From QCD to Nuclear Structure · nuclear structure theory for exotic nuclei bridging between...

From QCD to Nuclear Structure:

Interactions and Many-Body Techniques

Robert RothInstitut für Kernphysik

1

Overview

Robert Roth – TU Darmstadt – 10/2009

Motivation

Nuclear Interactions from QCD

Similarity Transformed Interactions

• Unitary Correlation Operator Method

• Similarity Renormalization Group

Computational Many-Body Methods

• No-Core Shell Model

• Importance Truncated NCSM

2

Nuclear Structure in the 21st Century


NuSTAR & friends@ FAIR

RIBF @ RIKEN,FRIB @ MSU,...

nuclearastrophysics

nuclei far-offstability

exotic modeshyper-nuclei,...

reliablenuclear structure theory

for exotic nuclei

bridging betweenlow-energy QCD and

nuclear structure theory

3

Theoretical Context


betterresolution/more

fundamental finite nuclei

few-nucleon systems

nucleon-nucleon interaction

hadron structure

quarks & gluons

deconfinement

Quantum

ChromoDynamics

NuclearStructure

4

Theoretical Context


betterresolution/more

fundamental

Quantum

ChromoDynamics

NuclearStructure

How to solve thequantum many-body

problem?

How to derive thenuclear interaction

from QCD?

5

Nuclear Interactionsfrom QCD

6

Nature of the Nuclear Interaction


∼ 1.6fm

ρ−1/30 = 1.8fm

NN-interaction is not fundamental

analogous to van der Waals interac-tion between neutral atoms

induced via mutual polarization ofquark & gluon distributions

acts only if the nucleons overlap, i.e. atshort ranges

genuine 3N-interaction is important

7

Nuclear Interaction from Lattice QCD


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0

NN

wa

ve

fu

nctio

n φ

(r)

r [fm]

1S03S1

-2 -1 0 1 2 -2-1

01

20.5

1.0

1.5

φ(x,y,z=0;1S0)

x[fm] y[fm]

φ(x,y,z=0;1S0)

0

100

200

300

400

500

600

0.0 0.5 1.0 1.5 2.0

VC

(r)

[Me

V]

r [fm]

-50

0

50

100

0.0 0.5 1.0 1.5 2.0

1S03S1 N

.Ishiietal.,PRL99,022001(2007)

first steps towards constructionof a nuclear interaction throughlattice QCD simulations

compute relative two-nucleonwavefunction on the lattice

invert Schrödinger equation toobtain local ‘effective’ two-nucleon potential

schematic results so far (un-physical quark masses, S-waveinteractions only,...)

8

Nuclear Interaction from Chiral EFT


EFT for relevant degreesof freedom (π,N) based onsymmetries of QCD

long-range pion dynamicstreated explicitly

short-range physics absorbedin contact terms

low-energy constants fitted toexperimental data (NN, πN)

hierarchy of consistent NN,3N,... interactions (includ-ing current operators)

2N forces 3N forces 4N forces

PSfrag repla ementsLO (Q00 )

NLO (Q22 )N2LO (Q33 )

N3LO (Q44 )

Meißner,NPA751(2005)149c

9

Realistic Nuclear Interactions


QCD ingredients

• chiral effective field theory

• meson-exchange theory

short-range phenomenology

• contact terms or parameterization ofshort-range potential

experimental two-body data

• scattering phase-shifts & deuteronproperties reproduced with highprecision

supplementary 3N interaction

• adjusted to spectra of light nuclei

ArgonneV18

CD Bonn

NijmegenI/II

ChiralN3LO

Argonne V18+ Illinois 2

Chiral N3LO+ N2LO

10

Argonne V18 Potential


(r) (r) ~L2

(r) S12 (r) (~L · ~S) (r) (~L · ~S)2

-100

0

100

.

[MeV]

0 1 2r [fm]

-100

0

100

.

