ManyBody Greens Functions For Atoms And Nuclei

1
Many-Body Greens Functions For Atoms And Nuclei

• C. Barbieri

Collaborators W. H. Dickhoff, D. Van Neck, D.
Rohe, L. Lapikás, I. Sick, M. Hjorth-Jensen, C.
Giusti, F. D. Pacati, G. Martínez-Pinedo, K.
Langanke
From quarks to the nuclear many-body problem
in honor of E. Osnes
2
The Stairway To Heaven
nuclear structure/ many-body
nucleon-nucleon force
3
The Stairway To Heaven

• Wish to predict properties of nuclei from the
  A-body Hamiltonian

4
The Stairway To Heaven

• Wish to predict properties of nuclei from the
  A-body Hamiltonian

• ab-initio approaches Monte Carlo methods (A
  12), no-core shell model, coupled cluster,
  

5
The Stairway To Heaven

• Wish to predict properties of nuclei from the
  A-body Hamiltonian

• ab-initio approaches Monte Carlo methods (A
  12), no-core shell model, coupled cluster,
  
• alternatives ? ? many-body Greens
  functions - starts from nucleon-nucleon force ?
  ab-initio approach! most of this
  talk - phonons as degrees of freedom ?
  phenomenological input possible
• - optical potential (disp. opt. mod. ?DOM) and
  quasiparticle-DFT (QP-DFT)

6
One-hole spectral function -- example
?red ? S(h)
independent particle picture
10-50
correlations
pm MeV/c
Em MeV
Saclay data for 16O
? distribution of momentum (pm) and energies (Em)
7
Greens functions in many-body theory

• One-body Greens function (or propagator)
  describes the motion of quasi- particles and
  holes
• this contains all the structure information
  probed by nucleon transfer (spectral function)

8
One-hole spectral function -- example
?red ? S(h)
independent particle picture
10-50
correlations
pm MeV/c
Em MeV
Saclay data for 16O
? distribution of momentum (pm) and energies (Em)
9
Dyson-Schwinger equation

• In diagrammatic form
• ? it leads to a 1-body equation

, free particle propagator , correlated
propagator , irreducible self-energy
??
10
Quasiparticle (QP-)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? ? see prev. talk (Van Neck) and Phys.
Rev. A74, 042501 (2006)
11
Extracting background QP-DFT functionals

• first attempt to extract the background

? GW calculations on small atoms
Phys. Rev. A74, 062503 (2006)
? need accurate ab-initio calculations of QP
properties, from small atoms/molecules to the
electron gas!!
12
Why a Faddeev (F-)RPA?

• 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

W
G
F-RPA !!
FRPA CB, D. Van Neck, W.H.Dickhoff, Phys. Rev.
A76, 052503 (2007)
13
Coupling single particle to collective modes

• Non perturbative expansion of the self-energy
• Explicit correlations enter the three-particle
  irreducible propagators

ExtendedHartree Fock
?? 2p1h/2h1p configurations

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

? particle ? hole
PRC63, 034313 (2001) PRC65, 064313 (2002) PRA76,
052503 (2007)
14
FRPA Faddeev summation of RPA propagators

• Both pp (ladder) andph (ring) modes included
• Pauli exchange at 2p1h/2h1p level
• All order summation through a set of Faddeev
  equations

where
TDA
RPA
References CB, et al., Phys. Rev. C63, 034313
(2001) Phys. Rev. A76, 052503 (2007)
15
Self-consistent Greens function approach
pp-RPA
ph-RPA
optical potential
16
A comparison with coupled-cluster theory
F-RPA, F-TDA ?
A.B.Trofimov, J. Schirmer, J. Chem. Phys. 123,
144115 (2005).
17
Binding energies for Atoms
Phys. Rev. A76, 052503 (2007). CB and van
Neck, work in progress
(preliminary)
Energies in Hartree / Relative to the experiment
in mH cc-pV(TQ)Z bases, extrapolated as EX
E?AX-3 (? 5mH accuracy)
18
Valence Ionization Energies
Preliminary
Systematic improvement of ionization energies
when including RPA propagators about 4mH for
valence orbits
Energies in Hartree/ Difference w.r.t. the
experiment in mH cc-pV(TQ)Z basis, extrapolated
F-TDA
F-RPA
19
Valence Ionization Energies relativity!!
Preliminary
Estimates of relativistic corrections based on
calculations by the Heidelberg group (J. Chem.
Phys. 121 (2004) 8782.
20
Applications to Nuclei

• Strong short-range cores require renormalizing
  the interaction
• G-matrix, VUCOM, Lee Suzuki, Bloch-Horowitz,
  Vlow-k,
• Long-range correlations ? FRPA !!

