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Heiko HergertDepartment of Physics, The Ohio State University
News from the In-Medium Similarity Renormalization Group
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Outline
• Similarity Renormalization Group
• In-Medium SRG for Closed-Shell Nuclei
• Open-Shell Nuclei from the Multi-Reference IM-SRG
• Ab Initio Shell-Model Hamiltonians
• Outlook
Similarity Renormalization Group
Review: S. Bogner, R. Furnstahl, and A. Schwenk, Prog. Part. Nucl. Phys. 65 (2010), 94
E. Anderson, S. Bogner, R. Furnstahl, and R. Perry, Phys. Rev. C82 (2011), 054001E. Jurgenson, P. Navratil, and R. Furnstahl, Phys. Rev. C83 (2011), 034301R. Roth, S. Reinhardt, and H. H., Phys. Rev. C77 (2008), 064003 H. H. and R. Roth, Phys. Rev. C75 (2007), 051001
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Similarity Renormalization Group
• evolved Hamiltonian
• flow equation:
• choose to achieve desired behavior, e.g. decoupling of momentum or energy scales
• consistently evolve observables of interest
Basic Concept
continuous unitary transformation of the Hamiltonian to band-diagonal form w.r.t. a given “uncorrelated” many-body basis
( ) = ( ) †( ) ≡ + ( )
( ) =η( ), ( )
, η( ) =
( ) †( ) = −η†( )
η( )
In-Medium SRG for Closed-Shell Nuclei
H. H., S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, and A. Schwenk, Phys. Rev. C 87, 034307 (2013)
K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev. Lett. 106, 222502 (2011)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Decoupling in A-Body Space
excitations relative to reference state: normal-ordering
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Decoupling in A-Body Space
aim: decouple reference state (0p-0h) from excitations
(∞)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Normal Ordering
• second quantization:
• particle- and hole density matrices:
• define normal-ordered operators recursively:
• algebra is simplified significantly because
• Wick’s theorem gives simplified expansions (fewer terms!) for products of normal-ordered operators
λ =
−→ δ , ∈ , ξ = λ − δ −→ − δ ≡ −( − )δ
...... = † . . . † . . .
...... = : ...
... : +λ : ...... : +
+λ λ − λ λ
: ...
... : + + . . .
: ...... :
=
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Normal-Ordered Hamiltonian
Normal-Ordered Hamiltonian
= +
: : +
: : +
: :
E0 = + +
two-body formalism with in-medium contributions from
three-body interactions
f = + +
Γ = +
W =
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Choice of Generator
=
: :: :
= − ¯
=
: :: :
∼
Off-Diagonal Hamiltonian & Generator
≡ + , ≡
: : + , ≡
:
: +
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow Equations
0-body Flow
1-body Flow
= +
= + + +
~ 2nd order MBPT for H(s)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow Equations
2-body Flow
= + - -
+ + + -
s channel t channel u channel
ladders rings
O(N6) scaling (before particle/hole distinction)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Decoupling
off-diagonal couplings are rapidly driven to zero
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
105 104 103 102 101 100 101600
580
560
540
520
s
EMeV
40CaEEMBPT2
, λ = . − , =
non-perturbative resummation of MBPT series
(correlations)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Hamiltonians
• NN: chiral interaction at N3LO (Entem & Machleidt)
• 3N: chiral interaction at N2LO (cD,cE fit to 3H energy & half-life)
Initial Hamiltonian
• NN + 3N-induced: start with initial NN Hamiltonian, keep two- and three-body terms
• NN + 3N-full: start with initial NN + 3N Hamiltonian, keep two- and three-body terms
SRG-Evolved Hamiltonians
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
He4 O16 O24 Ca40 Ca48 Ni48 Ni56
9
8
7
6EA
MeV
Λfm12.22.01.9
Results: Closed-Shell Nuclei
NN + 3N-ind.