[MeV]

0 1 2r [fm]

0 1 2r [fm]

(S, T)

(1,0)

(1,1)

(0,0)

(0,1)

11

Similarity Transformed

Interactions

12

Why Transformed Interactions?


Realistic Interactions

generate strong correlationsin many-body states

short-range central & tensorcorrelations most important

Many-Body Methods

rely on truncated many-nucleon Hilbert spaces

not capable of describingshort-range correlations

extreme: Hartree-Fock basedon single Slater determinant

Similarity Transformation

adapt realistic potential to theavailable model space

conserve experimentally con-strained properties (phaseshifts)

13

Deuteron: Manifestation of Correlations


0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

0.12

.

r ϕ

L

L = 0

L = 2

exact deuteron solutionfor Argonne V18 potential

ρ(2)S=1,MS=0

(~r)

ρ(2)S=1,MS=±1

(~r)

short-range repulsionsupresses wavefunctionat small distances r

central correlations

tensor interactiongenerates L=2 admixture

to ground state

tensor correlations

14


Unitary CorrelationOperator Method (UCOM)

H. Feldmeier et al. — Nucl. Phys. A 632 (1998) 61

T. Neff et al. — Nucl. Phys. A713 (2003) 311

R. Roth et al. — Nucl. Phys. A 745 (2004) 3

R. Roth et al. — Phys. Rev. C 72, 034002 (2005)

15

Unitary Correlation Operator Method


Correlation Operator

define a unitary operatorC to describe the effect ofshort-range correlations

C = exp[−i G] = exph− i∑

<j

gj

i

Correlated States

imprint short-range cor-relations onto uncorre-lated many-body states eψ= Cψ

Correlated Operators

adapt Hamiltonian touncorrelated states(pre-diagonalization)

eO = C† O C

eψO eψ′=ψC† O Cψ′=ψ eOψ′

16

Unitary Correlation Operator Method


explicit ansatz for unitary transformationoperator motivated by the physics of

short-range correlations

Central Correlator Cr

radial distance-dependentshift in the relative coordi-nate of a nucleon pair

gr =1

2

s(r) qr + qr s(r)

qr =12

~rr· ~q+ ~q ·

~r

r

Tensor Correlator CΩ

angular shift depending on the ori-entation of spin and relative coor-dinate of a nucleon pair

gΩ =3

2ϑ(r)(~σ1 · ~qΩ)(~σ2 · ~r) + (~r↔~qΩ)

~qΩ = ~q−~r

rqr

C = CΩCr = exp− i∑

<jgΩ,j

exp− i∑

<jgr,j

s(r) and ϑ(r) are optimized for the initial potential

17

Correlated States: The Deuteron


1 2 3 4 5r [fm]

0

0.05

0.1

0.15

.

r ϕ

L = 0

1 2 3 4 5r [fm]

0

0.05

0.1

0.15

.

r C

r

ϕ

1 2 3 4r [fm]

0

0.05

0.1

0.15

0.2

.

[fm]

s(r)central

correlations

1 2 3 4 5r [fm]

0

0.05

0.1

0.15

.

r C

ΩCr

ϕ

L = 2

L = 0

1 2 3 4r [fm]

0

0.02

0.04

0.06

0.08

.

ϑ(r)

tensorcorrelations

only short-range tensorcorrelations treated by CΩ

18

Correlated Interaction: VUCOM


3S1

01

23

45

0

1

23

45

-60

-40

-20

0

20

12

34

5

01

23

45

0

1

23

45

-60

-40

-20

0

20

12

34

5

q [fm−1]

q′

[fm−1]

3S1−3D1

01

23

45

0

1

23

45

-40

-20

0

12

34

5

01

23

45

0

1

23

45

-40

-20

0

12

34

5

q [fm−1]

q′

[fm−1]

VAV18

VUCOM

q(LS)JT q′(L′S)JTpre-diagonalization

of Hamiltonian

19


Similarity RenormalizationGroup (SRG)

