Applications of propagator theory to atoms and nuclei · Applications of propagator theory to atoms and nuclei C. Barbieri ... atoms molecules solids Many-body problem: to predict

50 Years of Coupled Cluster Theory50 Years of Coupled Cluster Theory INT Seattle, June 30INT Seattle, June 30--July 2, 2008July 2, 2008

Applications of propagator Applications of propagator theory to atoms and nucleitheory to atoms and nuclei

C. Barbieri

Symposium on “50 Years of Coupled Cluster Theory”

Collaborators: W. H. Dickhoff, D. Van Neck, G. Martínez-Pinedo, K. Langanke, M. Hjorth-Jensen, C. Giusti, F. D. Pacati


Nuclear structure in the 21Nuclear structure in the 21stst centurycentury

nucleinucleinucleonnucleon--nucleon nucleon

forceforce


Examples of ManyExamples of Many--fermoinfermoin Systems in NatureSystems in Nature

nucleinuclei

solidssolidsmoleculesmoleculesatomsatoms

Many-body problem: to predict properties of a system governed by the A-body Hamiltonian:

...2p

1

2

++=+= ∑∑<=

A

jiij

A

i

i Vm

VTH


[Saclay data for 16O]Em [MeV]

σred ≈ S(h)

p m[M

eV/c]

10-50

0p1/20p3/2

0s1/2

OneOne--hole spectral function hole spectral function ---- exampleexample

correlationscorrelations

∑ −− −−⟩ΨΨ⟨=n

An

Am

Ap

Anmm

h EEEcEpSm

))((||),( 10

20

1)( || δdistribution of momentum (pm) and energies (Em)

independentparticle picture


One-body Green’s function (or propagator) describes the motion of quasi- particles and holes:

…this contains all the structure information probed by nucleon transfer (spectral function):

)(ωαβg

GreenGreen’’s functions in manys functions in many--body theorybody theory

∑ ±± −±⟩ΨΨ⟨==n

An

AAAn EEcgS ))((||)(Im1)( 1

02

01 || ωδω

πω αααα

m

∑ +−−⟩ΨΨ⟩⟨ΨΨ⟨

= +

+++

nAA

n

AAn

An

A

iEEcc

ηωβα

)(||||

01

011

0

∑ −−−⟩ΨΨ⟩⟨ΨΨ⟨

+ −

−−+

kAk

A

AAk

Ak

A

iEEcc

ηωαβ

)(||||

10

011

0


Why manyWhy many--body body GreenGreen’’s functionss functions????

• “ab-initio” approach• hierarchy of equations—can improve systematically• Linked diags extensivity• Self-consistency: “no” reference

•Closely related to spectroscopy experiments

•“phonons” as degrees of freedom phenomenology

APLLICATIONS:•Faddeev RPA• optical potential (disp. opt. mod. ≡DOM)• quasiparticle-DFT (QP-DFT)


Spectral function: Spectral function: atmosatmos vsvs nucleinuclei

1

10

100

-60 -50 -40 -30 -20 -10 00.1

1

10

10055

Co

55Ni

esp(MeV)

Spec

tral

Str

engt

h (%

)

Ef

Ef

p1/2

p3/2

f5/2

f7/2

Ne+Phys. Rev. A76, 052503 (2007)


[Saclay data for 16O]Em [MeV]

σred ≈ S(h)

p m[M

eV/c]

10-50

0p1/20p3/2

0s1/2

OneOne--hole spectral function hole spectral function ---- exampleexample

correlationscorrelations

∑ −− −−⟩ΨΨ⟨=n

An

Am

Ap

Anmm

h EEEcEpSm

))((||),( 10

20

1)( || δdistribution of momentum (pm) and energies (Em)

independentparticle picture


DysonDyson--SchwingerSchwinger equationequation

= + Σ Σ

, free particle propagator

, correlated propagator

, “irreducible” self-energy

In diagrammatic form:

it leads to a 1-body equation:

)(')();',(')(2ˆ )/()/(*)/(

2

rrrrrdrm

p qhqpqhqpqhqp rrrrrr ψωψωψ =Σ+ ∫


QuasiparticleQuasiparticle (QP(QP--)DFT in two words)DFT in two words……

Basic idea: • separate the quasiparticle peak from spectral function• model background as a functional of density

DETAILS? Van Neck et al., and Phys. Rev. A74, 042501 (2006).

