new horizons in ab initio nuclear structure...

New Horizons in

Ab Initio Nuclear Structure Theory

Robert Roth

1

Ab Initio Nuclear Structure

Robert Roth – TU Darmstadt – 09/2013

Nuclear Structure Observables

Low-Energy Quantum Chromodynamics

LatticeQCD

quarks&gluononalattice

LatticeEFT

nucleons&pionsonalattice

Energ

y-D

ensityFunct.

guidedbychiralEFT

Chiral EFT Hamiltoniansconsistent NN,3N,... interactions & current operators

Chiral Effective Field Theorybased on relevant degrees of freedom & symmetries of QCD

Exact Solutionssolve nuclear many-

body problem with

converged truncations

Controlled Approx.treat many-body prob-

lem with controlled & im-

provable approximations

Similarity Transformationsphysics-conserving unitary transformation to

adapt Hamiltonian to limited model space

2





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with







FEW-BODYPATH

inject few-bodyphilosophy with relaxedaccuracy goals into the

many-body world

3





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with con-

verged truncations

Controlled Approx.treat many-body problem

with controlled & improv-

able approximations



4

Nuclear Interactions from Chiral EFT


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

■ chiral EFT background: talks byEpelbaum, Krebs, Gegelia,...

■ standard Hamiltonian:

● NN at N3LO:Entem & Machleidt, 500 MeV cutoff

● 3N at N2LO:Navrátil, A=3 fit, 500 MeV cutoff

■ alternatives:

● modified 3N interaction at N2LO(cutoff & LECs variations)

● consistent Hamiltonians at N2LO(NN: Epelbaum, POUNDERs-opt.)

● consistent Hamiltonians at N3LO(LENPIC Collaboration)

● Δ-full chiral EFT,...

NN 3N 4N

LO

NLO

N2LO

N3LO

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5





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with con-

verged truncations



able approximations



6

Similarity Renormalization Group


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

continuous transformation drivingHamiltonian to band-diagonal formwith respect to a uncorrelated basis

■ unitary transformation of Hamiltonian and other observables

Hα = Uα†HUα Oα = Uα†OUα

■ evolution equations for Hα and Uα depending on generator ηαd

dαHα = [ηα,Hα]

d

dαUα = −Uαηα

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

ηα = (2μ)2[Tint,Hα]

simplicity and flexibilityare great advantages of

the SRG approach

solve SRG evolutionequations using two-,

three- & four-body matrixrepresentation

7

SRG Evolution in Three-Body Space


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

α = 0.000 fm4Λ = ∞ fm−1

Jπ = 12

+

, T = 12, ̵hΩ = 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[M

eV]

8

SRG Evolution in Three-Body Space


α = 0.320 fm4Λ = 1.33 fm−1

Jπ = 12

+

, T = 12, ̵hΩ = 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[M

eV]

suppression ofoff-diagonal coupling =̂pre-diagonalization

significantimprovementof convergence

behavior

9

Hamiltonian in A-Body Space


■ evolution induces n-body contributions H[n]α to Hamiltonian

Hα = H[1]α +H

[2]α +H

[3]α +H

[4]α +H

[5]α + . . .

■ truncation of cluster series formally destroys unitarity and in-variance of energy eigenvalues (independence of α)

■ flow-parameter α provides diagnostic tool to assess neglectedhigher-order contributions

SRG-Evolved Hamiltonians

NNonly use initial NN, keep evolved NN

NN +3Nind use initial NN, keep evolved NN+3N

NN +3Nfull use initial NN+3N, keep evolved NN+3N

NN +3Nfull +4Nind use initial NN+3N, keep evolved NN+3N+4N

10





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with




able approximations



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 Nmxh̵Ω

■ 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

12

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[M

eV]

16ONNonly

α = 0.04 fm4

h̵Ω = 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[M

eV]

● IT-NCSM

+ full NCSM

13

4He: Ground-State Energies


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

NNonly

2 4 6 8 10 12 14 16 ∞Nmx

-29

-28

-27

-26

-25

-24

-23

.

E[M

eV]

NN+3Nind

2 4 6 8 10 12 14 ∞Nmx

Exp.

