new horizons in ab initio nuclear structure theory

New Horizons in

Ab Initio Nuclear Structure Theory

Robert Roth

1

Ab Initio Nuclear Structure

Robert Roth – TU Darmstadt – 01/2013

Nuclear Structure Observables

NuclearLatticeSim

.

chiralEFTonlattice

ExactAb-Initio

Solutions

few-bodyetal. Exact Ab-Initio

Solutions

few-body, no-core

shell model, etc.

Approx. Many-Body Methods

controlled & im-

provable schemes

Energ

y-D

ensity-

FunctionalTheory

guidedbychiralEFT

Similarity Transformationsphysics-conserving transform. of observables

Chiral Interactionsconsistent & improvable NN, 3N,... interactions

Chiral Effective Field Theorysystematic low-energy effective theory of QCD

Low-Energy Quantum Chromodynamics

2

Ab Initio Nuclear Structure


Nuclear Structure Observables

NuclearLatticeSim

.

chiralEFTonlattice

ExactAb-Initio

Solutions

few-bodyetal.

Exact Ab-InitioSolutions

few-body, no-core

shell model, etc.

Approx. Many-Body Methods

controlled & im-

provable schemes

Energ

y-D

ensity-

FunctionalTheory

guidedbychiralEFT

Similarity Transformationsphysics-conserving transform. of observables

Chiral Interactionsconsistent & improvable NN, 3N,... interactions

Chiral Effective Field Theorysystematic low-energy effective theory of QCD

Low-Energy Quantum Chromodynamics

PREDICTIO

NVALID

ATIO

Nthere...

...andbackagain

3

Nuclear Interactions

from Chiral EFT

4

Nuclear Interactions from Chiral EFT


Weinberg, van Kolck, Machleidt, Entem, Meissner, Epelbaum, Krebs, Bernard,...

low-energy effective field theoryfor relevant degrees of freedom (π,N)based on symmetries of QCD

long-range pion dynamics explicitly

short-range physics absorbed in con-tact terms, low-energy constants fit-ted to experiment (NN, πN,...)

hierarchy of consistent NN, 3N,...interactions (plus currents)

NN 3N 4N

LO

NLO

N2LO

N3LO

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many ongoing developments

• 3N interaction at N3LO, N4LO,...

• explicit inclusion of Δ-resonance

• YN- & YY-interactions

• formal issues: power counting,renormalization, cutoff choice,...

5

Chiral NN+3N Hamiltonians


standard Hamiltonian:

• NN at N3LO: Entem / Machleidt, 500 MeV cutoff

• 3N at N2LO: Navrátil, local, 500 MeV cutoff, fit to T1/2(3H) and E(3H, 3He)

standard Hamiltonian with modified 3N:

• NN at N3LO: Entem / Machleidt, 500 MeV cutoff

• 3N at N2LO: Navrátil, local, with modified LECs and cutoffs, refit to E(4He)

consistent N2LO Hamiltonian:

• NN at N2LO: Epelbaum et al., 450,...,600 MeV cutoff

• 3N at N2LO: Epelbaum et al., nonlocal, 450,...,600 MeV cutoff

consistent N3LO Hamiltonian:

• coming soon...

6

Similarity

Renormalization Group

Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)

Roth, Neff, Feldmeier — Prog. Part. Nucl. Phys. 65, 50 (2010)

Roth, Reinhardt, Hergert — Phys. Rev. C 77, 064033 (2008)

Hergert, Roth — Phys. Rev. C 75, 051001(R) (2007)

7

Similarity Renormalization Group


Wegner, Glazek, Wilson, Perry, Bogner, Furnstahl, Hergert, Roth, Jurgenson, Navratil,...

continuous transformation drivingHamiltonian to band-diagonal form

with respect to a chosen basis

unitary transformation of Hamiltonian (and other observables)

eHα = Uα†HUα

evolution equations for eHα and Uα depending on generator ηαd

dαeHα =ηα, eHα

d

dαUα = −Uαηα

dynamic generator: commutator with the operator in whoseeigenbasis H shall be diagonalized

ηα = (2μ)2Tint, eHα

simplicity and flexibilityare great advantages of

the SRG approach

solve SRG evolutionequations using two- &

three-body matrixrepresentation

8

SRG Evolution in Three-Body Space


perform SRG evolution for three-body Jacobi-HO matrix elements

0 . . . 18 20 22 24 26 28

(E, )

28

26

24

22

20

18

..

