Nuclear density functional theory with a semi-contact 3-body interaction

Denis Lacroix IPN Orsay

Outline

Infinite matter Results

Energy density function and symmetry breaking

Symmetry restoration and configuration mixing: difficulties within EDF

Problems with ra terms in Beyond Mean-Field calculation.

Proposal of a new EDF based on semi-contact three-body interaction

Collaboration: M. Bender, K. Bennaceur, T. Duguet. G. Hupin

Mean-Field Theories

Configuration Mixing within Energy Density Functional

Sly4-Bender,Bertsch,Heenen, PRL (2005)

E ex

p – E

th (M

eV)

E ex

p - E

th

Sly4-Bertsch,Sabbey, Uusnakki PRC (2005)

Energy scale!300 1000 1600

20 40 60 10080 120 140 160Neutron number N

(Skyrme, Gogny, …)To make the functional predictive,

we use and abuse of symmetry breaking

Pairing Correlations

Odd-even effects

Quadrupole Correlation

-Rotational

Bands

Surface Vibration

Translation

Rotation

Particle number

…

Single Reference (SR)-Mean-Field

Configuration Mixing within Energy Density Functional

Sly4-Bender,Bertsch,Heenen, PRL (2005)

E ex

p – E

th (M

eV)

E ex

p - E

th

Sly4-Bertsch,Sabbey, Uusnakki PRC (2005)

Energy scale!300 1000 1600

20 40 60 10080 120 140 160Neutron number N

(Skyrme, Gogny)

Multi- Ref. (MR)-GCM

Bender et al, PRC74 (2006)

74KrMean-Field

Energy

0+

0+0+

2+

2+ 2+

4+

4+

6+

8+

Corr

elati

on

Ener

gy

GCM is not so easy to use in EDF

M=91939597999

M=9

M=99

Divergence

Jump

Multi- Ref. (MR)-GCM

M: number ofMesh points

Application of conf. mixing in EDFNeeds to be regularized

Practical and conceptual difficulties in Configuration Mixing within EDF (I) Lacroix, et al, PRC79 (2009), (II) Bender et al PRC79 (2009), (III) Duguet et al, PRC79 (2009).

Correction is possible

Before correction

Corrected

SIII force

with

Example: particle number projection

or

with

Projection on Good particle number

Some uncontrolled and not understood spuriouscontributions persists M. Bender (private communication)

Duguet et al, PRC79 (2009).

The EDF energy will depend on the integration contour and becomes ill defined

We can use r, r2, r3, r4, r5 but not ra !!

The specific problem of the density dependent term ra

Example: particle number projection

Transition density

Is a complex number

Is multivalued in the complex plane

Problem: ra is very useful and practical

Interacting fermions in different regimes of density

Fictitious equation of state

Low

den

sity

regi

me

Satu

ratio

n de

nsity

regi

me

At low density

Bulgac, Forbes, …

High

den

sity

regi

me

Lee-Yang formula

Unitary gas

Galitskii formula

At saturation

Strategy : restart from true interaction to mimic density dependent term

Nuclear EDF phenomenology

Effective interaction1-body, 2-body, 3-body

?

Strategy 1

Hamiltonian1-body, 2-body, 3-body

Many-body technology

Single-Ref.

Multi-Ref.

Back to the original strategy

Constraints: - start from a Hamiltonian - no ra

Finite range ?

Zero-range(requires many terms)

Back to the construction of effective HamiltonianWith finite range interaction

Goal: mimic the proper dependence of the energy around saturation

i.e.

2-body interaction

c1 and c2 are free parameters

Infinite matter case

Convenient only for

Need for at least 3-body interaction

Semi-contact 3-body interaction

Effective Hamiltonian with 2-body and 3-body

Semi-contact 3-body interaction

Jacobi coordinate(r,R)

Idea: take a zero range in ROur starting point:

Lacroix, Bennaceur, Phys. Rev. C91 (2015) and (antisymmetrization)

EDF based on semi-contact 3-body interaction

Our starting point:

Lacroix, Bennaceur, Phys. Rev. C91 (2015)

Infinite matter case

EDF based on semi-contact 3-body interaction

EDF = 2-body + 3-body (semi-contact)

zero-rangeor

finite-range

Lacroix, Bennaceur, Phys. Rev. C91 (2015).

Illustration I : Skyrme (2-body)+3-body

Properties after a global fit:

EDF based on semi-contact 3-body interaction

EDF = 2-body + 3-body (semi-contact)

zero-rangeor

finite-range

Illustration II : Gogny (2-body)+3-body

Lacroix, Bennaceur, Phys. Rev. C91 (2015).

Properties after a global fit:

Some remarks: towards application to nuclei

Complete expression

Two difficulties :

First step : simplified interaction

Neglect the exchange with the third particle

Equivalent to the result of a density dependent finite range 2-body interaction

is not on the mesh

Exchange with the third particles

Simplified 2-body interaction: some remarks

Can we still mimic a ra behavior ? Yes

2-body functional

Simplified 2-body interaction: some remarks

We still have the difficulty

A natural solution

The functional is then equivalent to the one derived from the effective density dependent 2-body interaction:

Same as F. Chappert (PhD) with a=1Chappert, Pillet, Girod, Berger, Phys. Rev. C92 (2015)

Summary

To mimic density-dependent term, we proposed a new 3-body interaction

Gives a good description on infinite systems In different spin isospin channels

M=91939597999

M=9

M=99

DivergenceJump

Combining Nuclear EDF and conf. Mean-field leads to specific problems

Several solutions have been proposed

-regularization of divergences -New EDF with 2-body density matrix

Future dev: application to finite systems

One solution is to go back to effective HamiltonianAnd keep consistency between mean-field and pairing