21
Binding energy 4He case
C. B., to be published
binding energy (Migdal-Galitski-Koltun)
Based on the intrinsic Hamiltonian
Hint T V Tint
22
Binding energy 4He case
C. B., to be published
NOTE self-consistency in the mean field only
Preliminary
700 KeV far from the exact result
Based on the intrinsic Hamiltonian
Hint T V Tint
23
Quasiparticle spectrum of 16O (i.e.17F)
experiment
SCGF/Fadd
1- (7.1MeV)
3- (6.1MeV)
0 (6.0MeV)
spectrumof 16O
0 (g.s.)
24
Quasiparticle spectrum of 16O (i.e.17F)
coupling a proton to 3- and 1- phonons in 16O
experiment
without 3- and 1-
SCGF/Fadd
1- (7.1MeV)
3- (6.1MeV)
0 (6.0MeV)
spectrumof 16O
0 (g.s.)
25
Quasiparticle spectrum of 16O (i.e.17F)
experiment
without 3- and 1-
without 3- , 1- and 0
SCGF/Fadd
1- (7.1MeV)
3- (6.1MeV)
0 (6.0MeV)
spectrumof 16O
0 (g.s.)
26
Treating short-range corr. with a G-matrix

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

Q
P

??(?)



F-RPA
G(?)
(long-range effects)



F-RPA
G(?)
27
Treating short-range corr. with a G-matrix

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

G(?)

Near EF long-range / SM-like physics? stronger
eff. interaction
Deeply bound orbits binding! the HF
mean-field is weaker
28
Single neutron levels around 16O with FRPA
(AV18)

• particle-hole gap accurate with a G-matrix with
  ?-dependence
• p3/2-p1/2 spin-orbit splitting close to the VMC
  estimates 3.4MeV S. Pieper et al. PRL70
  (93) 2541, using AV14

CB, Phys. Lett. B643, 268 (2006)
29
Two-phonons in (D)RPA (explicit 2p2h)
RPA
Dressed RPA

• account for spectral distribution of qp and qh

• ph states described in terms of MF orbits
• includes correlations in the g.s.

Contributions from 2p2h

• conservation laws and dressing, together,
  require additional 2p2h diagrams
• (Baym-Kadanoff theorem)

Brand et al. Nucl. Phys. A509, 1 (1990)
screening diagrams
30
One- and two-phonons in 16O
C.B., W.H.Dickhoff, PRC68, 014311 (2003)
States with a strong p-h character are only
slightly modified by 2-phonons ones 3MeV
attraction missing ? N-body forces?
clustering? Several new levels arise as
two-phonon states Anharmonicity effects are not
strong for this nucleus but still present
(splitting of multiplets)
experiment
Two-phononDRPA
Dressed RPA
31
Stability with dressed propagators
Two-phonon (D)RPA (16O)
Neutron s.p. spectra for 1b GF, vs h.o. length bHO

• The results for the low energy excitations become
  more stable when dressing (self-consistency) is
  included.
• dressing improves convergence by including
  selected contributions from higher np-nh
  excitations

Undressed input (h.o. wave functions)
Dressed
Ecut max energy of two-phonon configurations
32
Conclusions and Outlook

• Self-Consistent Greens Functions (SCGF), in the
  Faddeev RPA (FRPA) approximation are well suited
  to describe the coupling between particle 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, toward large
  asymmetries/driplines not discussed in this
  talk.

work in progress
33
Conclusions and Outlook

• Self-Consistent Greens Functions (SCGF), in the
  Faddeev RPA (FRPA) approximation are well suited
  to describe the coupling between particle 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, toward large
  asymmetries/driplines not discussed in this
  talk.

work in progress
THANKS for your attention!
34
Spectral strength for valence orbits(i.e.,
spectroscopic factors)
35
Experimental spectroscopic factors
NIKHEF
MSU/NSCL
PRL93, 042501 (2004)
Nucl. Phys. A553 (1993) 297c
36
Spectral function 48Ca (e,ep) 47K
d3/2
NIKHEF G. Kramer, Thesis
0.56
mostly d5/2
Includes low-energy mixing in the Fadd-RPA
scheme ? long-range correlations high-energy
mixing ? short-range correlations
d3/2
0.58
(2j1) spectroscopic factor
d5/2
S1/2
(N3LO based G-matrix)
CB, to be published
Ex (MeV) in 47K
37
Example of calculated spectral function around
56Ni
Preliminary
pf
pfh
p
Spectral Strength ()
Ef
Ef
esp(MeV)
Fadd-RPA perturbation self-energy
F-RPA
Gmtx based on N3LO
38
Spectroscopic factors for sp states around 56Ni
Preliminary
57Cu
57Ni
55Ni
55Co
55Ni
57Ni
exp. value estimated for f7/2 knockout from
57Ni
Spectral Strength ()
C.B.,Hjorth-Jensen, work in progress
esp(MeV)
39
Asymmetry dependence F-RPA estimate