Phys. Rev. C 87, 034307 (2013), arXiv: 1212.1190 [ nucl-th]
He4 O16 O24 Ca40 Ca48 Ni48 Ni5610
9
8
7
6
EAMeV Λfm1
2.22.01.9
NN + 3N-full (400)
experiment
validate chiralHamiltonians
Open-Shell Nuclei from theMulti-Reference IM-SRG
H. H., S. Binder, A. Calci, J. Langhammer, and R. Roth, Phys. Rev. Lett 110, 242501 (2013)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
A
EMeV
AOE3Max14Λ1.9 fm1
IMSRG
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
A
EMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
A
EMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
CCSD
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
A
EMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
CCSD
CCSDT
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
AEMeV
AOE3Max14Λ1.9 fm1
IMSRG
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
AEMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
AEMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
CCSD
10 12 14 16 18 20 22 24 26175
150
125
100
75
50
25
0
AEMeV
AOE3Max14Λ1.9 fm1
IMSRG
ITNCSM
CCSD
CCSDT
NN + 3N-full (400)
Phys. Rev. Lett. 110, 242501 (2013)
Results: Oxygen Chain
• reference state: number-projected Hartree-Fock-Bogoliubov vacuum (pairing correlations)
• consistent results from different many-body methods
NN + 3N-ind.
no refit of3N interaction
exp.
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
12 14 16 18 20 22 24 26
175
150
125
100
75
50
A
EMeV
AO3NMeV ΛSRGfm1350 1.9 ... 2.2400450
Variation of Scales
• variation of initial 3N cutoff only
• diagnostics for chiral interactions
• dripline at A=24 is robust under variationsNN + 3N-full (400)
Phys. Rev. Lett. 110, 242501 (2013)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
34 36 38 40 42 44 46 48 50 52 54 56 58 60 62500
450
400
350
300
250
A
EMeV
ACa
Λfm12.21.92.21.92.21.9
MRIMSRG
CCSD
CRCC2,3
NN + 3N-full (400)
exp.
Calcium and Nickel Isotopes
48505254565860626466687072747678808284868890
700
600
500
400
A
EMeV
ANi
Λfm12.21.92.21.92.21.9
MRIMSRG
CCSD
CRCC2,3
NN + 3N-full (400)
P r e
l i m
i n
a r y
!
• improved free-space 3N SRG evolution in higher partial waves necessary (S. Binder et al., arXiv:1312.5685 [nucl-th])
• calculations for pf-shell nuclei in progress, heavier nuclei in reach
= , =
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
38 40 42 44 46 48 50 52 54 56 58 60 6210
0
10
20
30
40
A
S 2nMeV
ACa
Λfm12.21.9
Calcium and Nickel Isotopes
38 40 42 44 46 48 50 52 54 56 58 60 6210
0
10
20
30
40
A
S 2nMeV
ACa
Λfm12.21.9
Gor'kov prelim. 2.0
NN + 3N-full (400)
exp.
= , =IM-SRG:
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 9010
0
10
20
30
40
50
AS 2
nMeV
ANi
Λfm12.21.9
P r e l i
m i n a r y
!NN + 3N-full (400)
deformation (?)
• 2n separation energies too high (consistent with growing over-binding)
• (sub-)shell closures too pronounced - 3N tensor/spin-orbit ?
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Calcium and Nickel Isotopes
38 40 42 44 46 48 50 52 54 56 58 60 6210
0
10
20
30
40
A
S 2nMeV
ACa
Λ2.2 ... 1.9 fm1
3N MeVc350400
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 9010
0
10
20
30
40
50
AS 2
nMeV
ANi
Λ2.2 ... 1.9 fm1
3N MeVc350400
NN + 3N-full
exp. P r e l i
m i n a r y
!NN + 3N-full
= , =IM-SRG:
• 2n separation energies too high (consistent with growing over-binding)
• (sub-)shell closures too pronounced - 3N tensor/spin-orbit ?