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

Bogner et al. — Phys. Rev. C 75, 061001(R) (2007)

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

20

Similarity Renormalization Group


flow evolution of the Hamiltonian toband-diagonal form with respect to

uncorrelated many-body basis

Flow Equation for Hamiltonian

evolution equation for Hamiltonian

eH(α) = C†(α)HC(α) →d

dαeH(α) =η(α), eH(α)

dynamical generator defined as commutator with the oper-ator in whose eigenbasis H shall be diagonalized

η(α)2B=

12μ

~q2, eH(α)

UCOM vs. SRG

η(0) has the same structureas UCOM generators gr & gΩ

21

SRG Evolution: The Deuteron


01

23

45

0

1

23

45

-60

-40

-20

0

20

12

34

5

01

23

45

0

1

23

45

-40

-20

0

12

34

5

3S1

3S1−

3D1

VSRG(q,q′)

VSRG(q,q′)

q′

q′

q

q

0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

0.12

.

rϕL=0

SRG

rϕL=2

SRG

Argonne V18

strongoff-diagonalcontribu-tions

short-rangecentral & tensor

correlations

22

SRG Evolution: The Deuteron


01

23

45

0

1

23

45

-60

-40

-20

0

20

12

34

5

01

23

45

0

1

23

45

-40

-20

0

12

34

5

3S1

3S1−

3D1

VSRG(q,q′)

VSRG(q,q′)

q′

q′

q

q

0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

0.12

.

rϕL=0

SRG

rϕL=2

SRG

α = 0.1000 fm4

UCOM vs. SRG

extract the UCOMcorrelation functionss(r) and ϑ(r) fromthe SRG evolvedwavefunctions

23


No-Core Shell Model

Roth et al. — Phys. Rev. C 72, 034002 (2005)

Roth & Navrátil — in preparation

24

No-Core Shell Model: Basics


special case of a full configuration interaction (CI) scheme

many-body basis: Slater determinantsν

composed of har-

monic oscillator single-particle statesΨ=∑

ν

Cν

ν

model space: spanned by basis statesν

with unperturbed ex-

citation energies of up to NmxℏΩ

exact factorization of intrinsic and CM component is possible

numerical solution of eigenvalue problem for Hint within NmxℏΩmodel space via Lanczos methods

model spaces of up to 109 basis states are used routinely

increase Nmx until convergence is observed

25

4He: NCSM Convergence


VAV18

20 40 60 80ℏΩ [MeV]

-20

0

20

40

.

E[MeV]

Nmx

0 2 4 6

8

10

1214

16EAV18Eexp

VUCOMMIN, ϑ = 0.09 fm3

20 40 60 80ℏΩ [MeV]

-25

-20

-15

-10

VSRGα = 0.03 fm4

20 40 60 80ℏΩ [MeV]

ϑ or α adjusted such that 4He binding energy is reproduced

26

Tjon-Line and Correlator Range


-8.6 -8.4 -8.2 -8 -7.8 -7.6E(3H) [MeV]

-30

-29

-28

-27

-26

-25

-24

.

E(4He)[MeV]

AV18Nijm II

Nijm I

CD Bonn

NN + 3NExp.

Tjon-line: E(4He) vs.E(3H) for phase-shiftequivalent NN-interactions

27

Tjon-Line and Correlator Range


-8.6 -8.4 -8.2 -8 -7.8 -7.6E(3H) [MeV]

-30

-29

-28

-27

-26

-25

-24

.

E(4He)[MeV]

AV18Nijm II

Nijm I

CD Bonn

VUCOM(AV18)Exp.

increasingCΩ-range

Tjon-line: E(4He) vs.E(3H) for phase-shiftequivalent NN-interactions

change of CΩ-correlatorrange results in shift alongTjon-line

minimize net3N interactionby choosing

correlator close toexperimental point

this VUCOMis used in thefollowing

28

10B: Hallmark of a 3N Interaction?


