�1

100

101400 LV

20 40 60 80 100E� [MeV]

10�3

10�2

10�1

100

101800 LV

R(E2)[e2fm

4MeV�1]

16O

, N

N-o

nly,

α=

0.08

fm

4 , H

F ba

sis,

ħΩ

=20

MeV

, e

max

=12

, N

max

=4

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§ first coefficient provides transition matrix element

§ construct discrete strength distribution

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§ works with simplest Lanczos algorithm (no reorthogonalization, Lanczos vectors discarded)

§ same computational reach as regular NCSM

§ no ad-hoc truncations, convergence in Nmax and Lanczos iterations can be demonstrated explicitly

§ full convergence of individual transitions in the relevant energy regime after ~800 iterations

§ full access to fine structure of giant resonances

§ full access to below-threshold features

29

10�3

10�2

10�1

100

101100 LV

10�3

10�2

10�1

100

101200 LV

10�3

10�2

10�1

100

101400 LV

20 40 60 80 100E� [MeV]

10�3

10�2

10�1

100

101800 LV

R(E2)[e2fm

4MeV�1]

16O

, N

N-o

nly,

α=

0.08

fm

4 , H

F ba

sis,

ħΩ

=20

MeV

, e

max

=12

, N

max

=4ab initio approach to

strength distributions with many advantages



30

Discrete Strength Distribution

10�2

10�1

10016O

10�2

10�1

100

10�2

10�1

100

101

10�2

10�1

10018O

10�2

10�1

100

10�2

10�1

100

101

10�2

10�1

10020O

10�2

10�1

100

10�2

10�1

100

101

10�2

10�1

10022O

10�2

10�1

100

10�2

10�1

100

101

0 10 20 30 40 50 60E� [MeV]

10�2

10�1

10024O

0 10 20 30 40 50 60E� [MeV]

10�2

10�1

100

0 10 20 30 40 50 60E� [MeV]

10�2

10�1

100

101

R(E0)[e2fm

4MeV�1]

R(E1)[e2fm

2MeV�1]

R(E2)[e2fm

4MeV�1]

isoscalar E0 isovector E1 isoscalar E2

chiral

NN

+3N

(400

), α=

0.08

fm

4 ,

HF

basi

s, ħΩ

=20

MeV

, e

max

=12

, N

max

=8/

9, 1

000L

V• discrete strength distribution, even above particle-emission threshold, since we work with bound-state basis

• escape width: cannot be described, expected to be a few 100 keV in resonance region

• spreading width: fully included through configuration mixing, dominant in resonance region

• description of escape width needs continuum basis (Gamow)

• however, typical experimental resolution for unstable isotopes ~1 MeV, i.e., escape width is not resolved

• traditional approach: fold discrete strength distribution with Lorentzian of ~1 MeV width

0.20.40.60.81.0 16O

0.2

0.4

0.6

2

468

10

0.2

0.4

0.6

0.8 18O

0.1

0.2

0.3

2

4

6

8

0.5

1.0

1.5 20O

0.1

0.2

0.3

2468

10

0.5

1.0

1.5

2.0 22O

0.2

0.4

0.6

5

10

15

0 10 20 30 40 50 60E� [MeV]

0.5

1.0

1.5

2.0 24O

0 10 20 30 40 50 60E� [MeV]

0.2

0.4

0.6

0 10 20 30 40 50 60E� [MeV]

5

10

15

R(E0)[e2fm

4MeV�1]

R(E1)[e2fm

2MeV�1]

R(E2)[e2fm

4MeV�1]



31

Strength Distribution

chiral

NN

+3N

(400

) &

N2L

O-S

AT,

α=

0.08

fm

4 ,

HF

basi

s, ħΩ

=20

MeV

, e

max

=12

, N

max

=8/

9, 1

MeV

sm

earing




Comparison with RPA and SRPA

§ collective excitations traditionally described in RPA or SRPA

§ RPA (1p1h) cannot describe fragmentation, therefore, go to SRPA (2p2h)

§ NCSM shows much more fine structure than SRPA and resolves notorious problem with pathological SRPA energy-shifts

32

10�2

100

16O

10�2

100

10�2

100

10�2

100

10�2

100

10�2

100

10 20 30 40 50 60E� [MeV]

10�2

100

10 20 30 40 50 60E� [MeV]

10�2

100

10 20 30 40 50 60E� [MeV]

10�2

100

R(E0)[e2fm

4MeV�1]

R(E1)[e2fm

2MeV�1]

R(E2)[e2fm

4MeV�1]


NC

SM

RP

AS

RP

A

chiral

NN

+3N

(400

), α=

0.08

fm

4 ,

HF

basi

s, ħΩ

=20

MeV

, e

max

=12

, N

max

=8/

9


Conclusions

§ hybrids built on the NCSM enable comprehensive access to ground and excited states of arbitrary open-shell nuclei

§mass reach: A≲30 if large Nmax is needed: NAT-NCSM, SF-NCSMA≲70 if small Nmax is sufficient: IM-NCSM, NCSM-PT

§more hybrids: NCSM with Continuum, HORSE,...

33

IM-SRG decoupling



NCSMreference state




NAT-NCSM IM-NCSMNCSM-PTNCSM

ground state


SF-NCSM


� S. Alexa, D. Derr, M. Dupper, A. Geißel, S. Graf, T. Hüther, J. Kaltwasser, M. Knöll, L. Mertes, J. Müller, S. Schulz, C. Stumpf, T. Tilger, K. Vobig, R. WirthTechnische Universität Darmstadt

� A. Tichai CEA Saclay

� P. Navrátil TRIUMF, Vancouver

� H. HergertNSCL / Michigan State University

� J. Vary, P. MarisIowa State University

� E. Epelbaum, H. Krebs & the LENPIC CollaborationUniversität Bochum, …

Epilogue

§ thanks to my group and my collaborators

34

Epilogue

■ thanks to my group & my collaborators

● S. Binder, J. Braun, A. Calci, S. Fischer,E. Gebrerufael, H. Spiess, J. Langhammer, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, R. Wirth, K. VobigInstitut für Kernphysik, TU Darmstadt

● P. NavrátilTRIUMF Vancouver, Canada

● J. Vary, P. MarisIowa State University, USA

● S. Quaglioni, G. HupinLLNL Livermore, USA

● P. PiecuchMichigan State University, USA

● H. HergertOhio State University, USA

● P. PapakonstantinouIBS/RISP, Korea

● C. ForssénChalmers University, Sweden

● H. Feldmeier, T. NeffGSI Helmholtzzentrum

JUROPA LOEWE-CSC EDISON

COMPUTING TIME

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New Horizons for the No-Core Shell Model · No-Core Shell Model & Friends §complementarity of...

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