IM-SRG + Shell Model
S. K. Bogner, H. H., J. D. Holt, A. Schwenk, S. Binder, A. Calci, J. Langhammer, R. Roth, arXiv:1402:1407 [nucl-th]
K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev. C 85, 061304(R) (2012)
H. Hergert - The Ohio State University - ESNT Program “Radioactive Ion Beam Experiments and Three-Nucleon Forces”, CEA Saclay, 04/07/14
Valence Space Decoupling
valence particle states
hole states(core)
non-valence particle states
H. Hergert - The Ohio State University - ESNT Program “Radioactive Ion Beam Experiments and Three-Nucleon Forces”, CEA Saclay, 04/07/14
Valence Space Decoupling
• use White-type generator with off-diagonal Hamiltonian
(∞)
= , , , ,
,
,
H. Hergert - The Ohio State University - ESNT Program “Radioactive Ion Beam Experiments and Three-Nucleon Forces”, CEA Saclay, 04/07/14
MBPT NN+3N
IM-SRG NN
IM-SRG NN+3N
Expt.
0
1
2
3
4
5
6
Ener
gy (
MeV
)
23O
5/2+
1/2+
3/2+
1/2+
5/2+
5/2+
1/2+
3/2+
3/2+
1/2+
5/2+
3/2+
(5/2+)
1/2+
(3/2+)
From Oxygen...
MBPT NN+3N
IM-SRG NN
IM-SRG NN+3N
Expt.0
1
2
3
4
5
6
7
8
Ener
gy (
MeV
)
22O
0+
2+
2+
2+
0+
0+
(2+)
2+
2+
(0+)0
+
0+
4+
4+ (4
+)
4+
2+
0+
2+
0+
3+
3+
3+
3+
0+
ℏΩ variation
arXiv: 1402.1407 [nucl-th], [figures by J. Holt]
• 3N forces crucial
• IM-SRG improves on finite-order MBPT effective interaction
• competitive with phenomenological calculations
H. Hergert - The Ohio State University - ESNT Program “Radioactive Ion Beam Experiments and Three-Nucleon Forces”, CEA Saclay, 04/07/14
MBPT USDb IM-SRG NN+3N-full
Expt.
0
1
2
3
4
5
Ener
gy (M
eV)
0+ 0+
2+
4+
0+
2+
2+
2+
2+
0+ 0+ 0+ 0+
4+
2+
2+
4+2+
26Ne
MBPT USDb IM-SRG NN+3N-full
Expt.
0
1
2
3
4
Ener
gy (M
eV)
5/2+
9/2+
5/2+5/2+
9/2+
7/2+
5/2+
5/2+
3/2+
5/2+
3/2+
3/2+ 3/2+
1/2+ 1/2+ 1/2+ (1/2+)
7/2+
3/2+
7/2+
1/2+3/2+
(5/2+)
(3/2+)
25Ne
... Into the sd-Shell
MBPT USDb IM-SRG NN+3N-full
Expt.
0
1
2
3
4
5
Ener
gy (M
eV)
5/2+5/2+
1/2+
5/2+
7/2+
7/2+
3/2+
1/2+
1/2+
9/2+
5/2+
9/2+3/2+
1/2+ 5/2+ 5/2+ (5/2+)
9/2+
1/2+
3/2+ 3/2+
3/2+
(1/2+)
(9/2+)(3/2+)
(3/2+)
(5/2+)
25F
MBPT USDb IM-SRG NN+3N-full
Expt.
0
1
2
3
4
Ener
gy (M
eV)
3+
4+
1+
4+
4+
3+
2+
2+2+
1+ 1+ 1+ 1+
3+
4+2+
2+1+
3+
2+
3+
1+
26F
Conclusions
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Conclusions & Outlook
• IM-SRG is an efficient new Ab-initio framework, suitable for closed- and open-shell, medium-mass & heavy nuclei
• new method for the derivation of shell-model interactions
• easy access to spectra, odd nuclei, intrinsic deformation
• new perspectives for old (?) problems: evolution of long-range correlations, construction of density functionals...