-2

-1

0

1

2

3

4

5

6

7

.

E−E3+[MeV]

VUCOM Exp

3+

1+

0+

2+

4+

0ℏω 2ℏω 4ℏω 6ℏω 8ℏω

ℏω = 18MeV

-62.1 -64.7

29

10B: Hallmark of a 3N Interaction?


-2

-1

0

1

2

3

4

5

6

7

.

E−E3+[MeV]

VUCOM Exp CDBonn ChiralNN NN+NNN

3+

1+

0+

2+

4+

0ℏω 2ℏω 4ℏω 6ℏω 8ℏω

ℏω = 18MeV

8ℏω 6ℏω

-62.1 -64.7

-56.8

-56.3

-64.0

VUCOM gives correctlevel orderingwithout any 3N

interaction

30


Importance TruncatedNo-Core Shell Model

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)

31

Importance Truncated NCSM


converged NCSM calcu-lations are essentially re-stricted to p-shell

full 10 or 12ℏΩ calculationfor 16O not yet feasible(basis dimension > 1010)

0 2 4 6 8 10121416182022Nmx

-120

-100

-80

-60

-40

.

E[MeV]

+ full NCSM

16OVUCOM

ℏΩ = 22MeV

ImportanceTruncation

reduce NCSM spaceto the relevant basis

states using an a prioriimportance measurederived from MBPT

0 2 4 6 8 10121416182022Nmx

-120

-100

-80

-60

-40

.

E[MeV]

IT-NCSM(seq)

IT-NCSM(seq)

similar strategies havefirst been developed and

applied in quantum chemistry:

configuration-selectivemultireference CI

Buenker & Peyerimhoff (1974);

Huron, Malrieu & Rancurel (1973);

many others since then...

32

Importance Truncation: General Idea


given an initial approximationΨreffor the target state within a

limited reference space MrefΨref=∑

ν∈Mref

C(ref)ν

ν

measure the importance of individual basis stateν

/∈ Mref

via first-order multiconfigurational perturbation theory

κν = −

ν

Hint

Ψref

εν − εref

construct importance-truncated space M(κmin) spanned by ba-sis states with |κν | ≥ κmin

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

33

Importance Truncation: Iterative Scheme


non-zero importance measure κν only for states which differ fromΨref

by 2p2h excitation at most

IT-NCSM[] or IT-CI[]

simple iterative scheme for arbi-trary many-body model spaces

start withΨref=0

➊ construct importance truncatedspace containing up to 2p2h ontop ofΨref

➋ solve eigenvalue problem

➌ use dominant components ofeigenstate (|Cν | ≥ Cmin) as newΨref, goto ➊

IT-NCSM(seq)

sequential update scheme for aset of NmxℏΩ spaces

start with Nmx = 2 eigenstatefrom full NCSM as initial

Ψref

➊ construct importance truncatedspace for Nmx + 2

➋ solve eigenvalue problem

➌ use dominant components ofeigenstate (|Cν | ≥ Cmin) as newΨref, goto ➊

full NCSM modelspace is recovered in thelimit (κmin, Cmin)→ 0 in

IT-NCSM(seq) andIT-NCSM[conv]

34

Threshold Dependence


-105

-104

-103

-102

-101

-100

.

Eint[M

eV]

16OIT-NCSM(seq)

VUCOMℏΩ = 22 MeV

Nmx = 8

00.020.040.06

.

J

0 2 4 6 8 10 12 14κmin [10

−5]

00.020.040.06

.

T

do calculations for a se-quence of importancethresholds κmin

observables show smooththreshold dependence

systematic approach to thefull NCSM limit

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

35

Constrained Threshold Extrapolation


16O, IT-NCSM(seq)VUCOM, ℏΩ = 22MeV, Nmx = 16

0 2 4 6 8 10 12 14 16κmin [10

−5]

-140

-130

-120

-110

.