.

0

.

(E′,′)

α = 0 fm4

0 . . . 18 20 22 24 26 28

(E, )

α = 0.04 fm4

0 . . . 18 20 22 24 26 28

(E, )

α = 0.16 fm4

Jπ=

1 2

+,T=

1 2,ℏΩ=28MeV

0 2 4 6 8 10 12 14 16 18 20Nmx

-8

-6

-4

-2

.

E[MeV]

exp.

0 2 4 6 8 10 12 14 16 18 20Nmx

0 2 4 6 8 10 12 14 16 18 20Nmx

NCSM calculation3H ground state

suppress off-diagonal couplingimprove convergence behavior

9

Hamiltonian in A-Body Space


evolution induces n-body contributions eH[n]α

to Hamiltonian

eHα = eH[1]α+ eH[2]

α+ eH[3]

α+ eH[4]

α+ . . .

truncation of cluster series inevitable — formally destroys unitarityand invariance of energy eigenvalues (independence of α)

SRG-Evolved Hamiltonians

NN only: start with NN initial Hamiltonian and keep two-bodyterms only

NN+3N-induced: start with NN initial Hamiltonian and keep two-and induced three-body terms

NN+3N-full: start with NN+3N initial Hamiltonian and keep two-and all three-body terms

α-variation provides adiagnostic tool to assessthe contributions of omittedmany-body interactions

10

Importance Truncated

No-Core Shell Model

Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)

Navrátil, Roth, Quaglioni — Phys. Rev. C 82, 034609 (2010)

Roth — Phys. Rev. C 79, 064324 (2009)

Roth, Gour & Piecuch — Phys. Lett. B 679, 334 (2009)

Roth, Gour & Piecuch — Phys. Rev. C 79, 054325 (2009)

Roth, Navrátil — Phys. Rev. Lett. 99, 092501 (2007)

11

No-Core Shell Model


Barrett, Vary, Navratil, Maris, Nogga, Roth,...

NCSM is one of the most powerful anduniversal exact ab-initio methods

construct matrix representation of Hamiltonian using a basis of HOSlater determinants truncated w.r.t. HO excitation energy NmxℏΩ

solve large-scale eigenvalue problem for a few extremal eigenvalues

all relevant observables can be computed from the eigenstates

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

adaptive importance truncation extends the range of NCSM by reduc-ing the model space to physically relevant states

we have developed a parallelized IT-NCSM/NCSM code capable ofhandling 3N matrix elements up to E3mx = 16

12

Importance Truncated NCSM


Roth, PRC 79, 064324 (2009); PRL 99, 092501 (2007)

converged NCSM calcula-tions essentially restrictedto lower/mid p-shell

full 10ℏΩ calculation for16O getting very difficult(basis dimension > 1010)

0 2 4 6 8 10 12 14 16 18 20Nmx

-150

-140

-130

-120

-110

.

E[MeV]

16ONN-only

α = 0.04 fm4

ℏΩ = 20MeV

ImportanceTruncation

reduce model spaceto the relevant basis

states using an a prioriimportance measurederived from MBPT

0 2 4 6 8 10 12 14 16 18 20Nmx

-150

-140

-130

-120

-110

.

E[MeV]

IT-NCSM

+ full NCSM

13

4He: Ground-State Energies


Roth, et al; PRL 107, 072501 (2011)

NN only

2 4 6 8 10 12 14 16 ∞Nmx

-29

-28

-27

-26

-25

-24

-23

.

E[MeV]

NN+3N-induced

2 4 6 8 10 12 14 ∞Nmx

Exp.