efficient extension to observables (Magnus expansion)
study of medium-mass & heavy isotopic chains with chiral Hamiltonians
excited states
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Three-Nucleon Force Wishlist
• efficiency improvements for production codes
• cutoff variation
• basis sizes & oscillator parameters (e.g., for extrapolation)
• propagation of uncertainties
• input matrix elements: N3LO (...), Δ-full chiral EFT
• proper understanding of consistent NN/3N cutoffs/regularization
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Acknowledgments
S. Bogner, T. MorrisNSCL, Michigan State University
S. Binder, A. Calci, K. Hebeler, J. D. Holt, J. Langhammer, R. Roth, A. SchwenkTU Darmstadt, Germany
R. J. Furnstahl, S. Koenig, S. More, R. J. PerryThe Ohio State University
P. PapakonstantinouIBS / Rare Isotope Science Project, S. Korea
H. Hergert - NSCL, Michigan State University - TRIUMF Theory Seminar, 20/01/11
Acknowledgments
Thanks to my collaborators:
R. Roth, P. Papakonstantinou, A. Günther, S. Reinhardt, S. Binder, A. Calci, J. LanghammerInstitut für Kernphysik, TU Darmstadt
S. BognerNSCL, Michigan State University
Acknowledgments
H. Hergert – NSCL, Michigan State University — Research Discussion, 09/09/2010
Thanks to my collaborators on this work:
• R. Roth, P. Papakonstantinou, A. Gunther, S. Reinhardt,S. Binder, A. Calci, J. LanghammerInstitut fur Kernphysik, TU Darmstadt
• T. Neff, H. FeldmeierGesellschaft fur Schwerionenforschung (GSI)
42
Wednesday, January 19, 2011
Supplements
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Normal-Ordering & Wick’s Theorem
• define elementary contractions of a one-body operator w.r.t. a given reference state as
• define normal-ordered operators recursively through all possible internal contractions:
• Wick’s Theorem: products of normal-ordered operators can be expanded in terms of external contractions alone
...... = : ...
... : +λ : ...... : +
+λ λ − λ λ
: ...
... : + + . . .
: ...... :: ...
... : = (− ) − λ : ... ...... ... :
+ (− ) − ξ : ... ...... ... : + . . .
≡ † , λ ≡
, ξ ≡ λ − δ
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Choice of Generator
• Wegner:
• White: (J. Chem. Phys. 117, 7472)
: approx. 1p1h, 2p2h excitation energies
• “imaginary time”: (Morris, Bogner)
• off-diagonal matrix elements are suppressed like (Wegner), (White), and (imaginary time)
• g.s. energies ( ) differ by
η =
,
−
→ ∞
η =
( ) : : +
( )
:
: +
η =
: : +
: : +
,
−−| |
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Generalized Normal Ordering
• generalized Wick’s theorem for arbitrary reference states(Kutzelnigg & Mukherjee)
• define irreducible n-body density matrices of reference state:
. . .
• irreducible densities give rise to additional contractions:
. . .
ρ = λ + λ λ − λ λ
ρ = λ + λ λ + λ λ λ +
: ...... ::
...... : −→ λ
: ...... ::
...... : −→ λtwo-body flow unchanged,
O(N6) scaling preserved
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow Equations
0-body Flow
=
( − )η − η
+
η − η
¯ ¯
1-body Flow
=
η − η+
η − η( − )
+
η − η( ¯ ¯ + ¯ )
~ 2nd order MBPT for H(s)
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow Equations
2-body Flow
only linked diagrams contribute, IM-SRG size-extensive
=
η + η − η − η − η − η + η + η
+
η − η( − − )
+
( − )
η − η−η − η
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow: Diagrams
(δ ) ∼
( δ ) ∼
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
In-Medium SRG Flow: Diagrams
& manymore...
non-perturbative resummation
(δ ) ∼
( δ ) ∼
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Particle-Number Projected HFB State
• HFB ground state is a superposition of states with different particle number:
• calculate one- and two-body densities (project only once):
• work in natural orbitals (= HFB canonical basis):
=
= , ± ,...
,
≡
≡
π
π
φ φ(ˆ− )
λ =
, λ =
− λ λ + λ λ
λ = δ= δ
, ≤ ≤
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
20 25 30 35 40750
745
740
735
730
725
MeVEMeV
48Ca
Results
converged g.s. energies between CCSD and -CCSD(T)
need 3N forces
[gray lines: CCSD by S. Binder, ]= , =
20 25 30 35 40
596
594
592
590
588
586
584
582
MeV
EMeV
40Ca
VUCOM as a Pairing Force: Sn Isotopes
H. Hergert – NSCL, Michigan State University — CEA Bruyeres-le-Chatel, 27/05/2010
50 54 58 62 66 70 74 78 82N
0
0.5
1
1.5
.
