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