Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Phase-space representation for nuclear potentials
Dennis Weber1,2 , Hans Feldmeier2,3 , Thomas Neff2
1

ExtreMe Matter Institute EMMI and Research Division,
2
GSI Helmholtzzentrum für Schwerionenforschung3Frankfurt Institute for Advanced Studies

ARIS2014
2nd Conference on Advances in Radioactive Isotope ScienceTokyo, Japan
June 1-6, 2014

theoryThomas Neff

ARIS

June 6, 20141

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

SummaryOutline

Nucleon-nucleon interaction
Realistic NN potentials
Unitary transformations and effective interactions
Phase-space representation
Kirkwood representation
Momentum dependence in phase-space representaion
Results for realistic NN potentials

Thomas NeffARIS

June 6, 2014

2Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

NN interaction and correlations
Realistic NN potentials
describing two-nucleon properties (scattering, deuteron) with high accuracy
different potentials available, e.g.
Argonne V18 Wiringa, Stoks, Schiavilla, PRC 51, 38 (1995)
N3 LO from Chiral effectiv field theory Entem, Machleidt, PRC 68, 041001 (2003)
S = 1, T = 0200
AV18VNN [MeV]

100

r12

strong dependence on theorientation of the spins due to the tensor
force → tensor correlations

0the nuclear force will induce strong
short-range correlations in the nuclear
wave function

r12-100-200
0.0

0.5

1.0

1.5

repulsive core:
nucleons can not get closer than ≈ 0.5
fm→ central correlations

2.0

2.5

3.0

r12 [fm]

Thomas Neff

ARIS

June 6, 20143

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

SummaryNN interaction and correlations
Realistic NN potentials
describing two-nucleon properties (scattering, deuteron) with high accuracy
different potentials available, e.g.
Argonne V18 Wiringa, Stoks, Schiavilla, PRC 51, 38 (1995)
N3 LO from Chiral effectiv field theory Entem, Machleidt, PRC 68, 041001 (2003)
S = 1, T = 0
200
AV18VNN [MeV]

100

r12

strong dependence on theorientation of the spins due to the tensor
force → tensor correlations

0the nuclear force will induce strong
short-range correlations in the nuclear
wave function

r12-100-200
0.0

0.5

1.0

1.5

repulsive core:
nucleons can not get closer than ≈ 0.5
fm→ central correlations

2.0

2.5

3.0

[fm]
User12unitarytransformations to obtain “soft” effective
realistic interaction

Thomas Neff

ARIS

June 6, 20143

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Unitary Correlation Operator Method
NN interaction induces strong central and tensor correlations→ Many-body methods working with (superpositions of) Slater determinantsrequire huge model space sizes
Unitary Correlation Operator Method (UCOM):
Roth, Neff, Feldmeier, Prog. Part. Nucl. Phys. 65, 51 (2010)

Unitary transformation C imprints correlation on simple model state |Ψ〉|Ψ̂〉 = C |Ψ〉 = CΩ Cr |Ψ〉Work with transformed operators and simple model states, e.g.
〈Ψ̂| H |Ψ̂0 〉=〈Ψ| C† HC |Ψ0 〉 =: 〈Ψ| Heﬀ |Ψ0 〉Heﬀ for local NN potential (e.g. Argonne V18) contains quadratic momentum
dependence replacing short-range repulsion and short-range tensor:
 2~~2pp†
Heﬀ = Cr+ V(r) Cr =+ Veﬀ (r, p)
2μ2μWeber, Feldmeier, Hergert, Neff, Phys. Rev. C 89, 034002 (2014)

Thomas Neff

ARISJune 6, 20144

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Similarity Renormalization Group
Similarity Renormalization Group (SRG):
Bogner, Furnstahl, Perry, Phys. Rev. C, 75, 061001 (2007)

Evolve Hamiltonian and unitary transformation matrixdHαdα= (2μ)2 [ [Tint , Hα ], Hα ]→Hα = U†α HUαUnitary transformation Uα to obtain “soft” effective realistic interaction
SRG drives the Hamiltonian towards a band-diagonal structure
Performed in matrix element representation → momentum dependence ?
AV18: 〈k(L0)J; T| Vα |k 0 (L0)J; T〉 in MeVfm3

α = 0.00fm4 (bare)Thomas Neff

ARIS

June 6, 20145

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

SummarySimilarity Renormalization Group
Similarity Renormalization Group (SRG):
Bogner, Furnstahl, Perry, Phys. Rev. C, 75, 061001 (2007)

Evolve Hamiltonian and unitary transformation matrixdHαdα= (2μ)2 [ [Tint , Hα ], Hα ]→Hα = U†α HUαUnitary transformation Uα to obtain “soft” effective realistic interaction
SRG drives the Hamiltonian towards a band-diagonal structure
Performed in matrix element representation → momentum dependence ?
AV18: 〈k(L0)J; T| Vα |k 0 (L0)J; T〉 in MeVfm3

α = 0.01fm4
Thomas NeffARIS

June 6, 20145

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Similarity Renormalization Group
Similarity Renormalization Group (SRG):
Bogner, Furnstahl, Perry, Phys. Rev. C, 75, 061001 (2007)

Evolve Hamiltonian and unitary transformation matrix
dHαdα= (2μ)2 [ [Tint , Hα ], Hα ]→Hα = U†α HUαUnitary transformation Uα to obtain “soft” effective realistic interaction
SRG drives the Hamiltonian towards a band-diagonal structure
Performed in matrix element representation → momentum dependence ?
AV18: 〈k(L0)J; T| Vα |k 0 (L0)J; T〉 in MeVfm3

α = 0.04fm4
Thomas Neff

ARIS

June 6, 20145

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Representation of NN interactions

effective realistic NN potentials given usually as momentum matrix elements
can be used in many-body calculations with shell model or plane wave basis
but not for Fermionic Molecular Dynamics and cluster models
matrix element representation not intuitive, not transparent

Other representation to study and visualize NN potentials
How does the potential look in position space?
What happens to the repulsive core?
What is the range of the interaction?
What is the momentum dependence?