Sh(E

m) [Data: 12C, D. Rohe

Habilitation thesis]NB[ρ]


QuasiparticleQuasiparticle (QP(QP--)DFT in two words)DFT in two words……

QP-DFT equation (generalized eigenvalue problem):

density matrix:removal energy matrix:

• one still solves a one-body (HF-like) equation• generalizes Kohn-Sham (KS) eq. to two functionals (KS for =0

and )• energy, density, and QP properties (sp. energies and spect. factors!)

background contributions (B)are functionals of density!

DETAILS? Van Neck et al., and Phys. Rev. A74, 042501 (2006).


Extracting the QPExtracting the QP--DFT background functionalsDFT background functionals……

first attempt to extract the background:

[Phys. Rev. A74, 062503 (2006)]

GW calculationson small atoms

need accurate “ab-initio” calculations of QP properties,from small atoms/molecules to the electron gas!!


• Electron gas screening of Coulomb need RPACorrelation energies (GW):

• Finite systems QP and ionization energiesGW does NOT work need 3rd order PT minimumADC(3), Heidelberg (chem.) group ≈ F-TDA

Why a Faddeev (FWhy a Faddeev (F--)RPA?)RPA?

GW

FRPA: CB, D. Van Neck, W.H.Dickhoff, Phys. Rev. A76, 052503 (2007)

F-RPA!!


• Non perturbative expansion of the self-energy:

• Explicit correlations enter the “three-particle irreducible”propagators:

=Σ* (2p1h) R(2h1p)R

PRC63, 034313 (2001)PRC65, 064313 (2002)PRA76, 052503 (2007)

II(pp)

Π (ph)Π

g

(ph)

“Extended”Hartree Fock ≥ 2p1h/2h1p configurations

Coupling single particle to collective modesCoupling single particle to collective modes

•Both pp/hh (ladder) andph (ring) response included•Pauli exchange at 2p1h/2h1p level

≡ particle≡ hole


FRPA: Faddeev summation of RPA propagatorsFRPA: Faddeev summation of RPA propagators

TDA

RPA

•Both pp/hh (ladder) andph (ring) response included•Pauli exchange at 2p1h/2h1p level

•All order summation through a set of Faddeev equations

where:

References: CB, et al., Phys. Rev. C63, 034313 (2001); Phys. Rev. A76, 052503 (2007)


pp-BSE

ph-BSE

Faddeev 2h1peq.

)ωg((pp)Γ

(ph)Π

propagator

one-hole spectral function

two-hole spectral function

excitation spectrum

Dysonand 2p1h

input(0)g

SelfSelf--consistent Greenconsistent Green’’s functions function approachapproach

pp-RPA

ph-RPA

II(pp)

Π (ph)Π

g

(ph)

optical potential

FULL self consistency in mid size bases in now POSSIBLE: 16O , 8 shells ~ CB, Phys. Lett. B643, 268 (2006)


Characteristics of FRPA and CCCharacteristics of FRPA and CC

A.B.Trofimov, J. Schirmer, J. Chem. Phys. 123, 144115 (2005).

F-RPA, F-TDA ≈


Binding energies for AtomsBinding energies for Atoms

-200.043-200.058-200.055-199.617Mg:

-128.928-128.917-128.913-128.547Ne:

-14.667-14.643-14.643-14.573Be:

-2.904-2.903-2.903-2.860He:

Exp.FRPAFTDAHF

426

281

+94

+44

-12

+15

+24

+1

-15

+11

+24

+1

Energies in Hartree /Relative to the experiment in mH

cc-pV(TQ)Z bases, extrapolated as EX = E∞+AX-3 (≈ 5mH accuracy)

Phys. Rev. A76, 052503 (2007).

