Ab Initio Nuclear Structure from Chiral NN+3N HamiltoniansRobert Roth – TU Darmstadt – 04/2013 Nuclear Structure Low-Energy QCD 2. From QCD to Nuclear Structure Robert Roth –

Ab Initio Nuclear Structure

from Chiral NN+3N Hamiltonians

Robert Roth

1

From QCD to Nuclear Structure

Robert Roth – TU Darmstadt – 04/2013

Nuclear Structure

Low-Energy QCD

2



Nuclear Structure

Low-Energy QCD

NN+3N Interactionfrom Chiral EFT

chiral EFT based on the rel-evant degrees of freedom &symmetries of QCD

provides consistent NN, 3N,...interaction plus currents

3



Nuclear Structure

Low-Energy QCD


Unitary / SimilarityTransformation

adapt Hamiltonian to trun-cated low-energy model space

• tame short-range correlations• improve convergence behavior

transform Hamiltonian & ob-servables consistently

4



Nuclear Structure

Low-Energy QCD



Exact & Approx.Many-Body Methods

accurate solution of the many-body problem for light & inter-mediate masses (NCSM, CC,...)

controlled approximations forheavier nuclei (MBPT,...)

all rely on truncated modelspaces & benefit from unitarytransformation

5



Nuclear Structure

Low-Energy QCD



Exact & Approx.Many-Body Methods

How can we extend current ab-initiomethods to describe open-shell anddeformed nuclei?

How can we include the effects ofthree-nucleon forces in a computa-tionally efficient manner?

How can we describe the onset ofpairing in nuclei within various ab-initio frameworks?

Benchmarking and accuracy: canwe develop reliable theoretical errorestimates?

How can we bridge structure and re-actions in a consistent fashion?

How can we generate reliable predic-tions for the drip-lines?

6

Nuclear Interactions

from Chiral EFT

7

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,...

8

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...

9

Similarity

Renormalization Group

Roth, Calci, Langhammer, Binder — in preparation (2013)

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)

10

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- & four-body matrixrepresentation

11

SRG Evolution in Three-Body Space


chiral NN+3NN3LO + N2LO, triton-fit, 500 MeV

α = 0.000 fm4

Λ =∞ fm−1

Jπ =1

2

+, T =

1

2, ℏΩ = 28MeV

3B-Jacobi HO matrix elements

0 → E → 18 20 22 24 26 28(E, )

28

26

24

22

20

18

↓E′

↓

0

.

(E′ ,′)

NCSM ground state 3H

0 2 4 6 8 101214161820Nmx

-8

-6

-4

-2

0

2

.

E[MeV]

12

SRG Evolution in Three-Body Space


α = 0.320 fm4

Λ = 1.33 fm−1

Jπ =1

2

+, T =

1

2, ℏΩ = 28MeV

3B-Jacobi HO matrix elements

0 → E → 18 20 22 24 26 28(E, )

28

26

24

22

20

18

↓E′

↓

0

.

(E′ ,′)

NCSM ground state 3H

0 2 4 6 8 101214161820Nmx

-8

-6

-4

-2

0

2

.

E[MeV]

suppression ofoff-diagonal coupling= pre-diagonalization

significantimprovementof convergence

behavior

13

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

14

Sounds easy, but...


➊ initial 3B-Jacobi HO matrix elements of chiral 3N interactions

• direct computation using Petr Navratil’s ManyEff code (N2LO)

• conversion of partial-wave decomposed moment-space matrix ele-ments of Epelbaum et al. (N2LO, N3LO,...)

➋ SRG evolution in 2B/3B space and cluster decomposition

• efficient implementation using adaptive ODE solver & BLAS;largest JT-block takes a few hours on single node

➌ transformation of 2B/3B Jacobi HO matrix elements into JT-coupledrepresentation

• transform directly into JT-coupled scheme; highly efficient implemen-tation; can handle E3mx = 16 in JT-coupled scheme

➍ data management and on-the-fly decoupling in many-body codes

• optimized storage scheme for fast on-the-fly decoupling; can keep allmatrix elements up to E3mx = 16 in memory; suitable for GPUs

15

Importance Truncated

No-Core Shell Model



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)

16

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

17

Importance Truncated NCSM


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

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

full Nmx = 10 calculationfor 16O 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

similar strategies havefirst been developed and

applied in quantum chemistry:

configuration-selectivemultireference CI

Buenker & Peyerimhoff (1974);

Huron, Malrieu & Rancurel (1973);

and others...

