Effective interactions for SM calculationsLecture 4: Phenomenology of realistic effective Hamiltonians

E. Caurier, G. Martinez-Pinedo,

F. Nowacki, A. Poves and K. Sieja

ISOLDE Shell Model Course for Non-Practitioners

ISOLDE-CERN, October 2013

F. Nowacki ISMCNP 14-18/10/2013 1 / 30

Effective interaction

Derivation of effective Hamiltonian from realistic NN potentials1 NN potentials2 Renormalization of the NN interactions3 Calculations of effective operators

Monopole and multipole Hamiltonians

Phenomenological corrections to effective Hamiltonians

F. Nowacki ISMCNP 14-18/10/2013 2 / 30

What we need:

We need an effective interaction for SM based on NN force.

We need to renormalize the repulsive part of the NN force.

This can be achieved by calculation of the G-matrix, or Vlowk , SRG

interaction.

The next step is the computation of a model space effective

interaction and/or operator. Such interactions are two-body in

character.

Finally, applications to nuclear systems using Shell Model,

Green’s function methods, Coupled-Cluster etc.

F. Nowacki ISMCNP 14-18/10/2013 3 / 30

Realistic effective interactions

Exp. KB KLS Bonn A Bonn B Bonn C

2+1 excitation energy

44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17

〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437

56Ni model space (f 72p 3

2)16

56Ni 2.70 0.39 0.31 0.43 0.42 0.42

〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019

〈np3/2〉 4.5 5.2 5.7 5.2 5.0

The realistic interactions do not reproduce the shell closure

N or Z=28back

F. Nowacki ISMCNP 14-18/10/2013 4 / 30

Realistic effective interactions

Exp. KB KLS Bonn A Bonn B Bonn C

2+1 excitation energy

44Ca 1.16 1.45 1.43 1.31 1.25 1.2646Ca 1.35 1.45 1.42 1.26 1.22 1.2348Ca 3.83 1.80 1.60 1.23 1.30 1.4150Ca 1.03 1.41 1.35 1.27 1.10 1.17

〈(f 72)8|ΨGS〉 0.468 0.381 0.214 0.345 0.437

56Ni model space (f 72p 3

2)16

56Ni 2.70 0.39 0.31 0.43 0.42 0.42

〈(f 72)16|ΨGS〉 0.04 0.015 0.018 0.011 0.019

〈np3/2〉 4.5 5.2 5.7 5.2 5.0

The realistic interactions do not reproduce the shell closure

N or Z=28back

F. Nowacki ISMCNP 14-18/10/2013 4 / 30

Effective 2-body HamiltoniansImportant features

/ Effective 2-body Hamiltonians derived from realistic potentials fail to

reproduce correctly the spectroscopy of many-body systems and bulk

properties of nuclei-no right binding, no double shell closures...

, The matrix elements depend very little on the potential used

(Argonne, Paris, Bonn, N3LO...) and the method of regularization

(G-matrix, Vlowk , ...). The link to phase shifts is then nearly model

independent.

,/ NN potentials are nowadays "perfect". We need a NNN force.

F. Nowacki ISMCNP 14-18/10/2013 5 / 30

What we can do ?

we have solved the problem of s.p. energies replacing them by the

experimental ones

we can treat all the matrix elements as free parameters and fit

them to the many-body data (e.g. USD, USDb interactions)

or we can proceed in a more general manner: understand the

physics behind the set of the matrix elements and change only

some of them

F. Nowacki ISMCNP 14-18/10/2013 6 / 30

Outline

Separation of the effective Hamiltonian: monopole and multipole

3-body forces: corrections to the monopole Hamiltonian

Some useful definitions

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Separation of the effective HamiltonianMonopole and multipole

From the work of M. Dufour and A. Zuker (PRC 54 1996 1641)

Separation theorem:

Any effective interaction can be split in two parts:

H = Hmonopole +Hmultipole

Hmonopole: spherical mean-field

Zresponsible for the global saturation properties and for the evolution

of the spherical single particle levels.

Hmultipole: correlator

Zpairing, quadrupole, octupole...