Eλ[M

eV]

−0.5

0

0.5

λ =

1

1.5

2

estimate energy contributionof excluded states perturba-tively Δexcl(κmin)

simultaneous fit of combinedenergy

Eλ(κmin)

= Eint(κmin) + λΔexcl(κmin)

for set of λ-values with the con-straint Eλ(0) = Eextrap

robust threshold extrapola-tion with error bars determinedby variation of the λ set

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

Calculation”

36

4He: Importance-Truncated NCSM


-28

-26

-24

-22

.

E[MeV]

0 4 8 12 16 20 24Nmx

102103104105106107

.

Dmx

4HeVUCOM

ℏΩ = 40MeV

+ full NCSM IT-NCSM(seq)

sequential IT-NCSM(seq):single importance update us-ing (Nmx−2)ℏΩ eigenstate asreference

reproduces exact NCSM re-sult for all Nmx

reduction of basis by morethan two orders of magnitudew/o loss of precision

37

4He: Importance-Truncated NCSM


20 25 30 35 40 45 50ℏΩ [MeV]

-28

-27

-26

-25

-24

-23

-22

.

E[M

eV]

Nmx = 4

6

8

101214

4HeVUCOM

+ full NCSM IT-NCSM(seq)

reproduces exact NCSMresult for all ℏΩ and Nmx

importance truncation &threshold extrapolation isrobust

no center-of-mass con-tamination for any Nmx

and ℏΩ

38

16O: Importance-Truncated NCSM


-120

-100

-80

-60

-40

.

E[MeV]

0 2 4 6 8 10121416182022Nmx

102

104

106

108

1010

.

Dmx

16OVUCOM

ℏΩ = 22MeV

+ full NCSM IT-NCSM(seq), Cmin = 0.0005

IT-NCSM(seq), Cmin = 0.0003

IT-NCSM(seq) provides excel-lent agreement with fullNCSM calculation

dimension reduced by sev-eral orders of magnitude

possibility to go way beyondthe domain of the full NCSM

39

16O: Importance-Truncated NCSM


-140

-120

-100

-80

-60

.

E[MeV]

0 2 4 6 8 10121416182022Nmx

102

104

106

108

1010

.

Dmx

16OVSRG(N3LO)

λ = 2.66 fm−1

α = 0.02 fm4

ℏΩ = 24MeV


SRG-evolved N3LO poten-tial provides a much betterconvergence behavior

nevertheless, Nmx ≤ 8 calcu-lations are not sufficient

non-exponential behavior ob-served with VUCOM is reallydue to interaction

40

12C: IT-NCSM for Open-Shell Nuclei


-80

-60

-40

-20

.

E[MeV]

0 2 4 6 8 10121416182022Nmx

103104105106107108109

.

Dmx

12CVUCOM

ℏΩ = 24MeV


IT-NCSM(seq), Cmin = 0.0003

excellent agreement withfull NCSM calculations

IT-NCSM(seq) works just aswell for non-magic / open-shell nuclei

all calculations limited byavailable two-body matrix ele-ments & CPU time only

41

12C: IT-NCSM for Excited States


-80

-70

-60

-50

.

E[MeV]

0+2+

0 2 4 6 8 10121416182022Nmx

103104105106107108109

.

Dmx

12CVSRG(N3LO)

λ = 2.66 fm−1

α = 0.02 fm4

ℏΩ = 24MeV


target ground & excitedstates simultaneously

separate importance measureκ(n)ν

for each target state

basis state is included if|κ(n)

ν| ≥ κmin for any n

dimension of importance trun-cated space grows linearlywith # of target states

42

12C: IT-NCSM for Spectroscopy


PRELIMINARY2.000

2.005

2.010

2.015

.

J(2+)

3.4

3.6

3.8

4.0

4.2

.

[efm

2]

Q(2+)

0 2 4 6 8 10 12 14 16 18 20κmin [10

−5]

16.016.517.017.518.0

.