18

Importance Truncation: Basic Idea


starting point: approximation |Ψref,m⟩ for the target stateswithin a limited reference space Mref

|Ψref,m⟩ =∑

ν∈Mref

C(ref,m)ν

|ν⟩

measure the importance of individual basis state |ν⟩ /∈ Mref

via first-order multiconfigurational perturbation theory

κ(m)ν= −⟨ν |H |Ψref,m⟩

Δεν

construct importance-truncated space M(κmin) from all basisstates with |κ(m)

ν| ≥ κmin for any m

solve eigenvalue problem in importance truncated spaceMIT(κmin) and obtain improved approximation of target state

importance measure onlyprobes 2p2h excitations ontop of Mref for a two-body

Hamiltonianembed into iterativescheme to access full

model space

19

Importance Truncation: Iterative Scheme


property of Nmx-truncated space: step from Nmx to Nmx + 2requires 2p2h excitations at most

sequential calculation for a range of NmxℏΩ spaces:

do full NCSM calculations up to a convenient Nmx

use components of eigenstates with |C(m)ν| ≥ Cmin as initial |Ψref,m⟩

➊ consider all states |ν⟩ /∈ Mref from an Nmx + 2 space and addthose with |κ(m)

ν| ≥ κmin to importance-truncated space MIT

➋ solve eigenvalue problem in MIT

➌ use components of eigenstate with |C(m)ν| ≥ Cmin as new |Ψref,m⟩

➍ goto ➊

full NCSM space isrecovered in the limit(κmin, Cmin)→ 0

20

Threshold Extrapolation


-148.5

-148.0

-147.5

-147.0

-146.5

-146.0

.

[MeV]

Hint

0.00

0.01

0.02

0.03

.

[MeV] Hcm

0 2 4 6 8 10

κmin × 105

0

0.01

0.02

0.03

.

J

16ONN-only

α = 0.04 fm4

ℏΩ = 20MeV

Nmx = 8

repeat calculations for asequence of importancethresholds κmin

observables show smooththreshold dependence andsystematically approach thefull NCSM limit

use a posteriori extrapo-lation κmin → 0 of observ-ables to account for effect ofexcluded configurations

21

Threshold Extrapolation


-148.5

-148.0

-147.5

-147.0

-146.5

-146.0

.

[MeV]

Hint

0 2 4 6 8 10

κmin × 105

-155.0

-154.0

-153.0

-152.0

-151.0

-150.0

.

[MeV]

Hint

16ONN-only

α = 0.04 fm4

ℏΩ = 20MeV

Nmx = 8

Nmx = 12

repeat calculations for asequence of importancethresholds κmin

observables show smooththreshold dependence andsystematically approach thefull NCSM limit

use a posteriori extrapo-lation κmin → 0 of observ-ables to account for effect ofexcluded configurations

22

Constrained Threshold Extrapolation


16ONN-only, α = 0.04 fm4

ℏΩ = 20MeV, Nmx = 12

0 2 4 6 8 10κmin × 105

-158

-156

-154

-152

-150

.

Eλ[M

eV]

1.0

0.75

0.5

0.25

λ = 0

for free: importance selectiongives perturbative energy cor-rection Δexcl(κmin) accountingfor excluded states

formal property

Δexcl(κmin)→ 0 for κmin→ 0

auxiliary parameter λ defininga family of energy sequences

Eλ(κmin) = E(κmin) + λΔexcl(κmin)

simultaneous extrapolationfor family of λ-values with con-straint Eλ(0) = Eextrap

proposed byBuenker & Peyerimhoff (1975):“Energy Extrapolation in CI

Calculations”