!(3
)n
[MeV
]
! [ fm4]
! 0.03! 0.04" 0.06# 0.10
Gogny D1S +VUCOM
" stability of gaps for wide range of !: stable 1S0 matrix elements
" residual reduction of gaps through contributions from higher partial waves
39
eMax
81012
,λ = . −
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Isotopic “Chains”
• “closed-shell” Ni and Sn isotopes can give insight into isovector interaction...
• ... but complete isotopic chains would be preferable, i.e., devise an approach to open-shell nuclei
20 25 30 35 40660
650
640
630
MeV
EMeV
48Ni
20 25 30 35 40900
890
880
870
860
MeV
EMeV
56Ni
20 25 30 35 401320
1300
1280
1260
1240
1220
1200
MeV
EMeV
78Ni
VUCOM as a Pairing Force: Sn Isotopes
H. Hergert – NSCL, Michigan State University — CEA Bruyeres-le-Chatel, 27/05/2010
50 54 58 62 66 70 74 78 82N
0
0.5
1
1.5
.
!(3
)n
[MeV
]
! [ fm4]
! 0.03! 0.04" 0.06# 0.10
Gogny D1S +VUCOM
" stability of gaps for wide range of !: stable 1S0 matrix elements
" residual reduction of gaps through contributions from higher partial waves
39
eMax
81012
,λ = . − , NN only
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
CCSD/ -CCSD(T), , G. Hagen et al., PRL 109, 032502 (2012) -CCSD(T), , S.Binder et al., arXiv:1211.4748 [nucl-th] & PRL 109, 052501 (2012)
λ = ∞λ = . − . −
6 8 10 12 14
370
360
350
340
330
320
310
eMax
EMeV
Ca40E3Max14
28 MeVΛfm12.22.01.9
Results: Closed-Shell Nuclei
NN + 3N-ind.
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
6 8 10 12 14
500
450
400
350
eMax
EMeV
Ca40E3Max14
28 MeV
NN + 3N-full (400)
6 8 10 12 14
500
450
400
350
eMax
EMeV
Ca40E3Max12
28 MeV
NN + 3N-full (500)
Results: Closed-Shell Nuclei
6 8 10 12 14
500
450
400
350
eMax
EMeV
Ca40E3Max14
28 MeV
Λfm12.22.01.9
NN + 3N-ind.
validate chiral Hamiltonians
CCSD/ -CCSD(T), , G. Hagen et al., PRL 109, 032502 (2012) -CCSD(T), , S.Binder et al., arXiv:1211.4748 [nucl-th] & PRL 109, 052501 (2012)
λ = ∞λ = . − . −
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Extrapolation
20 25 30 35377
376
375
374
MeV
EMeV
40CaE3Max14
=
( + / )
=
( + / )/
max./UV momentum:
radial extent:
simultaneous ultraviolet & infrared extrapolation:
( , ) = ∞ + exp− /
+ exp (− ∞ )
20 25 30 35380
360
340
320
300
280
260
MeV
EMeV
40CaE3Max14
eMax68101214
(R. Furnstahl, G. Hagen & T. Papenbrock, PRC 86,031301 (2012))
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Multi-Reference Flow Equations
0-body flow:
1-body flow:
=
η − η+
η − η( − )
+
η − η( ¯ ¯ + ¯ )
+
η − ηλ +
η − η
λ
−
η − ηλ +
η − η
λ
=
( − )η − η
+
η − η
¯ ¯
+
λ +
η − ηλ
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Multi-Reference Flow Equations
=
η + η − η − η − η − η + η + η
+
η − η( − − )
+
( − )
η − η−η − η
2-body flow:
2-body flowunchanged
H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014
Decoupling
∼ ,
λ ,
λ , . . .
∼
,
λ ,
λ
,
λ
, . . .
∼ . . .
• truncation in irreducible density matrices
• number of correlated vs. total pairs, triples, ... (caveat: highly collective reference states)
• perturbative analysis (e.g. for shell-model like states)
• verify for chosen multi-reference state when possible