+ CB and van Neck, work in progress

(preliminary)


Valence Ionization EnergiesValence Ionization Energies

-0.579-1.075-9.160

-0.578-1.065-9.199

-0.580-1.087-9.213

-0.585-1.159-9.519

-0.590-1.276-9.570

Ar: 3p3s2p

-0.281-2.12

-0.277-2.130

-0.270-2.130

-0.274-2.146

-0.253-2.281

Mg: 3s2p

-0.793-1.782

-0.803-1.795

-0.808-1.803

-0.763-1.750

-0.850-1.931

Ne: 2p2s

-0.343-4.533

-0.322-4.540

-0.323-4.544

-0.320-4.620

-0.309-4.733

Be: 2s1s

-0.904-0.900-0.902-0.906-0.918He: 1sExp.FRPAFTDA2ndHF

-11201-410

+28-161

-57-149

+34-200

-14

-6-84

-359

+7-26

+30+32

+23-87

-2

-1-13-53

+11-10

-15-21

+20-11

+2

+1+10-39

+4-10

-10-15

+21-7

+4

F-TDA F-RPA

Energies in Hartree/Difference w.r.t. the experiment in mH

cc-pV(TQ)Z basis,extrapolated

Systematic improvement of ionization energies when including RPA propagators:about 4mH for valence orbits

Preliminary


Applications to NucleiApplications to Nuclei

• Strong short-range cores require “renormalizing” the interaction:– G-matrix, VUCOM, Lee Suzuki, Bloch-Horowitz, Vlow-k, …

• Long-range correlations FRPA !!


Binding energy Binding energy –– 44He caseHe case

( ))(

221 )(

2

0 ωωδω βααβαβ

αβ

εhA S

mk

dEF

⎥⎥⎦

⎤

⎢⎢⎣

⎡+= ∑∫

−

∞−

binding energy(Migdal-Galitski-Koltun)

Based on the intrinsic Hamiltonian: Hint = T + V – Tint

R(2p1h)Σ (ω) = R(2h1p)

[C. B., to be published]


Binding energy Binding energy –– 44He caseHe case

Based on the intrinsic Hamiltonian: Hint = T + V – Tint

(av18, Λ=1.9 fm-1)Fadd-Yak-29.19-28.49(7)Vlow-k

HH-28.57(av18, Iϑ=0.09fm3)

NCSM-28.4-27.90(7)VUCOM

Exact:GF:

Preliminary

≈700 KeV far from the exact result

R(2p1h)Σ (ω) = R(2h1p) NOTE: self-consistency in the mean field only

[C. B., to be published]


• The short-range core can be treated by summing ladders outside the model space:

Treating shortTreating short--range range corrcorr. with a G. with a G--matrixmatrix

Σ (ω) =G(ω)

+ +

(long-range effects)

F-RPA + …

+ F-RPA= + …G(ω)

)(2/)(

ˆ)( 22 ω

ηωω G

imkkQVVGba ++−

+=

…

Q

P


• The short-range core can be treated by summing ladders outside the model space:

Treating shortTreating short--range range corrcorr. with a G. with a G--matrixmatrix

G(ω)=

Near EF: long-range / SM-like physicsstronger eff. interaction

Deeply bound “orbits”: binding!the HF mean-field is weaker


-25

-20

-15

-10

-5

0

5

10

Es.

p. [

MeV

]

d3/2

s1/2d5/2

p1/2

p3/2BHF 1

st itr. SCGF

G-matrix

exp.