NN+3Nfull

2 4 6 8 10 12 14 ∞Nmx

● ◆ ▲ ∎ ★

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

̵hΩ = 20MeV

14

16O: Ground-State Energies


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

NNonly

2 4 6 8 10 12 14 ∞Nmx

-180

-160

-140

-120

-100

-80

.

E[M

eV]

NN+3Nind

2 4 6 8 10 12 ∞Nmx

Exp.

NN+3Nfull

2 4 6 8 10 12 ∞Nmx

● ◆ ▲ ∎ ★



̵hΩ = 20MeV

in progress:

explicit inclusion of

induced 4N clear signature ofinduced 4N originating

from initial 3N

15

16O: Ground-State Energies


Roth, et al; PRL 107, 072501 (2011); PRL 109, 052501 (2012)

NNonly

2 4 6 8 10 12 14 ∞Nmx

-180

-160

-140

-120

-100

-80

.

E[M

eV]

NN+3Nind

2 4 6 8 10 12 ∞Nmx

Exp.

NN+3Nfull

2 4 6 8 10 12 ∞Nmx

● ◆ ▲ ∎ ★



̵hΩ = 20MeV

simple fix:3N interaction with

400 MeV cutoff, cE refittedto 4He ground state

16

Ground States of Oxygen Isotopes


■ oxygen isotopic chain has received significant attention anddocuments the rapid progress over the past years

Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL 105, 032501 (2010)

■ 2010: shell-model calculations with 3N effects highlighting therole of 3N interaction for drip line physics

Hagen, Hjorth-Jensen, Jansen, Machleidt, Papenbrock, PRL 108, 242501 (2012)

■ 2012: coupled-cluster calculations with phenomenologicaltwo-body correction simulating chiral 3N forces

Hergert, Binder, Calci, Langhammer, Roth, PRL 110, 242501 (2013)

■ 2013: ab initio IT-NCSM with explicit chiral 3N interactions...

17



Hergert et al., PRL 110, 242501 (2013)

NN+3Nind(chiral NN)

2 4 6 8 10 12 14 16 18Nmx

-160

-140

-120

-100

-80

-60

-40

.

E[M

eV]

12O

14O

16O18O20O22O24O26O

NN+3Nfull(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 h̵Ω

18



Hergert et al., PRL 110, 242501 (2013)

NN+3Nind(chiral NN)

12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[M

eV]


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

19

Spectroscopy of Carbon Isotopes


Forssen et al., JPG 40, 055105 (2013); Roth et al., in prep.

0+ 1

2+ 1

2+ 1

2 4 6 8 Exp.0

2

4

6

8

.

E[M

eV]

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[M

eV]

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+3NfullΛ3N = 400MeV

α = 0.08 fm4

h̵Ω = 16MeV

20

Spectroscopy of Carbon Isotopes


INPR

OGRES

S2 4 6 8

0

1

2

3

4

5

6

7

8

.

B(E2,2+→0+)[e

2fm

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+)[e

2fm

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+3NfullΛ3N = 400MeV

α = 0.08 fm4

h̵Ω = 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.

21





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with







22

Frontier: Medium-Mass Nuclei


advent of novel ab initio many-body approachesapplicable in the medium-mass regime

Hagen, Papenbrock, Dean, Piecuch, Binder,...

■ coupled-cluster theory: ground-state parametrized by exponentialwave operator applied to single-determinant reference state

● truncation at doubles level (CCSD) plus triples corrections (Λ-CCSD(T))

● equations of motion for excited states and near-closed-shell nuclei

Bogner, Tsukiyama, Schwenk, Hergert,...

■ in-medium SRG: complete decoupling of particle-hole excitations frommany-body reference state through SRG evolution

● normal-ordered evolving A-body Hamiltonian truncated at two-body level

● both closed- and open-shell ground states; excitations via EOM or SM

Barbieri, Soma, Duguet,...

■ self-consistent Green’s function approaches and others...

control

lingand

quantif

ying th

e

uncerta

inties d

ueto v

arious t

runcat

ions is

major

challen

ge

23

CCSD with Explicit 3N Interactions


Roth, et al., PRL 109, 052501 (2012); Binder et al., PRC 87, 021303(R) (2013)

NN+3Nfull

-130

-120

-110

-100

.