NN+3N-full

2 4 6 8 10 12 14 ∞Nmx

Î

α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4

Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1

ℏΩ = 20MeV

14

12C: Ground-State Energies


Roth, et al; PRL 107, 072501 (2011)

NN only

2 4 6 8 10 12 14 ∞Nmx

-110

-100

-90

-80

-70

-60

.

E[MeV]

NN+3N-induced

2 4 6 8 10 12 ∞Nmx

Exp.

NN+3N-full

2 4 6 8 10 12 ∞Nmx

Î



ℏΩ = 20MeV

15

16O: Ground-State Energies


Roth, et al; PRL 107, 072501 (2011)

NN only

2 4 6 8 10 12 14 ∞Nmx

-180

-160

-140

-120

-100

-80

.

E[MeV]

NN+3N-induced

2 4 6 8 10 12 ∞Nmx

Exp.

NN+3N-full

2 4 6 8 10 12 ∞Nmx

Î



ℏΩ = 20MeV

in progress:

explicit inclusion of

induced 4Nclear signature of

induced 4N originatingfrom initial 3N

16

16O: Lowering the Initial 3N Cutoff


standard

500 MeV

2 4 6 8 1012141618Nmx

-150

-145

-140

-135

-130

-125

-120

-115

-110

.

E[MeV]

cD= −0.2cE = −0.205

reduced 3N cutoff(cE refit to 4He binding energy)

450 MeV

2 4 6 8 1012141618Nmx

cD= −0.2cE = −0.016

400 MeV

2 4 6 8 1012141618Nmx

cD= −0.2cE = 0.098

350 MeV

2 4 6 8 1012141618Nmx

cD= −0.2cE = 0.205

Î

α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4

Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1NN+3N-full

ℏΩ = 20MeV

lowering theinitial 3N cutoff

suppresses induced4N terms

17

Spectroscopy of 12C


Roth, et al; PRL 107, 072501 (2011)

NN only

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

0

2

4

6

8

10

12

14

16

18

.

E[MeV]

NN+3N-induced

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

NN+3N-full

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

12CℏΩ = 16MeV

α = 0.04 fm4 α = 0.08 fm4

Λ = 2.24 fm−1 Λ = 1.88 fm−1

18

Spectroscopy of 12C


Roth, et al; PRL 107, 072501 (2011)

NN only

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

0

2

4

6

8

10

12

14

16

18

.

E[MeV]

NN+3N-induced

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

NN+3N-full

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 0

2+ 1

4+ 0

2 4 6 8 Exp.Nmx

12CℏΩ = 16MeV

α = 0.04 fm4 α = 0.08 fm4

Λ = 2.24 fm−1 Λ = 1.88 fm−1

spectra largelyinsensitive toinduced 4N

19

The Bottom Line...


beyond the lightest nuclei, SRG-induced 4N contributions af-fect the absolute energies (but not the excitation energies)

with the inclusion of the leading 3N interaction we already obtain agood description of spectra (and ground states)

breakthrough in computation, transformation and managementof 3N matrix-elements

applications: spectroscopy of p- and sd-shell nuclei and groundstates with reduced initial 3N cutoff

next-generation SRG: include induced 4N contributions or sup-press many-body terms with modified SRG-generators

next-generation chiral 3N: use consistent chiral Hamiltoniansand propagate uncertainties to many-body observables

20

Ab Initio Calculations

for p- and sd-Shell Nuclei

21

Spectroscopy of Carbon Isotopes


PRELI

MINARY

0+ 1

2+ 1

2+ 1

2 4 6 8 Exp.0

2

4

6

8

.

E[MeV]

10C

0+ 0

0+ 0

0+ 0

1+ 0

1+ 1

2+ 0

2+ 02+ 1

4+ 0

2 4 6 8 Exp.0

2

4

6

8

10

12

14

16

12C

0+ 1

0+ 1

0+ 1

1+ 1

2+ 1

2+ 1

2+ 14+ 1

2 4 6 8 Exp.0

2

4

6

8

10

14C

0+ 2

0+ 2

2+ 2

2+ 23? 24+ 2

2 4 6 Exp.Nmx

0

1

2

3

4

.