Single neutron levels around Single neutron levels around 1616O with FRPAO with FRPA

(AV18)

p-h gap:

16.616.5Ed3/2-Ep1/2

12.412.2Es1/2-Ep1/2

5.083.1Ep1/2-Ep3/2

6.123.5Ed3/2-Ed5/2

p-h gap:

Exp.[MeV]Theory(MeV)

[CB, Phys. Lett. B643, 268 (2006)]

• particle-hole gap accurate with a G-matrix with ω-dependence

•p3/2-p1/2 spin-orbit splitting close to the VMC estimates ≈3.4MeV[S. Pieper et al. PRL70 (’93) 2541, using AV14]


pp-BSE

ph-BSE

Faddeev 2h1peq.

)ωg((pp)Γ

(ph)Π

propagator

one-hole spectral function

two-hole spectral function

excitation spectrum

Dysonand 2p1h

input(0)g

SelfSelf--consistent Greenconsistent Green’’s functions function approachapproach

pp-RPA

ph-RPA

II(pp)

Π (ph)Π

g

(ph)

optical potential


• 16O(e,e’pn)14N• initial wave function from SCGF

• Pavia model for final state interactions

• pB ≡ q – p1 – p2

(q,ω)p1

p2

Correlations form twoCorrelations form two--nucleon knock outnucleon knock out

14N, 1+2

14N, 1+2

pB(MeV/c)pB(MeV/c)

ONLY short-range correlations included

full SCGF(two-hole RPA)

• two orders of magnitude from long range correlations !!

dσ[(fm

)4(sr)

-3]

FRPA!!


Experiment: MAMITheory: SCGF/Pavia scattering model

• Test run, low energy resolution:

• The 1+2 final state dominates –

tensor correlations!

• long-range correlations in the two-hole wave function are critical

[D. Middleton, et al. Eur. J. Phys. A29, 261 (2006)]

ProtonProton--neutron knockout: neutron knockout: 1616O(e,eO(e,e’’pn)pn)1414NN

1+g.s.

1+2.

0+1.

2+1.

1414N , E

N , E

xx

[MeV/c]mp0 50 100 150 200 250 300 350

]3 sr2 [p

b/MeV

nΩd pΩd e’Ωd pdT e’

/dTσ8 d

-110

1

10

210

<9)/3.95MeVxO(e,e’pn): (2<E16

DW-NN

DW

data

1-body

1-body+seag

π1-body+seag+

Δ+π1-body+seag+

<9)/3.95MeVxO(e,e’pn): (2<E16

[MeV/c]mp0 50 100 150 200 250 300 350

]3 sr2 [p

b/MeV

nΩd pΩd e’Ωd pdT e’

/dTσ8 d

-110

1

10

210

DW-NN

DW

data

1-body

1-body+seag

π1-body+seag+

Δ+π1-body+seag+

<9)/3.95MeVxO(e,e’pn): (2<E16

0 100 200 300

101

101

102

1

pB(MeV/c)

dσ

+ c.o.m. correction (C.Giusti et al, nuclt-th:0706.0636C.Giusti et al, to be published)


Conclusions and OutlookConclusions and Outlook

• Self-Consistent Green’s Functions (SCGF), in the Faddeev RPA (FRPA) approximation are well suited to describe the coupling betweenparticle and collective modes of a many-body system.

• Ab-initio applications:• accurate ionization energies for atoms• coherent description of atoms/e- gas, possible?• convergent calculations in nuclei

• Linked to developments of:• quasiparticle (QP-)DFT to treat fragmentation as (partially

occupied) single particle + a background

• dispersive optical model (DOM). Data driven (and theory constrained) extrapolation of elastic nucleon scattering, towardlarge asymmetries/driplines [not discussed in this talk].

work in progress…

……THANKS for your attentionTHANKS for your attention!!

Documents

Applications of propagator theory to atoms and nuclei · Applications of propagator theory to atoms and nuclei C. Barbieri ... atoms molecules solids Many-body problem: to predict