E[M

eV]

16Oh̵Ω = 20 MeV

-170

-160

-150

-140

-130

-120

-11024Oh̵Ω = 20 MeV

4 6 8 10 12emx

-380

-360

-340

-320

-300

-280

-260

.

E[M

eV]

40Cah̵Ω = 24 MeV

4 6 8 10 12emx

-500

-450

-400

-350

-300

-25048Cah̵Ω = 28 MeV

CCSD-3B

(● ◆ ▲)

CCSD-NO2B

(◯ ◇ △)

● α = 0.02 fm4

◆ α = 0.04 fm4

▲ α = 0.08 fm4

HF basis

E3mx = 12

Λ3N = 400 MeVnormal-ordering

approximation reproducesexplicit-3N results to

better than 1%

24



Hergert et al., PRL 110, 242501 (2013)

NN+3Nind(chiral NN)

12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[M

eV]


12 14 16 18 20 22 24 26

A

AO

experiment

● IT-NCSM


25



Hergert et al., PRL 110, 242501 (2013)

NN+3Nind(chiral NN)

12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[M

eV]


12 14 16 18 20 22 24 26

A

AO

experiment

● IT-NCSM

◆ MR-IM-SRG


26



Hergert et al., PRL 110, 242501 (2013)

NN+3Nind(chiral NN)

12 14 16 18 20 22 24 26

A

-180

-160

-140

-120

-100

-80

-60

-40

.

E[M

eV]


12 14 16 18 20 22 24 26

A

AO

experiment

● IT-NCSM

◆ MR-IM-SRG

▼ CCSD▲ Λ-CCSD(T)


different many-bodyapproaches using sameNN+3N Hamiltonian give

consistent results minor differences are consis-tent with uncertainty analysis

27

Next Step: Calcium Isotopes


INPR

OGRES

S

NN+3Nind NN+3Nfull

-10

-9

-8

-7

-6

.

E/A[M

eV]

α-dependence α-dependence

36Ca 40Ca 48Ca 52Ca 54Ca-10

-9

-8

-7

.

E/A[M

eV]

E3mx-dependence

36Ca 40Ca 48Ca 52Ca 54Ca

E3mx-dependence

● α = 0.02 fm4

◆ α = 0.04 fm4

▲ α = 0.08 fm4

CCSD

E3mx = 14

Λ3N = 400 MeV

● E3mx = 14

◆ E3mx = 16

▲ E3mx = 18

CCSD

α = 0.04 fm4

Λ3N = 400 MeV

28

Next Step: Nickel Isotopes


INPR

OGRES

S

NN+3Nind NN+3Nfull

-10

-9

-8

-7

-6

.

E/A[M

eV]

α-dependence α-dependence

48Ni 56Ni 60Ni 62Ni 68Ni-10

-9

-8

-7

.

E/A[M

eV]

E3mx-dependence

48Ni 56Ni 60Ni 62Ni 68Ni

E3mx-dependence

● α = 0.02 fm4

◆ α = 0.04 fm4

▲ α = 0.08 fm4

CCSD

E3mx = 14

Λ3N = 400 MeV

● E3mx = 14

◆ E3mx = 16

▲ E3mx = 18

CCSD

α = 0.04 fm4

Λ3N = 400 MeV

control

lingand

quantif

ying th

e

uncerta

inties d

ueto v

arious t

runcat

ions is

major

challen

ge

29





LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT




body problem with







30

Ab Initio Hyper-Nuclear Structure


Hyper-Nuclear Structure Observables


LatticeQCD


LatticeEFT


Energ

y-D

ensityFunct.

guidedbychiralEFT

Chiral EFT Hamiltoniansconsistent NN,3N,YN,YY,... interactions & current operators



body problem with







31

Motivation: Hyper-Nuclear Structure


talks by Feliciello,...

■ precise data on ground states &spectroscopy of hyper-nuclei

talks by Hiyama, Nogga,...

■ ab initio few-body (A ≲ 4) andphenomenological shell modelor cluster calculations

talks by Haidenbauer,...