[e2fm

4]

B(E2,0+ → 2+)

12CVSRG(N3LO)

α = 0.02 fm4

ℏΩ = 24MeVNmx = 6

access to spectroscopic ob-servables via eigenstates

multipole moments, transi-tion strengths, transition form-factors, densities,...

simple threshold extrapolationessentially reproduces fullNCSM results

systematicspectroscopy in p-and sd-shell with

large NmxℏΩ spaces

43

7Li: IT-NCSM for Odd Nuclei


-40

-35

-30

-25

.

E[MeV]

3/2−1/2−

7/2−

0 2 4 6 8 10121416182022Nmx

0

2

4

6

.

E⋆[MeV]

7LiVSRG(N3LO)

α = 0.06 fm4

ℏΩ = 24MeV


IT-NCSM(seq) treats a groundstate & low-lying excitedstates for open- and closed-shell nuclei on the samefooting

excellent agreement withfull NCSM calculations in allcases

44

RGM & IT-NCSM: Ab Initio Reactions


with Petr Navrátil & Sofia Quaglioni (LLNL)

IT-NCSM wave function as input for RGM (Resonating GroupMethod) calculations of low-energy nucleon-nucleus scattering

using 3 lowest 7Li states

so-far up to Nmx = 14,here Nmx = 8

phase-shifts with full NCSMand IT-NCSM input agree

2 bound states for 8Li

4 resonances: 3+ and 1+

are known, 0+ and 2+ res-onances are predictions

45

IT-NCSM: Pros and Cons


fulfills variational principle & Hylleraas-Undheim theorem

no center-of-mass contamination induced by importance trun-cation in NmxℏΩ space

constrained threshold extrapolation κmin → 0 recovers contribu-tion of excluded configurations efficiently and accurately

open and closed-shell nuclei with ground and excited statescan be treated on the same footing

compatible with shell model: compute any observable fromwave functions in SM representation

• approximate size-extensivity after threshold extrapolation inIT-NCSM(seq) or IT-NCSM[conv] – no explicit npnh truncation

computationally still demanding

46


Other Options...

47

Other Options...


similarity transformedinteractions (e.g. VUCOM) provide

universal input for variousmany-body methods

exact few-body methods

coupled-cluster method

Hartree-Fock & many-body perturbation theory

RPA & Second-RPA

FMD with projection & configuration mixing

NCSM + Resonating Group Method

48

Hartree-Fock with VUCOM


-8

-6

-4

-2

0

.

E/A[MeV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

68Ni78Ni

88Sr90Zr

100Sn114Sn

132Sn146Gd

208Pb1

2

3

4

5

6

.

Rch[fm]

experiment HF

long-rangecorrelationsare missing

49

Perturbation Theory with VUCOM


-8

-6

-4

-2

0

.

E/A[MeV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

68Ni78Ni

88Sr90Zr

100Sn114Sn

132Sn146Gd

208Pb1

2

3

4

5

6

.

Rch[fm]

experiment HF HF+PT2 HF+PT2+PT3

long-rangecorrelations areperturbative

indications forpresence ofresidual 3N

interaction ???

50

Conclusions


three steps from QCD to the nuclear chart

• QCD-based nuclear interactions

• similarity transformed interactions (UCOM, SRG,...)

• computational many-body methods

exciting new developments in all three sectors

QCD-based description ofnuclear structure acrossthe whole nuclear chart is

within reach

51

Epilogue


thanks to my group & my collaborators

• S. Binder, A. Calci, B. Erler, A. Günther, M. Hild,H. Krutsch, J. Langhammer, P. Papakonstantinou,S. Reinhardt, F. Schmitt, N. VogelmannInstitut für Kernphysik, TU Darmstadt

• P. Navrátil, S. QuaglioniLawrence Livermore National Laboratory, USA

• H. Hergert, P. Piecuch, J. GourMichigan State University, USA

• H. Feldmeier, T. Neff,...Gesellschaft für Schwerionenforschung (GSI)

52

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