Pairing with Correlated Realistic NN Interactions Heiko Hergert Institut f ¨ ur Kernphysik, TU Darmstadt 1
Pairing with Correlated
Realistic NN Interactions
Heiko HergertInstitut fur Kernphysik, TU Darmstadt
1
Overview
The Unitary Correlation Operator Method
Hartree-Fock-Bogoliubov
• Particle-Number Projection
• Inclusion of 3N -Forces
Summary & Outlook
2
Motivation
Argonne V18 Deuteron Solution
MS = 01√2(∣∣↑↓⟩+∣∣↓↑⟩)
MS = ±1∣∣↑↑⟩,∣∣↓↓⟩
ρ(2)1,MS
(~r)
central correlations:two-body density is suppressed atlow distances
tensor correlations:angular distribution depends onthe relative spin alignments
use very large many-bodyHilbert spaces
⇒ high computationaleffort
use numerically affordableHilbert spaces and
treat strong correlationsexplicitly
or
.Central Correlator Cr
radial distance-dependent shift in therelative coordinate of a nucleon pair
Cr = exp(−iA∑
i,j
gr,ij[s(rij)])
Tensor Correlator CΩ
angular shift, depending on orientationof spin and relative coordinate
CΩ = exp(−iA∑
i,j
gΩ,ij[ϑ(rij)])
s(r) and ϑ(r) encapsulate thephysics of short-range correlations
3
Beyond Hartree-Fock
Hartree-Fock(Mean-Field)
Long-RangeCorrelations
perturbation theory,Brueckner-HF(HK 27.4),RPA(HK 32.9) ...
PairingCorrelations
Hartree-Fock-Bogoliubov
4
HFB Theory Overview
Bogoliubov Transformation
β†k =
∑
q
Uqkc†q + Vqkcq
βk =∑
q
U∗qkcq + V ∗
qkc†q
where βk, βk′
!=β†
k, β†k′
!= 0
βk, β†
k′
!= δkk′
HFB Densities & Fields
ρkk′ ≡⟨Ψ∣∣ c†
k′ck
∣∣Ψ⟩
= (V ∗V T )kk′
κkk′ ≡⟨Ψ∣∣ ck′ck
∣∣Ψ⟩
= (V ∗UT )kk′
Γkk′ =∑
qq′
(2
Atrel + v
)
kq′,k′q
ρqq′
∆kk′ =∑
qq′
(2
Atrel + v
)
kk′,qq′
κqq′
Energy
E[ρ, κ, κ∗] =
⟨Ψ∣∣H
∣∣Ψ⟩
⟨Ψ∣∣Ψ⟩ ≡
1
2(tr Γρ − tr ∆κ∗)
HFB Equations
(H − λN )
(U
V
)≡
(Γ − λ ∆
−∆∗ −Γ∗ + λ
)(U
V
)= E
(U
V
)
5
Particle Number Projection
Projected Energy
E(N0) =
⟨Ψ∣∣HPN0
∣∣Ψ⟩
⟨Ψ∣∣PN0
∣∣Ψ⟩ =
1
2π⟨PN0
⟩∫ 2π
0
dφ⟨Ψ∣∣Heiφ(N−N0)
∣∣Ψ⟩
Variation of Projected Energy
δE(N0) =1
2π⟨PN0
⟩∫ 2π
0
dφ⟨eiφ(N−N0)
⟩ δ⟨H⟩
φ−(E(N0) −
⟨H⟩
φ
)δ log
⟨eiφN
⟩
⟨H⟩
φ≡⟨HeiφN
⟩/⟨eiφN
⟩
Variation of Projected Energy
δE(N0) =1
2π⟨PN0
⟩∫ 2π
0
dφ⟨eiφ(N−N0)
⟩ δ⟨H⟩
φ−(E(N0) −
⟨H⟩
φ
)δlog
⟨eiφN
⟩
⟨H⟩
φ≡⟨HeiφN
⟩/⟨eiφN
⟩
Lipkin-Nogami + PAV
power series expansion
expansion coefficients not varied indeterminate / numerically unsta-
ble at shell closures
exact PNP after variation
VAP
higher (but managable) computa-tional effort
implement with care: subtle can-cellations between divergencesof direct, exchange, and pairingterms
Structure of HFB equations is preserved by both methods!
Flocard & Onishi, Ann. Phys. 254, 275 (1997, approx. PNP); Sheikh et al., Phys. Rev. C66, 044318 (2002, exact PNP)
6
Implementation
vary intrinsic energy Hint = H − Tcm
project on proton and neutron numbers simultaneously:PN0Z0
= PN0PZ0
consistent treatment of direct, exchange & pairing terms
VUCOM in ph- and pp-channel !
consistent treatment of direct, exchange & pairing terms
VUCOM in ph- and pp-channel !
include (anti-)pairing effects from intrinsic kinetic energy andCoulomb interaction
consistent treatment of direct, exchange & pairing terms
VUCOM in ph- and pp-channel !
include (anti-)pairing effects from intrinsic kinetic energy andCoulomb interaction
this level of consistency is crucial for particle-number projection
7
Sn Isotopes: Binding & Pairing Energies
-8
-7
-6
-5
-4
.
E/A
[M
eV]
ASn
100 104 108 112 116 120 124 128 132
A
-6
-4
-2
0
.
Ep
air[
MeV
]
emax = 14, lmax = 10 exp. HFB
8
Sn Isotopes: Binding & Pairing Energies
-8
-7
-6
-5
-4
.