■ chiral YN & YY interactions at(N)LO are available

■ constrain YN & YY interactionby ab initio hyper-nuclear struc-ture calculations

32

Application: Λ7Li


INPR

OGRES

S1+

3+

-45

-40

-35

-30

-25

.

E[M

eV]

1+

3+

2 4 6 8 10 Exp.Nmx

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

2 4 6 8 10 Exp.Nmx

6Li 7ΛLi

Jülich’04

NN @ N3LOEntem&MachleidtΛNN = 500 MeV

3N @ N2LONavratil

Λ3N = 500 MeVtriton fit

Jülich’04Haidenbauer et al.scatt. & hypertriton

α = 0.08 fm4

h̵Ω = 20 MeV

induced YNNnot included

33

Application: Λ7Li


INPR

OGRES

S1+

3+

-45

-40

-35

-30

-25

.

E[M

eV]

1+

3+

2 4 6 8 10 Exp.Nmx

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

2 4 6 8 10 Exp.Nmx

6Li 7ΛLi

chiral YN @ LOΛYN = 700 MeV

NN @ N3LOEntem&MachleidtΛNN = 500 MeV

3N @ N2LONavratil

Λ3N = 500 MeVtriton fit

YN @ LOHaidenbauer et al.ΛYN = 700 MeV

scatt. & hypertriton

α = 0.08 fm4

h̵Ω = 20 MeV

induced YNNnot included

hyper-nuclear structuresets tight constraints on

YN interaction

start investigating

sensitivity of spectra

on YN input

34

New Horizons...


■ nuclear structure theory connected to QCD via chiral EFT

● chiral EFT as universal, controlled and improvable starting point

● consistent and optimized interactions at N2LO, N3LO,...

● consistent similarity transformation of Hamiltonian and observables

■ innovations in ab initio many-body theory

● consistent inclusion of 3N (and 4N) interactions

● precision structure and spectroscopy in p- and sd-shell (IT-NCSM,...)

● access to the medium-mass regime (CC, IM-SRG,...)

● extension to ab initio hyper-nuclear structure

● bridge to reaction theory (NCSM/RGM, NCSMC)

● uncertainty quantification, error propagation, feedback cycle

■ many exciting applications ahead...

35

Epilogue


■ thanks to my group & my collaborators

● S. Binder, A. Calci, S. Fischer, E. Gebrerufael,H. Spiess, J. Langhammer, S. Reinhardt, 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. PapakonstantinouIPN Orsay, F

● C. ForssénChalmers University, Sweden

● H. Feldmeier, T. NeffGSI Helmholtzzentrum

JUROPA LOEWE-CSC HOPPER

COMPUTING TIME

36

1 - Title2 - Ab Initio Nuclear Structure3 - Ab Initio Nuclear Structure4 - Ab Initio Nuclear Structure5 - Nuclear Interactions from Chiral EFT6 - Ab Initio Nuclear Structure7 - Similarity Renormalization Group8 - SRG Evolution in Three-Body Space9 - SRG Evolution in Three-Body Space10 - Hamiltonian in A-Body Space11 - Ab Initio Nuclear Structure12 - No-Core Shell Model13 - Importance Truncated NCSM14 - 4He: Ground-State Energies15 - 16O: Ground-State Energies16 - 16O: Ground-State Energies17 - Ground States of Oxygen Isotopes18 - Ground States of Oxygen Isotopes19 - Ground States of Oxygen Isotopes20 - Spectroscopy of Carbon Isotopes21 - Spectroscopy of Carbon Isotopes22 - Ab Initio Nuclear Structure23 - Frontier: Medium-Mass Nuclei24 - CCSD with Explicit 3N Interactions25 - Ground States of Oxygen Isotopes26 - Ground States of Oxygen Isotopes27 - Ground States of Oxygen Isotopes28 - Next Step: Calcium Isotopes29 - Next Step: Nickel Isotopes30 - Ab Initio Nuclear Structure31 - Ab Initio Hyper-Nuclear Structure32 - Motivation: Hyper-Nuclear Structure33 - Application: 7Li34 - Application: 7Li35 - New Horizons...36 - Epilogue

new horizons in ab initio nuclear structure...

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