E[MeV]

16C

0+ 3

2+ 3

2+ 3

2 4 6 Exp.Nmx

0

1

2

3

4

18C

0+ 4

2+ 4

2 4 6 Exp.Nmx

0

1

2

3

4

20C

NN+3N-full

Λ3N = 400MeV

α = 0.08 fm4

ℏΩ = 16MeV

22

Spectroscopy of Carbon Isotopes


PRELI

MINARY

2 4 6 80

1

2

3

4

5

6

7

8

.

B(E2,2+→

0+)[e2fm

4] 10C

2 4 6 80

1

2

3

4

5

6

7

8

12C

2 4 6 80

1

2

3

4

5

6

7

8

14C

2 4 6Nmx

0

0.5

1

1.5

2

2.5

3

.

B(E2,2+→

0+)[e2fm

4] 16C

2 4 6Nmx

0

0.5

1

1.5

2

2.5

3

18C

2 4 6Nmx

0

1

2

3

4

5

20C

NN+3N-full

Λ3N = 400MeV

α = 0.08 fm4

ℏΩ = 16MeV

B(E2,2+1→ 0+

gs)

B(E2,2+2 → 0+gs)

NEW

EXPE

RIME

NT

ANL,Mc

Cutch

anetal.

NEW

EXPE

RIME

NT

NSCL, Petri e

t al.

NEW

EXPE

RIME

NT

NSCL, Voss e

t al.

NEW

EXPE

RIME

NT

NSCL, Petri e

t al.

23

Ground States of Oxygen Isotopes


Hergert, Binder, Calci, Langhammer, Roth; in prep.

PRELI

MINARY

NN+3N-induced(chiral NN)

2 4 6 8 10 12 14 16 18Nmx

-160

-140

-120

-100

-80

-60

-40

.

E[MeV]

12O

14O

16O18O20O22O24O26O

NN+3N-full(chiral NN+3N)

2 4 6 8 10 12 14 16 18Nmx

12O

14O

16O18O20O22O26O24O

Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ

24




PRELI

MINARY


12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[MeV]


12 14 16 18 20 22 24 26

A

AO

experiment

IT-NCSM


parameter-freeab initio calculations with

full 3N interactionshighlights predictive power

of chiral NN+3N Hamiltonians

25




PRELI

MINARY


12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[MeV]


12 14 16 18 20 22 24 26

A

AO

experiment

IT-NCSM IM-SRG (prel.)

(H. Hergert)


26




PRELI

MINARY


12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[MeV]


12 14 16 18 20 22 24 26

A

AO

experiment

IT-NCSM IM-SRG (prel.)

(H. Hergert)

È CCSD

Î Λ-CCSD(T)


different many-bodyapproaches using sameNN+3N Hamiltonian give

consistent results minor differences are

understood (NO2B, E3mx,...)

27

Ab Initio Calculations

for Heavy Nuclei

Roth, Binder, Vobig et al. — Phys. Rev. Lett. 109, 052501 (2012)

Binder, Langhammer, Calci et al. — arXiv:1211.4748

Hergert, Bogner, Binder et al. — arXiv:1212.1190

28

Coupled-Cluster Method


Coester, Kuemmel, Bischop, Dean, Piecuch, Walet, Papenbrock, Hagen, Binder,...

CC is one of the most efficientmethods for the description of ground statesof medium-mass or heavy closed-shell nuclei

many-body state parametrized as exponential wave operator appliedto single-determinant reference state

ref

ΨCC= Ωref= expT1 + T2 + T3 + · · ·+ TA

ref

truncation with respect to n-particle-n-hole excitation operators Tn

solve non-linear system of equations for the amplitudes in T1, T2, T3,...

extensions to near-closed-shell nuclei and excited states throughequations-of-motion methods

we have developed a parallelized CC code for CCSD and Λ-CCSD(T)

29

Inclusion of 3N Interactions


premium option: explicit 3N

• extend coupled-cluster equations for explicit 3N interactions

• CCSD-3B, Λ-CCSD(T)-3B are feasible, but much more expensive

low-cost option: normal-ordered two-body approximation

• write 3N interaction in normal-ordered form with respect to the actualA-body reference determinant (HF state)

V3N =∑

V3N

†††

=W0B +∑

W1B

†+∑

W2B

††

+∑

W3B

†††

• discard normal-ordered three-body term and use two-body coupled-cluster formalism

30

CCSD with Explicit 3N Interactions


Binder et al.; arXiv:1211.4748

NN+3N-induced

-130

-120

-110

-100

-90

.