Important property:

〈CS±1|H|CS±1〉 = 〈CS±1|Hmonopole|CS±1〉

F. Nowacki ISMCNP 14-18/10/2013 8 / 30

Effective HamiltonianMonopole and Multipole Hamiltonians

V = ∑JT

V JTijkl

[

(a+i a+

j )JT (ak al)

JT]00

In order to express the number of particles operators

ni = a+i ai ∝ (a+

i ai)0,

particle-hole recoupling :

V = ∑λτ

ωλτikjl

[

(a+i ak)

λτ (a+j al)

λτ]00

ωλτikjl ∝ ∑

JT

V JTijkl

i k λ

j l λ

J J 0

12

12 τ

12

12 τ

T T 0

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Effective HamiltonianMonopole and Multipole Hamiltonians

Hmonopole corresponds only to the terms λλλτ=00 and 01 which implies

that i = j and k = l and writes as

Hmonopole = ∑i

niεi +∑i≤j

ni .nj Vij

Hmultipole corresponds to all other combinations of λλλτ

F. Nowacki ISMCNP 14-18/10/2013 10 / 30

Effective HamiltonianMultipole Hamiltonian

Hmultipole can be written in two representations, particle-particle

Hmultipole = ∑ik<jlΓ

W Γijkl [(a

†

ia†

j)Γ(akal )

Γ]0,

where Γ = JT or in particle-hole

Hmultipole = ∑ik<jlΓ

√

(2γ +1)

√

(1+δij )(1+δkl )

4ω

γikjl

[(a†

iak)

γ (a†

jal )

γ ]0,

where γ = λτ

The W and ω matrix elements are related by a Racah transformation:

ωγikjl

= ∑Γ

(−)j+k−γ−Γ

{i j Γl k γ

}

W Γijkl(2J +1)(2T +1),

W Γijkl = ∑

γ

(−)j+k−γ−Γ

{i j Γl k γ

}

ωγikjl

(2λ +1)(2τ +1).

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Effective HamiltonianMultipole Hamiltonian

In the preceding expressions we can replace pairs of indices by a single one ij ≡ x ,

kl ≡ y , ik ≡ a et jl ≡ b, and diagonalise the matrices W Γxy and

fγab

= ωγab

√

(1+δij )(1+δkl )/4, by unitary transformations UΓxk ,u

γak

:

U−1WU = E =⇒ W Γxy = ∑

k

UΓxk UΓ

yk EΓk

u−1fu = e =⇒ fγab

= ∑k

uγak

uγbk

eγk,

then

Hmultipole = ∑k ,Γ

EΓk ∑

x

UΓxk Z+

xΓ ·∑y

UΓyk ZyΓ,

Hmultipole = ∑k ,γ

eγk

(

∑a

uγak

Sγa ∑

b

uγbk

Sγb

)0

[γ]1/2,

that we call representations E and e.

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Effective HamiltonianMultipole Hamiltonian

E-eigenvalue density for the KLS interaction in the pf+sdg major shells hω = 9. Each eigenvaluehas multiplicity [Γ].

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Effective HamiltonianMultipole Hamiltonian

e-eigenvalue density for the KLS interaction in the pf+sdg major shells hω = 9. Each eigenvaluehas multiplicity [γ].

0

5000

-4 -3 -2 -1 0 1 2 3 4 5

num

ber

of s

tate

s

e-eigenvalue density

2 0+4 0+

1 0+ 3 0-

1 1+

1 0-

Energy (Mev)

F. Nowacki ISMCNP 14-18/10/2013 14 / 30

Effective HamiltonianMultipole Hamiltonian

E-eigenvalue density for the KLS interaction in the pf+sdg major shells hω = 9, after removal ofthe five largest multipole contributions. Each eigenvalue has multiplicity [Γ].

0

2000

-10 -8 -6 -4 -2 0 2 4

num

ber

of s

tate

s

1 0+ 0 1+ 2 0+

Energy (Mev)

E-eigenvalue densityepsilon=2.0

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Multipole Hamiltonian

Hmultipole can be written in two representations, particle-particle andparticle-hole. Both can be brought into a diagonal form. When this isdone, it comes out that only a few terms are coherent, and those arethe simplest ones:

L = 0 isovector and isoscalar pairing

Elliott’s quadrupole

~σ~τ ·~σ~τ

Octupole and hexadecapole terms of the type rλ Yλ · rλ Yλ

Besides, they are universal (all the realistic interactions give similar

values) and scale simply with the mass number

Interaction particle-particle particle-holeJT = 01 JT = 10 λτ = 20 λτ = 40 λτ = 11