⇒ Phase-space representation

Thomas Neff

ARIS

June 6, 20146

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Phase-space representation
Kirkwood representation Kirkwood, Phys. Rev., 44, 31 (1933)
phase-space distribution for a given state |ϕ〉:~ = (2π)3/ 2 〈~~ p|~~ r〉ƒps (~r , p)r |ϕ〉〈ϕ|p〉〈phase-space representation of an operator O:~ = (2π)3/ 2 〈~~ p|~~ r〉
Ops (~r , p)r | O |p〉〈quantum expectation value (analogue to classical expression)
ZZ
O = 〈ϕ| O |ϕ〉 =d3 r∗~ Ops (~~d3 p ƒps(~r , p)r , p)

Study phase-space representation of effective NN interactions
Multipole expansion with Legendre polynomials PΛ :X
Ò~ =~ · p).
~Λ VΛ (r, p) PΛ (rbVps (~r , p)
ΛThomas NeffARISJune 6, 20147

Outline

NN InteractionUnitary Transformations

Phase-space Representation

Summary

Phase-space representation of local V = V(r)VΛ (r, p) (arb. units)nV(r) = exp −r2
2fm2olocal: V = V(r)→Thomas Neff

~Vps (~r , p)

ARIS=V(r)=Ò~ · p)~V(r)P0 (rbJune 6, 20148

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary~ 2 V(r) + V(r)p~2
Phase-space representation of V = 12 p

VΛ (r, p) (arb. units)nV(r) = exp −~ 2 V(r) + V(r)p~2quadratic momentum dependence: V = 12 p~→ Vps (~r , p)==Thomas NeffV(r) p2 −V(r) p2 −
ARIS

1212

V 00 (r) −V 00 (r) −V 0 (r)−r
V 0 (r)r

V 0 (r)r

r22fm2o~r~· pÒ~ · p)~ −
P0 (rbV 0 (r)June 6, 2014r

Ò~ · p)~rp P1 (rb9Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary~2
Phase-space representation of V = V(r) LVΛ (r, p) (arb. units)nV(r) = exp −r22fm2o~2quadratic angular momentum dependence: V = V(r) L~→ Vps (~r , p)==Thomas Neff

~ 2 + 2V(r) r~· p~V(r) (~r × p)
23ÒÒ
~ · p)~ + 2V(r)rp P1 (rb~ · p)
~ −V(r) (rp)2 P0 (rbARIS

23June 6, 2014

Ò~ · p)~V(r) (rp)2 P2 (rb10Outline

NN Interaction

Unitary Transformations

Phase-space Representation

SummaryBare potentials in phase-space representation
S=0, T=1: VΛ (r, p) in MeV
AV18N3 LO

Thomas NeffARISJune 6, 201411

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

UCOM Argonne potential in phase-space representation
S = 0, T = 1: VΛ (r, p) in MeV
AV18

AV18 UCOM

Thomas Neff

ARIS

June 6, 201412

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

SRG Argonne potential in phase-space representation
S = 0, T = 1: VΛ (r, p) in MeV
AV18

AV18 SRG

Thomas Neff

ARIS

June 6, 201413

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

SRG potentials in phase-space representation
S = 0, T = 1: VΛ (r, p) in MeV
AV18 SRG

N3 LO SRG

Thomas Neff

ARIS

June 6, 201414

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

SummaryLocal projection
Wendt et al. introduced the local projection of a potential
Wendt, Furnstahl, Ramanan, Phys. Rev. C 86, 014003 (2012)

This is the same as the phase space representation for Λ = 0 and p = 0:Voc (r) = VΛ=0 (r, p = 0)150150100bare Argonnebare N3LO

UCOMH0400L

N3LO, SRG Α=0.04 fm 4

100

SRG Α=0.01 fm 4

N3LO, SRG Α=0.20 fm 4Argonne, SRG Α=0.20 fm 4

SRG Α=0.20 fm 450

0-50-100

1,0V ps
Hr ,0L @MeVD

1,0V ps
Hr ,0L @MeVD

SRG Α=0.04 fm 4

500-500.0

Thomas Neff

0.5

1.0

1.5
r @fmD

2.0

2.5

ARIS

3.0

-100

0.0

0.5

1.0

1.5
r @fmD

June 6, 2014

2.0

2.5

3.015

Outline

NN Interaction

Unitary Transformations

Phase-space Representation

Summary

Summary

Realistic effective interactions
UCOM and SRG soften the interaction by unitary transformationtransformed interactions are non-local
no “intuitive” picture in partial wave matrix element representation
Phase-space Representation
investigate momentum-dependence
non-locality reflected in p- and Λ-dependence
form of regulators reflected in N3 LO phase-space representation
AV18 UCOM has quadratic momentum-dependence
AV18 and N3 LO SRG have more complicated momentum-dependence
Outlookextend phase-space representation to S = 1 channels
what can we learn for the operator representation for effective interactions

Thomas NeffARIS

June 6, 201416