E/A
[M
eV]
ASn
100 104 108 112 116 120 124 128 132
A
-6
-4
-2
0
.
Ep
air[
MeV
]
ASn
emax = 14, lmax = 10 exp. HFB PAV LN+PAV N VAP
9
3N Forces: HF Single-Particle Energies
-40
-30
-20
-10
0
10
.
ǫ[
MeV
]
100Sn
Iϑ [ fm3]
C3N [ GeV fm6]
0.09
–
0.20
2.5
Exp.
protons
0.09
–
0.20
2.5
Exp.
neutrons
occupied unoccupied
3N forceimproves
level density!(HK 32.8)
10
3N Forces: Pairing
A112 116 120 124
-8
-6
-4
-2
0
.
Ep
air[
MeV
] emax = 10, lmax = 8
HFB PAV
LN+PAV N VAP
v[2]kk′,qq′ → v
[2]kk′,qq′ + f ·
∑
rr′
v[3]kk′r,qq′r′ρHF
r′r , f =
13
expect. values12
fields
phenomenological VAP calculations: Epair ≃ 10 − 20 MeV(Stoitsov et al., nucl-th/0610061; Anguiano et al., Phys. Lett. B545 (2002), 62)
“correct” orderof magnitudewith realistic
NN int.
11
Summary
fully consistent HFB calculations with particle number projection,based on a Hamiltonian
using VUCOM — a “universal” phase-shift equivalent NN interaction
inclusion of 3N -forces fixes the two-body VUCOM’s problems withsingle-particle level density (→ A. Zapp, HK 32.8)
HFB results are encouraging
⇒contribute to a consistent description of different aspects ofnuclear structure based on VUCOM: HF(B), (Q)RPA, SRPA, MBPT,FMD, NCSM...
12
Outlook
Mean-Field(Hartree-Fock)
Long-RangeCorrelations
perturbation theory,Brueckner-HF(HK 27.4),RPA(HK 32.9) ...
PairingCorrelations
Hartree-Fock-Bogoliubov
QRPA
13
Outlook
Quasiparticle RPA (benchmark calculations in progress)
investigation of pn pairing & pn-QRPA
⇒ Isobaric Analog & Gamow-Teller Resonances
“deformed” UCOM-HFB (in progress)
symmetry restoration by projection (isospin, parity, angu-lar momentum)
14
Epilogue...
My Collaborators
R. Roth, P. Papakonstantinou, A. Zapp, P. HedfeldInstitut fur Kernphysik, TU Darmstadt
T. Neff, H. FeldmeierGesellschaft fur Schwerionenforschung (GSI)
N. PaarDepartment of Physics — Faculty of Science, University of Zagreb, Croatia
References
H. Hergert, R. Roth, nucl-th/0703006, submitted to Phys. Rev. C
N. Paar, P. Papakonstantinou, H. Hergert, and R. Roth, Phys. Rev. C74, 014318 (2006)
R. Roth, P. Papakonstantinou, N.Paar, H. Hergert, T. Neff, and H. Feldmeier, Phys. Rev. C73, 044312(2006)
http://crunch.ikp.physik.tu-darmstadt.de/tnp/
15
Optional
16
Correlated Interaction
Correlated Hamiltonian
H=T[1]+T[2]+V[2]+T[3]+V[3] +. . .
Correlated Hamiltonian
H = T[1] + VUCOM + V[3]UCOM + . . .
closed operator representation of VUCOM
in two-body approximation
⇒usable with arbitrary many-body basis
closed operator representation of VUCOM
in two-body approximation
⇒usable with arbitrary many-body basis
VUCOM is phase-shift equivalent to the un-derlying bare nucleon-nucleon interaction
closed operator representation of VUCOM
in two-body approximation
⇒usable with arbitrary many-body basis
VUCOM is phase-shift equivalent to the un-derlying bare nucleon-nucleon interaction
VUCOM is pre-diagonalized in momentumspace, i. e. high-momentum componentsare decoupled (similar to Vlow-k)
AV18
02
46
0
2
4
6-40
-20
0
20
24
6
02
46
0
2
4
6-40
-20
0
20
24
6
q [fm−1]
q′
[fm−1]
3S1
Vbare
VUCOM
17
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(4
He)
[MeV
]
AV18Nijm II
Nijm I
CD Bonn
VNN + V3N
Exp.
Tjon-line : E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
Data points: A. Nogga et al., Phys. Rev. Lett. 85, 944 (2000)
18
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(4
He)
[MeV
]
AV18Nijm II
Nijm I
CD Bonn
VUCOM(AV18)Exp.
increasingCΩ-range
Tjon-line : E(4He) vs. E(3H)for phase-shift equivalent NN-interactions
change of CΩ-correlator rangeresults in shift along Tjon-line
minimize netthree-body force
by choosing correlatorwith energies close to
experimental value
VUCOM
for calc. without3N force
Data points: A. Nogga et al., Phys. Rev. Lett. 85, 944 (2000)
19
4He: Convergence
AV18
20 40 60 80~ω [MeV]
-20
0
20
40
60
.
E[M
eV]
Nmax
0 2 4
6
8
10121416EAV18
NCSM code by P. Navratil [PRC 61, 044001 (2000)]
VUCOM
20 40 60 80~ω [MeV]
-20
0
20
40
60
.E
[MeV
]
4He
20 40 60 80
-30
-25
-20
-15
-10
omitted higher-ordercluster contributions
20