Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Skyrme EDFfor beyond mean-field calculations

K. Bennaceur

Université Claude Bernard Lyon 1,Institut de Physique Nucléaire de Lyon, CNRS/IN2P3,

F-69622 Villeurbanne cedex, France.

THEXO Workshop, ECT*, July 2013

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Outline

Skyrme mean-field and beyond mean-field calculations formasses and excitations

Standard effective Skyrme interaction: E = 〈T + V2(ρ0)〉

Standard effective EDF: E 6= 〈T + V2(ρ0)〉

Constaints from beyond mean-field calculations

Constaints from mean-field calculations

New Skyrme effective interaction: E = 〈T + V2 + V3+V4〉

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Microscopic calculation of contributions of some correlationson ground states energies

Mean-field approximation relies on a very simple wave functionfor the ground state of the system

Some beyond mean-field effects might be absorbed in aneffective way by refitting the coupling constants of the EDF orby using schematic corrections

Some ground state correlations can be microscopicallycalculated by going beyond the mean-field approximation by

the mechanism of symmetry breaking and restauration

the mixing of states along collective coordinates (GCM)

See M. Bender, G.F. Bertsch and P.-H. Heenen, PRC 73, 034322:

How large are these correlation energies ?

How much do they fluctuate ?

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

The standard (2-body) Skyrme interaction

Effective Skyrme interaction

Veff(r) = t0(

1 + x0Pσ)

δ(r) local

+ 12 t1

(

1 + x1Pσ)[

k′2 δ(r) + δ(r)k

2]

non local

+ t2(

1 + x2Pσ)

k′ · δ(r)k non local

+ 16 t3

(

1 + x3Pσ)

ρα

0 δ(r) density dep.

+ iW0 σ ·[

k′ × δ(r)k

]

spin-orbit

Sometimes complemented with a tensor term

Possibly complemented with a D-wave term

Higher order derivative terms ?

Other density dependent terms ?

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

From effective interaction to EDF (« T even » part)

H = K + H0 + H3 + Heff + Hfin + Hso + Hsg + Hcoul

H0 =1

4t0

[

(2 + x0)ρ20 − (2x0 + 1)

∑

q

ρ2q

]

=∑

T=0,1

CT [ρ0]ρ2T

H3 =1

24t3ρα

0

[

(2 + x3)ρ20 − (2x3 + 1)

∑

q

ρ2q

]

Heff =1

8[t1(2 + x1) + t2(2 + x2)]τ0ρ0 =

∑

T=0,1

Cτ

T τTρT

+1

8[t2(2x2 + 1) − t1(2x1 + 1)]

∑

q

τqρq

Hfin =1

32[t2(2 + x2) − 3t1(2 + x1)]ρ0∆ρ0 =

∑

T=0,1

C∆ρ

TρT∆ρT

+1

32[3t1(2x1 + 1) + t2(2x2 + 1)]

∑

q

ρq∆ρq

Hso = −W0

2

[

ρ0∇ · J0 +∑

q

ρq∇ · Jq

]

=∑

T=0,1

C∇JT ρT ∇ · JT

Hsg = −t1x1 + t2x2

16J

20 +

t1 − t2

16

∑

q

J2q =

∑

T=0,1

CJT J

2T

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

From effective interaction to EDF (« T even » part)

H = K + H0 + H3 + Heff + Hfin + Hso + Hsg + Hcoul

H0 =1

4t0

[

(2 + x0)ρ20 − (2x0 + 1)

∑

q

ρ2q

]

=∑

T=0,1

CT [ρ0]ρ2T

H3 =1

24t3ρα

0

[

(2 + x3)ρ20 − (2x3 + 1)

∑

q

ρ2q

]

Heff =1

8[t1(2 + x1) + t2(2 + x2)]τ0ρ0 =

∑

T=0,1

Cτ

T τTρT

+1

8[t2(2x2 + 1) − t1(2x1 + 1)]

∑

q

τqρq

Hfin =1

32[t2(2 + x2) − 3t1(2 + x1)]ρ0∆ρ0 =

∑

T=0,1

C∆ρ

TρT∆ρT

+1

32[3t1(2x1 + 1) + t2(2x2 + 1)]

∑

q

ρq∆ρq

Hso = −W0

2

[

ρ0∇ · J0 +∑

q

ρq∇ · Jq

]

=∑

T=0,1

C∇JT ρT ∇ · JT

Hsg = −t1x1 + t2x2

16J

20 +

t1 − t2

16

∑

q

J2q =

∑

T=0,1

CJT J

2T : SLy4

C∇J0 6= 3C∇J

1 : SLy10

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

From effective interaction to EDF: further simplifications

Different interaction used in the pairing channel

Vp(r) = V0

[

1 −

(

ρ0(r)

ρsat

)α′]

δ(r)

Unconvenient terms sometimes disregarded: s0 · ∆s0 , s1 · ∆s1 .