• Explorative, FRPA calculations show only a slight
  dependence of spect. factors on separation
  energies (asymmetry)
• in rough agreement with nuclear matter
  calculations
• collective modes may not be fully realistic

Occupation number at the Fermi surface (for
nucleonic matter)
n(0)
60Ca
Preliminary
28O
T.Frick, et al. nucl-th/0409067 Phys. Rev. C
(2005)
? (N-Z)/A
40
Conclusions and Outlook

• Self-Consistent Greens Functions (SCGF), in the
  Faddeev RPA (FRPA) approximation are well suited
  to describe the coupling between particle 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, toward large
  asymmetries/driplines not discussed in this
  talk.

work in progress
THANKS for your attention!
41
Self-consistent Greens function approach
pp-RPA
ph-RPA
optical potential
42
Correlations form two-nucleon knock out

• 16O(e,epn)14N
• initial wave function from SCGF
• Pavia model for final state interactions
• pB ? q p1 p2

14N, 12
14N, 12
FRPA!!
d? (fm)4(sr)-3
full SCGF(two-hole RPA)
ONLY short-range correlations included
pB(MeV/c)
pB(MeV/c)

• two orders of magnitude from long range
  correlations !!

43
Proton-neutron knockout 16O(e,epn)14N
D. Middleton, et al. Eur. J. Phys. A29, 261
(2006)

• Test run, low energy resolution
• The 12 final state dominates tensor
  correlations!
• long-range correlations in the two-hole wave
  function are critical

c.o.m. correction (C.Giusti et al,
nuclt-th0706.0636 C.Giusti et al, to be
published)
Experiment MAMI Theory SCGF/Pavia scattering
model
44
Results for the hole spectral function of 16O
s shell
p shell
d shell
Experiment from NIKHEF, Leuschner et. al.,
PRC59, 655 (94)
experiment
experiment
? Results from Faddeev expansion and SCGF
?-k(MeV)
?-k(MeV)
p1/2 (g.s.)
p3/2 (Ex?6.3MeV)
S1/2 (Ex?5.2MeV)
d5/2 (Ex?5.2MeV)
C.B. et. al., PRC65, 064313 (2002)
45
Results for the hole spectral function of 16O
s shell
C.B. and WD, PRC65, 064313 (02)
p shell
d shell
Experiment from NIKHEF, Leuschner et. al.,
PRC59, 655 (94)
experiment
experiment

• Results from Faddeev expansion and SCGF

?-k(MeV)
?-k(MeV)
46
Results for the hole spectral function of 16O
s shell
C.B. and WD, PRC65, 064313 (02)
p shell
d shell
Experiment from NIKHEF, Leuschner et. al.,
PRC59, 655 (94)
experiment
experiment

• Results from Faddeev expansion and SCGF

?-k(MeV)
?-k(MeV)
3-
(d5/2 p1/2-1) p1/2-1?d5/2? 3-
47
Results for the hole spectral function of 16O
s shell
C.B. and WD, PRC65, 064313 (02)
p shell
d shell
Experiment from NIKHEF, Leuschner et. al.,
PRC59, 655 (94)
experiment
experiment

• Results from Faddeev expansion and SCGF

?-k(MeV)
?-k(MeV)
3-
0
(d5/2 p1/2-1) p1/2-1?d5/2? 3-
4p4h?
48
Single neutron levels around 16O (G-mtx VUCOM)

• p3/2-p1/2 spin-orbit splitting for av18(14)
• 3.4 MeV, VMC PRL93 v14
• 4.5 MeV, CCSD, fixed ? G-mtx PRC06 v18
• 3.1 MeV, FRPA, G(?) 06 v18
• For VUCOM (not av18!) 4.4 MeV
• 3NF are (still) missing!

(AV18)
Theory
Exp.
CB, Phys. Lett. B643, 268 (2006)
49
Single neutron levels around 16O (G-mtx VUCOM)

• Particle-hole gap, better described by the 2-body
  G-matrix interaction

(AV18)
CB, Phys. Lett. B643, 268 (2006)