KB3 -4.75 -4.46 -2.79 -1.39 +2.46FPD6 -5.06 -5.08 -3.11 -1.67 +3.17

GOGNY -4.07 -5.74 -3.23 -1.77 +2.46

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Multipole Hamiltonian

γ eγ1 e

γ2 e

γ1+2 M 〈u1|M〉 〈u2|M〉 〈u1|u

′1〉 〈u2|u

′2〉 α2

11 1.77 2.01 3.90 στ .992 .994 .999 1.000 .9420 -1.97 -2.14 -3.88 r2Y2 .996 .997 1.000 1.000 .9510 -1.02 -0.97 -1.96 σ .880 .863 .997 .994 1.0421 -0.75 -0.85 -1.60 r2Y2τ .991 .998 .999 .997 .9440 -1.12 -1.24 -2.11 r4Y4

Γ EΓ1 EΓ

2 EΓ1+2 P 〈U1|P〉 〈U2|P〉 〈U1|U

′1〉 〈U2|U

′2〉 α2

01 -2.95 -2.65 -5.51 P01 .992 .998 1.000 .994 1.04810 -4.59 -4.78 -10.12 P10 .928 .910 .998 .997 .991

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Effective HamiltonianMonopole Hamiltonian: explicit form

Monopole Hamiltonian in the pn representation gives the average

energy of a configuration at fixed niτ ,njτ ′:

Hmonopole = ∑i ,τ

eiτniτ + ∑ij ,ττ ′

V ττ ′

ij

niτ(njτ ′ −δijδττ ′)

1+δijδττ ′

with

V ττ ′

kl =∑J V

J,ττ ′

klkl (2J +1)(1+(−1)Jδklδττ ′)

(2jk +1)(2jl +1−δklδττ ′)

V ττ ′

kl =1

2[V T=1

kl (1−δkl

2jk +1)+V T=0

kl (1+δkl

2jk +1)], V ττ

kl = V T=1kl ,

where ττ ′=pn.

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Effective HamiltonianMonopole Hamiltonian: explicit form

Monopole Hamiltonian in the isospin representation describes the

average energy of a configuration at fixed niTi :

Hmonopole = ∑i

eini +∑i≤j

aij

ni(nj −δij)

1+δij+∑

i≤j

bij1

1+δij

(

Ti ·Tj −3

4nδij

)

,

with aij =14(3V T=1

ij +V T=0ij ), bij = V T=1

ij −V T=0ij

Monopole interaction in the isospin representation:

V Tij =

∑J V JTijij (2J +1)[1− (−1)J+T δkl ]

∑J(2J +1)[1− (−1)J+T δkl ]

F. Nowacki ISMCNP 14-18/10/2013 19 / 30

Effective HamiltonianMonopole Hamiltonian: explicit form

Monopole Hamiltonian in the isospin representation describes the

average energy of a configuration at fixed niTi :

Hmonopole = ∑i

eini +∑i≤j

aij

ni(nj −δij)

1+δij+∑

i≤j

bij1

1+δij

(

Ti ·Tj −3

4nδij

)

,

with aij =14(3V T=1

ij +V T=0ij ), bij = V T=1

ij −V T=0ij

Monopole interaction in the isospin representation:

V Tij =

∑J V JTijij (2J +1)[1− (−1)J+T δkl ]

∑J(2J +1)[1− (−1)J+T δkl ]

! monopole energies in isospin and in pn formalism are equal ONLY

for closed shell configurations!

F. Nowacki ISMCNP 14-18/10/2013 19 / 30

Effective HamiltonianMonopole Hamiltonian

The evolution of effective spherical single particle energies with the

number of particles in the valence space can be extracted from

Hmonopole. In the case of identical particles the expresion is:

εj(n) = εj(n = 1) + ∑i

V 1ij ni

The monopole Hamiltonian Hmonopole also governs the relative position

of the various T-values in the same nucleus, via the terms:

bij Ti ·Tj

Even small defects in the centroids can produce large changes in the

relative position of the different configurations due to the appearance of

quadratic terms involving the number of particles in the different orbits

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Effective HamiltonianClosed shell vs 4p4h states in 56Ni

ECS = 16 εf + 16∗152 Vff

2p3/22p1/21f5/2P n

1f7/2

r

f

E4p4h = 12 εf + 4 εr

66Vff + 48Vfr + 6Vrr

2p3/22p1/21f5/2P n

1f7/2f

r

∆ E = 4(εf − εr ) + 48(Vff − Vfr ) + 6(Vff − Vrr )