Fit of the parameters on infinite nuclear matter properties on(ground state) properties of (doubly magic) nuclei

Good results for ground state properties of even-even nuclei(but how good is good ?)

Encouraging constrained HF and beyond mean-field results

Several annoying technical questions

Coupling constants of the time odd part of the functional Calculations sometimes do not converge for some functional How to deal with the density dependent terms in beyond

mean-field calculations ?

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Mean field and beyond

Beyond mean field calculations with a Skyrme EDF

Poles

and steps

in theprojected

energy

See: PRC 79, 044318, 044319 and 044320.

⇒ Density dependent term must be droped...

⇒ Three-body interaction ?

⇒ Four-body interaction ?

⇒ In Hartree, Fock and pairing terms...

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

New strong constraints on the EDF

The EDF must be derived from an interaction: E = 〈T + V 〉

No density dependence

All terms kept in the functional (Hartree, Fock and pairing)

Must give attractive pairing

But that’s not all:

First, mean-field calculations must give converged results

See:

A. Pastore, D. Davesne, K.B., J. Meyer and V. Hellemans,Phys. Scr. 2013, 014014.

A. Pastore, K.B., D. Davesne, J. Meyer,Int. J. of Modern Phys. E Vol. 21, No. 05, 1250040.

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Finite size instabilities in nuclei

Instabilities often experienced with the skyrme functionals

Ferromagnetic instabilities: (spin polarization) n ↑, p ↑ Isospin instabilities: neutron-proton segregation

Both: n ↑, p ↓

Example: isospin instability in 48Ca

0

0.02

0.04

0.06

0.08

0.1

0.12

-8 -6 -4 -2 0 2 4 6 8

[fm

-3]

r [fm]

np

0

0.02

0.04

0.06

0.08

0.1

0.12

-8 -6 -4 -2 0 2 4 6 8

[fm

-3]

r [fm]

np

0

0.02

0.04

0.06

0.08

0.1

0.12

-8 -6 -4 -2 0 2 4 6 8

[fm

-3]

r [fm]

np

C∆ρ

1= 15 MeV fm5 25 MeV fm5 35 MeV fm5 & 36 MeV fm5

∼ SLy5

T. Lesinski, K.B., T. Duguet, J. Meyer, PRC 74, 044315 (2006).

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Linear response – Instabilities in infinite nuclear matter

Response of the system to a perturbation given by

Q(α) =∑

a

eiq·ra Θ

(α)a ,

Θssa = 1a , Θvs

a = σa, Θsva = ~τa, Θvv

a = σa~τa

Response functions are given by

χ(α)(ω,q) =1

Ω

∑

n

|〈n|Q(α)|0〉|2(

1

ω − En0 + iη−

1

ω + En0 − iη

)

(Cf. C. Garcia–Recio et al., Ann. of Phys. 214 (1992) 293–340 )

• Predicts instabilitiesin finite size systems

• Easy to implement

• Negligible computation time

• Might be crucial with tensor,3- or 4-body terms

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Linear response as a tool for diagnosis

Pole of the response at E = 0 ≡ instability

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3 3.5 4

S=0,T=1sat. density

k

[

fm−1

]

ρ

[

fm−

3]

• T. Lesinski, K.B., T. Duguet, J. Meyer, PRC 74, 044315 (2006);

• D. Davesne, M. Martini, K.B., J. Meyer, Phys. Rev. C80, 024314 (2009),erratum: Phys. Rev. C 84, 059904(E) (2011).

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Linear response as a tool for diagnosis

Pole of the response at E = 0 ≡ instability

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3 3.5 4

S=0,T=1sat. density

k

[

fm−1

]

ρ

[

fm−

3]

at ρ ∼ 0.3 fm−3

appearance of domains

(S =0,T =1) with size ∼2π

2.7

• T. Lesinski, K.B., T. Duguet, J. Meyer, PRC 74, 044315 (2006);

• D. Davesne, M. Martini, K.B., J. Meyer, Phys. Rev. C80, 024314 (2009),erratum: Phys. Rev. C 84, 059904(E) (2011).

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

“Standard” EDFs, instabilities and Murphy’s law

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ [

fm-3

]

q [fm-1]

SLY5 - SNMS = 0, M = 0, I = 0S = 0, M = 0, I = 1S = 1, M = 0, I = 0S = 1, M = 0, I = 1S = 1, M = 1, I = 0S = 1, M = 1, I = 1

Murphy’s law: “Anything that can go wrong will go wrong”.