23

Uncertainty Quantification

in the IT-NCSM

24

Uncertainty Quantification


Importance Truncation

use sequence of (Cmin, κmin)-truncated model spaces

extrapolate to κmin → 0 us-ing poynomial ansatz or morerefined constrained extrapola-tion scheme

uncertainty estimate derivedfrom extrapolation protocol

systematic uncertainty ab-sent in full NCSM

Model-Space Truncation

use sequence of Nmx-trunc-ated model spaces

extrapolate to Nmx → ∞using exponential ansatz ormore elaborate extrapolationschemes

uncertainty estimate derivedfrom extrapolation protocol

same extrapolation uncer-tainties as in full NCSM

25

Comment on Cmin Truncation


-148.5

-148.0

-147.5

-147.0

-146.5

-146.0

.

E[M

eV]

0 2 4 6 8 10

κmin × 105

-155.0

-154.0

-153.0

-152.0

-151.0

-150.0

.

E[M

eV]

16ONN-only

α = 0.04 fm4

ℏΩ = 20MeV

Nmx = 8

Nmx = 12

Cmin = 1× 10−4

Î Cmin = 2× 10−4

Cmin = 3× 10−4

Cmin = 5× 10−4

truncation of reference state tocomponents with |Cν | ≥ Cmin

technical reason: importanceselection phase scales with(dimMref)

2

typically Cmin = 2× 10−4

practically noinfluence on thresholdextrapolated energies

26

Protocol: Simple κmin Extrapolation


-148.5

-148.0

-147.5

-147.0

-146.5

-146.0

.

E[M

eV]

0 2 4 6 8 10

κmin × 105

-155.0

-154.0

-153.0

-152.0

-151.0

-150.0

.

E[M

eV]

16ONN-only

α = 0.04 fm4

ℏΩ = 20MeV

Nmx = 8

Nmx = 12

perform IT-NCSM calculations forrange of κmin-values, typicallyκmin = 3,3.5, ...,10× 10−5

extrapolation κmin → 0 using poly-nomial Pp(κmin) fit to full κmin-set,typically of order p = 3

generate uncertainty band fromset of alternative extrapolations

• Pp−1 and Pp+1 extrapolations us-ing full κmin-range

• Pp extrapolations with lowest andlowest two κmin-points dropped

quote standard deviation as nomi-nal uncertainty

27

Protocol: Constrained κmin Extrapolation


16ONN-only, α = 0.04 fm4

ℏΩ = 20MeV, Nmx = 12

0 2 4 6 8 10κmin × 105

-157

-156

-155

-154

-153

-152

-151

.

Eλ[M

eV]

select a few λ-values to get sym-metrical approach towards com-mon Eextrap = Eλ(κmin = 0)

constrained simultaneous extrap-olation κmin → 0 using polynomialPp(κmin), typically of order p = 3

generate uncertainty band fromset of constrained extrapolations

• Pp−1 and Pp+1 extrapolations us-ing full κmin-range

• Pp extrapolations with lowest andlowest two κmin-points dropped

• Pp extrapolations with smallestand largest λ-set dropped

std. deviation gives uncertainty

28

Characterization of SRG-Evolved

NN+3N Hamiltonians



29

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

30

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

31

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 4N

(→ A. Calci)clear signature of

induced 4N originatingfrom initial 3N

32

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

33

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

34

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

35

Ab Initio IT-NCSM Calculations

for p- and sd-Shell Nuclei

36

Spectroscopy of Carbon Isotopes


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

37

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.

38

Ground States of Oxygen Isotopes


Hergert, Binder, Calci, Langhammer, Roth; arXiv:1302.7294

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

39





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

40





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

(→ H. Hergert)


41





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

(→ H. Hergert)

È CCSDÎ Λ-CCSD(T)

(→ S. Binder)


different many-bodyapproaches using sameNN+3N Hamiltonian give

consistent results minor differences are

understood (NO2B, E3mx,...)