E[MeV]

exp.

4 6 8 10 12emx

-170

-160

-150

-140

-130

-120

-110

.

E[MeV]

exp.

NN+3N-full

16OℏΩ = 20 MeV

4 6 8 10 12emx

24OℏΩ = 20 MeV

CCSD-3B( Î)

CCSD-NO2B( ◊ Í)

α = 0.02 fm4

◊ α = 0.04 fm4

Î Í α = 0.08 fm4

HF basis

E3mx = 12

Λ3N = 400 MeV

31




NN+3N-induced

-380

-360

-340

-320

-300

-280

-260

.

E[MeV]

exp.

4 6 8 10 12emx

-500

-450

-400

-350

-300

-250

.

E[MeV]

exp.

NN+3N-full

40CaℏΩ = 24 MeV

4 6 8 10 12emx

48CaℏΩ = 28 MeV

CCSD-3B( Î)

CCSD-NO2B( ◊ Í)

α = 0.02 fm4

◊ α = 0.04 fm4

Î Í α = 0.08 fm4

HF basis

E3mx = 12

Λ3N = 400 MeVNO2B approximationreproduces explicit-3Nresults with better than

1% accuracy

32




NN+3N-induced

4 6 8 10 12emx

-550

-500

-450

-400

-350

.

E[MeV]

exp.

NN+3N-full

4 6 8 10 12emx

56NiℏΩ = 28 MeV

CCSD-3B( Î)

CCSD-NO2B( ◊ Í)

α = 0.02 fm4

◊ α = 0.04 fm4

Î Í α = 0.08 fm4

HF basis

E3mx = 12

Λ3N = 400 MeV

E3mx truncation of 3N matrix elements has sig-nificant effects for A ¦ 60

many-body framework is ready to go to heaviernuclei... still cheap with NO2B approximation

33

Λ-CCSD(T) with NO2B Approximation



NN+3N-full

-130

-120

-110

-100

.

E[MeV]

16OℏΩ = 20 MeV

4 6 8 10 12emx

-380

-360

-340

-320

-300

-280

-260

.

E[MeV]

40CaℏΩ = 24 MeV

NN+3N-full

-170

-160

-150

-140

-130

-120

-11024OℏΩ = 20 MeV

4 6 8 10 12emx

-500

-450

-400

-350

-300

-25048CaℏΩ = 28 MeV

Λ-CCSD(T)( Î)

CCSD( ◊ Í)

α = 0.02 fm4

◊ α = 0.04 fm4

Î Í α = 0.08 fm4

HF basis

E3mx = 14

Λ3N = 400 MeV

all NO2B

34

Conclusions

35

Conclusions


new era of ab-initio nuclear structure and reaction theoryconnected to QCD via chiral EFT

• chiral EFT as universal starting point... propagate uncertainties &provide feedback

consistent inclusion of 3N interactions in similarity transfor-mations & many-body calculations

• breakthrough in computation & handling of 3N matrix elements

innovations in many-body theory: extended reach of exactmethods & improved control over approximations

• versatile toolbox for different observables & mass ranges

many exciting applications ahead...

36

Epilogue


thanks to my group & my collaborators

• S. Binder, A. Calci, B. Erler, E. Gebrerufael,H. Krutsch, J. Langhammer, S. Reinhardt, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, K. Vobig, R. WirthInstitut 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. Hergert, K. HebelerOhio State University, USA

• P. PapakonstantinouIPN Orsay, F

• C. ForssénChalmers University, Sweden

• H. Feldmeier, T. NeffGSI Helmholtzzentrum

JUROPA LOEWE-CSC HOPPER

COMPUTING TIME

37

new horizons in ab initio nuclear structure theory

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