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Effective HamiltonianMonopole Hamiltonian

table

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Effective HamiltonianMonopole Hamiltonian: quadratic effects

Two shells (i and j ) system:

Hmonopole = E0 + ni εi + njεj +ni (ni −1)

2Vii +

nj(nj −1)

2Vjj + ninjVij

It is possible to rewrite this equation and separate a global term H0 (depending only

on the total number of particles n = ni +nj ) from a linear term H1 and a quadratic term

H2 in ni et nj to get:

Hmonopole = E0 + n ε0 +n(n−1)

2W0

︸ ︷︷ ︸

+ Γij [ε1 +(n−1)W1]︸ ︷︷ ︸

+ Γ(2)ij W2︸ ︷︷ ︸

= H0 + H1 + H2

with

Γij =Djni −Dinj

Di +DjΓ(2)ij

=DiDj

2

(2ninj

DiDj−

ni(nj −1)

Di (Dj −1)−

ni (nj −1)

Dj (Di −1)

)

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Effective HamiltonianMonopole Hamiltonian: quadratic effects

nP

1p3/2

1p1/2

1d5/22s1/2

1d3/2

nP

1p3/2

1p1/2

1d5/22s1/2

1d3/2

0 1 2 3 40

5

10

15

20

25

30

35

40

45

lineairequadratique

kp-kh excitation energies for 16O

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3-body forcescorrections to the monopole Hamiltonian

The 2-body potentials are now ’perfect’. Exact calculations are

possible for light systems. The bad spectroscopy has to be then

related with the lack of 3-body forces.

Let’s add 3-body term to the monopole Hamiltonian

Hmonopole = ∑i

eini

︸ ︷︷ ︸

+∑i≤j

aijnij +∑i≤j

bijTij

︸ ︷︷ ︸

+∑ijk

aijknijk +∑ijk

bijkTijk

︸ ︷︷ ︸

,

1-body 2-body 3-body

where

nijk = ninjnk , Tijk = TiTjTk , etc.

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3-body forces I

One should remember that the 2-body force contributes to the 1-body

piece of the effective interaction, when we make summation over orbits

in the core:

∑c

aicninc = ni ∑c

aicnc ≡ niei

Similarly, the 3-body force contributes to 1-body and 2-body terms.

∑c

aijcninjnc = ninj ∑c

aijcnc ≡ ninjaij

∑cc′

aic′cnincncc′ = ni ∑cc′

aicc′ncnc′ ≡ niei

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3-body forces II

ZDifferent studies (no-core, coupled-cluster) suggest that the residual

3-body force is much smaller than 1-and 2-body parts of the 3-body

force. As a first step one should look to the contributions of the 3-body

terms to 1 and 2-body pieces.

Contributions from

3-body force to 4He

binding energy (from

Hagen et al. Phys. Rev.

C 76, 034302, 2007)

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Shell gapDefinition

The shell gap ∆ is defined as the difference of binding energies

(positively defined)

∆ = [BE(N)−BE(N−1)]− [BE(N+1)−BE(N)]

= 2BE(N)−BE(N+1)−BE(N−1)

ZThe uncorrelated shell gap is therefore equal to the difference of the

corresponding ESPE.

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Computing session

This afternoon:

you will learn what a Hamiltonian file contains

you will check yourself how calculated spectra differ depending on

the interaction used (tuned, not tuned)

you will do some mathematics on the paper, please bring

calculators!

you will learn the usage of the option 52 to calculate/change

monopoles

you will "repare" yourself a bad interaction

you will calculate ESPE

F. Nowacki ISMCNP 14-18/10/2013 29 / 30

Further reading

E. Caurier et al., The shell model as a unified view of the nuclear

structure, Rev. Mod. Phys. 77 (2005) 427.

M. Dufour, A.P. Zuker, The realistic collective nuclear Hamiltonian,

Phys. Rev. C52 (1996) 1641.

A.P. Zuker, Separation of the monopole contribution to the nuclear

Hamiltonian, nucl-th/9505012.

A.P. Zuker, Three body monopole corrections to the realistic

interactions, Phys. Rev. Lett.90 (2003) 042502.

A. Schwenk, A.P. Zuker, Shell model phenomenology of low

momentum interactions, Phys. Rev. C74 (2007) 061302.

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