42

Multi-Reference

Normal-Ordering Approximation

43

Motivation: Normal Ordering


avoid formal and computationalchallenges of including explicit 3Nterms in many-body calculations

circumvent formal extension of many-body method to includeexplicit 3N interactions

avoid the increase of computational cost caused by inclusionof explicit 3N interactions

normal-ordered two-body approximation works very well forclosed-shell systems (→ S. Binder)

can we do the same for open-shell nuclei?

44

Normal Ordering of 3N Interaction


starting point: three-body operator in second-quantized formwith respect to the zero-body vacuum |0⟩

V3N =1

36

∑

bc,bc

Vbc

bcAbcbc

Vbc

bc= ⟨bc|V3N |bc⟩ Abc

bc= †

†b

†ccb

single-reference normal ordering: assume reference state|SR⟩ given by a single Slater determinant

• standard toolbox: Wick theorem, contractions, etc.

multi-reference normal ordering: assume reference state|MR⟩ given by a superposition of Slater determinants

• generalized Wick theorem and n-tupel contractions proposed byMukherjee & Kutzelnigg (1997)

45

Multi-Reference Normal Ordering


three-body operator in normal-ordered form with respect tomulti-reference state |MR⟩

V3N =W +∑

c,c

WcceAcc+1

4

∑

bc,bc

Wbc

bceAbcbc+

1

36

∑

bc,bc

Wbc

bceAbcbc

where eA

indicates multi-reference normal ordered string of cre-

ation and annihilation operators (abstract concept)

matrix elements of normal-ordered n-body contributions in-volve one-, two- and three-body density matrices for |MR⟩

W =1

36

∑

bc,bc

Vbc

bcρbcbc

Wcc=1

4

∑

b,b

Vbc

bcρbb

Wbc

bc=∑

,

Vbc

bcρ

Wbc

bc= Vbc

bc

46

Multi-Reference Normal Ordering


discard normal-ordered three-body contribution to define thenormal-ordered two-body (NO2B) approximation

VNO2B =W +∑

c,c

WcceAcc+1

4

∑

bc,bc

Wbc

bceAbcbc

converted back into vacuum normal order with respect to |0⟩

VNO2B = V +∑

c,c

VccAcc+1

4

∑

bc,bc

Vbc

bcAbcbc

with new matrix elements

V =1

36

∑

bc,bc

Vbc

bc

ρbcbc− 18 ρ

ρbcbc+ 36 ρ

ρbbρcc

Vcc=1

4

∑

b,b

Vbc

bc

ρbb− 4 ρ

ρbb

Vbc

bc=∑

,

Vbc

bcρ

47

Single-Reference Normal Ordering


single-reference normal ordering is recovered by pugging indensity matrices for a single Slater-determinant

ρ= nδ

ρbb= ρ

ρbb− ρb

ρb

ρbcbc= ...

three-body operator in single-reference NO2B approximationconverted back into vacuum representation

VNO2B = V +∑

c,c

VccAcc+1

4

∑

bc,bc

Vbc

bcAbcbc

with simplified matrix elements

V =1

6

∑

bc

Vbcbc

nnbnc Vcc= −

1

2

∑

b

Vbcbc

nnb Vbc

bc=∑

Vbc

bcn

48

IT-NCSM with MR-NO2B Approximation


➊ perform NCSM with explicit 3N interaction for small Nmx

• ground state defines the reference state |MR⟩

• no explicit information on excited states enters

➋ compute zero-, one- and two-body matrix elements of MR-NO2Bapproximation

• density matrices for |MR⟩ can be precomputed and stored

• three-body density matrix is not need explicitly

➌ perform NCSM or IT-NCSM calculation up to large Nmx using MR-NO2B approximation

• same computational cost as a simple NN-only calculation

• larger model spaces become accessible

49

Benchmark: 6Li


0+

1+

2+

3+

-32

-30

-28

-26

-24

-22

-20

-18

.

E[MeV]

NN+3N-induced

0+

1+

2+

3+

2 4 6 8 10 12 Exp.Nmx

0

1

2

3

4

5

6

.