Instabilities are very difficult to detect with a code working on anharmonic oscillator basis

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

New Skyrme effective interaction for mean-field and beyondmean-field calcualtions

We need

An interaction (no density dependence)

That can be used in all channels (attractive pairing)

Stable in all spin/isospin channel

Previous work (thesis of J. Sadoudi):

Finite size instability can be avoided and correctreproduction of masses can be achieved with 2- and 3-bodyterms

A 4-body might help

Properties in the pairing channel not considered at thattime.

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Skyrme effective interaction with 2-, 3- and 4-body terms

(Cf J. Sadoudi thesis, CEA Saclay)

Two-body effective interaction

Veff = t0(

1 + x0Pσ)

δ(r) local

+ 12 t1

(

1 + x1Pσ)(

k′2 δ(r) + δ(r)k

2)

non local

+ t2(

1 + x2Pσ)

k′ · δ(r)k non local

+ iW0 σ ·[

k′ × δ(r)k

]

spin-orbit

Complemented it with

3u0 δ(r12)δ(r13)

+ 32 u1

(

1 + y1Pσ)

[

δ(r12)δ(r13)k212 + k

′212 δ(r12)δ(r13)

]

+3u2

(

1 + y21Pσ

12 + y22Pσ

13

)

k′12 · δ(r12)δ(r13)k12

+v0 δ(r12)δ(r13)δ(r14)

And possibly: tensor, D-wave and 3-body spin-orbit...

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Interactions with 2-, 3- and 4-body terms

Stability: How do the 3- and 4-body terms change the picture ?

→ They generate nicer figures !

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ [

fm-3

]

q [fm-1]

2+3+4 - SNMS = 0, M = 0, I = 0S = 0, M = 0, I = 1S = 1, M = 0, I = 0S = 1, M = 0, I = 1S = 1, M = 1, I = 0S = 1, M = 1, I = 1

No hint in the Landau parameters !

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Tentative fit with simplified interaction: SLyMR0

See: Phys. Scr. 2013 014013J. Sadoudi, M. Bender, K.B., D. Davesne, R. Jodon, T. Duguet

3- and 4-body terms reduced to

3u0 δ(r12)δ(r13) + v0 δ(r12)δ(r13)δ(r14)

Infinite nuclear matter properties

ρsat = 0.152 fm−3, E/A = −15.04 MeV, K∞ = 264.2 MeV, m∗/m = 0.47, J = 23 MeV.

Allows to test the mean-field and beyond mean-field machinery:Beyond mean-field calculations (B. Bally, M. Bender) inprogress...

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

SLyMR0: Results

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

SLyMR0: Results for 25Mg

5

2

Calculation for odd-even nucleus with blocking andprojection on N, Z and J

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Work in progress: SLyMR1 interaction

Interaction with 2-, 3- and 4-body terms

Used both in the ph and pp channels

Perfeclty stable !

Incompressibility too high for the moment

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

ρ [

fm-3

]

q [fm-1]

SNMS = 0, M = 0, I = 0S = 0, M = 0, I = 1S = 1, M = 0, I = 0S = 1, M = 0, I = 1S = 1, M = 1, I = 0S = 1, M = 1, I = 1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

q [fm-1]

PNMS = 0, M = 0S = 1, M = 0S = 1, M = 1

ρsat = 0.16 fm−3 E/A = −16.1 MeV K∞ = 315 MeV m∗/m = 0.62

aI = 28.8 MeV L = 49.7 MeV Q = −376 MeV ∆m∗

m= 0.29 > 0

F0 = −0.12 F ′

0 = 0.46 G0 = −0.36 G′

0 = 0.26F1 = −1.14 F ′

1 = 1.37 G1 = −0.63 G′

1 = 0.16

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Properties of SLyMR1

Attractive pairing but too weak...

Pairing is built from all terms of the interaction, where theattraction comes from ? What can be tuned to enhance it ?

What about surface properties ?

How the 2- and 3-body gradient terms act on the surface ?→ Constraint on surface energy will be added

Four-body term: is it really needed ?Life is complicated and the four-body term may have driven usto a bad region of parameters→ Attempt to make a fit without the 4-body term.

Skyrme EDF for

beyond MF

K. Bennaceur

Introduction

Skyrme EDFs

New constaints

New Skyrme

interaction

Collaboration

• IPN Lyon: K. Bennaceur, D. Davesne, R. Jodon, J. Meyer

• CENBG: B. Avez, B. Bally, M. Bender, J. Sadoudi

• IRFU: T. Duguet

• ULB: V. Heelemans, P.H. Heenen, A. Pastore, M. Martini

• UW: T. Lesinski