E[MeV]

0+

1+

2+

3+

-32

-30

-28

-26

-24

-22

-20

-18NN+3N-full

0+

1+

2+

3+

2 4 6 8 10 12 Exp.Nmx

0

1

2

3

4

5

6

6LiΛ3N = 500 MeV

α = 0.08 fm4

ℏΩ = 20 MeV

explicit 3N

50

Benchmark: 6Li


0+

1+

2+

3+

-32

-30

-28

-26

-24

-22

-20

-18

.

E[MeV]

NN+3N-induced

0+

1+

2+

3+

2 4 6 8 10 12 Exp.Nmx

0

1

2

3

4

5

6

.

E[MeV]

0+

1+

2+

3+

-32

-30

-28

-26

-24

-22

-20

-18NN+3N-full

0+

1+

2+

3+

2 4 6 8 10 12 Exp.Nmx

0

1

2

3

4

5

6

6LiΛ3N = 500 MeV

α = 0.08 fm4

ℏΩ = 20 MeV

explicit 3N

©MR-NO2BNrefmx

= 2,4,6

51

Benchmark: 12C


0+

0+

0+

1+

2+

4+

-90

-85

-80

-75

-70

-65

-60

-55

-50

.

E[MeV]

NN+3N-induced

0+

0+

0+

1+

2+

4+

2 4 6 8 10 Exp.Nmx

02468

101214

.

E[MeV]

0+

0+

0+

1+

2+

4+

-95

-90

-85

-80

-75

-70

-65

NN+3N-full

0+

0+

0+

1+

2+

4+

2 4 6 8 10 Exp.Nmx

02468

101214

12CΛ3N = 500 MeV

α = 0.08 fm4

ℏΩ = 20 MeV

explicit 3N

©MR-NO2BNrefmx

= 2,4

MR-NO2B approximationworks surprisingly well forground states as well as

excited states

robust with respect tovariations of the reference

state, i.e. Nrefmx

52

Challenge: 10B


-54

-52

-50

-48

-46

-44

-42

-40

-38

.

E[MeV]

NN+3N-induced

0+

1+

1+

3+

2 4 6 8 10 Exp.Nmx

0

0.5

1.0

1.5

2.0

2.5

.

E[MeV]

0+

1+

1+

3+-64

-62

-60

-58

-56

-54

-52

-50

-48

NN+3N-full

0+

1+

1+

3+

2 4 6 8 10 Exp.Nmx

0

0.5

1.0

1.5

2.0

2.5

10BΛ3N = 500 MeV

α = 0.08 fm4

ℏΩ = 16 MeV

explicit 3N

©MR-NO2BNrefmx

= 2,4

53

Reflections

54

Questions I


How can we extend current ab-initio methods to describe open-shell and deformed nuclei?

• IT-NCSM describes open and closed-shell nuclei on the same footing

• MR-IM-SRG is a true open-shell approach for medium/heavy masses(→ H. Hergert), others are following

How can we include the effects of three-nucleon forces in a com-putationally efficient manner?

• explicit 3N interactions are used in IT-NCSM very efficiently

• CCSD and ΛCCSD(T) is available with explicit 3N (→ S. Binder)

• single- and multi-reference normal ordering provide robust approxima-tions at reduced cost

• this does not imply that any kind of ‘summation over the third parti-cle’ is accurate

55

Questions II


How can we describe the onset of pairing in nuclei within variousab-initio frameworks?

• any ab initio approach for open shells has to describe pairing

Can we develop reliable theoretical error estimates?

• defining element of any ab initio approach

• uncertainty quantification case by case within the many-body ap-proach, not just guessing

• uncertainty quantification also necessary for the chiral EFT inputs

How can we bridge structure and reactions in a consistent fashion?

• NCSM/RGM and NCSMC with 3N are on their way (→ P. Navratil)

How can we generate reliable predictions for the drip-lines?

• simply do all of the above...

56

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, 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

LENPICLow-Energy NuclearPhysics International

Collaboration

JUROPA LOEWE-CSC HOPPER

COMPUTING TIME

57

Documents

Ab Initio Nuclear Structure from Chiral NN+3N HamiltoniansRobert Roth – TU Darmstadt – 04/2013 Nuclear Structure Low-Energy QCD 2. From QCD to Nuclear Structure Robert Roth –