1 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Safe EDF for SR and MR calculations M. Bender, T. Duguet, D. Lacroix FIDIPRO Workshop,

1Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008

Safe EDF for SR and MR calculations

M. Bender, T. Duguet, D. Lacroix

FIDIPRO Workshop, Jyvaskyla, Oct. 9th-10th 2008

tduguet


Multi-Reference: generalities and difficulties

Single Reference (SR) Multi- Ref. (MR)

Bender, Duguet, Int. J. Mod. Phys. E16 (2007)

M=91939597999

Example: Particle Number Restoration

with

PNR (or PNP) on a mesh M=9

M=99

Divergence

Jump


The problem is everywhere

Angular momentum projection

Zdunczuk et al, nucl-th/0610118.

Particle number projection

SIII

Sly4

Gogny force is also pathological


Early warnings in the PNR case

The problem disappears if the same interaction (with exchange and coulomb treated properly) in the pp and ph channels

First indication of the role of Generalized Wick Theorem (GWT)

Dobaczewski et al, PRC76 (2007)

Analysis in the complex plane for the PNP case

Bender and Duguet, Int. J. Mod. Phys. E16 (2007)

In the PNP case, by avoiding the use of GWT, one can cure the divergence problem even when different effective interaction are used in different channels.

Anguiano et al, NPA696 (2001)

Divergences occur in PNR when a level crosses the Fermi energy, with occupation 0.5 and =/2 at this point

Tajima et al, NPA542 (1992)Early warnings


Construction of the EDF: Single-Ref. The “nuclear physics strategy”

Energy Density Functional case

Breaking the link with the Hamiltonian

-Technical issues: coulomb, exchange…

- Introduction of new terms-Different interactions in ph and pp channels

Hamiltonian case

Standard Wick Theorem


Construction of EDF for configuration mixing:the “nuclear physics strategy”

EDF caseHamiltonian case

Sin

gle

-R

, ,

Standard Wick Theorem

,,…

Practical difficulties

Origin of divergences

Mu

lti-

R

,,…

Generalized Wick Theorem

Balian, Brezin, Nuovo Cimento 64 (1969)


Identification and cure to pathologies(I) Lacroix, et al, General solution to MR

pathologies,arXiv/0809.2041

Sources of pathologies

Self-Interaction (SR+MR)• Not dramatic a priori• Need to be

characterized

SI

Self-Pairing (SR+MR)• Not dramatic a priori• Need to be

characterized

SP

Misuse of Gen. Wick Th. (GWT) within EDF method (MR only)

• Divergences, sharp steps, smooth steps plus kinkCure to MR

Find an Alternative to GWT

Identify critical terms

Remove pathologies


Starting point:

The Balian-Brezin (Thouless) strategy

valid for

Start from

Work on the transformation

The Bloch-Messiah-Zumino Strategy

Work directly on A and B D

C

Simplify the connection (valid if )

Ring and Schuck BookDonau PRC(1998), Dobaczewski PRC(2000)

Methods for connecting different quasi-particle vacua(I) Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041


Interest of the Bloch-Messiah-Zumino techniqueSome Theorem made simple

(Onishi-Yoshida)Overlaps

Thouless Theorem

if We define

Expectationvalues

with

SR

MR


Sources of problems:

Lacroix, Duguet, Bender (2007)

Correcting for the misuse of the GWTand sources of difficulties

Notations

with and

!!!Condition!!!

term

term

MR

MR Divergences

Exchange (self-interaction)

Condition for

SR

SR

Summary of pathologies

SR- self-interaction

SR-self-inter/self-pairing

}

}

MR self-inter/self-pairing

MR self-inter/self-pairing

Diverg.

Diverg.


Correction of spurious contributions

Identify problems and correct MR calculations without breaking the actual strategy of EDFLimited to integer power of densities (Generalization to k-body interactions)

Valid also for mixing Slater DeterminantsShould correct divergences observed in Zdunczuk et al, nucl-th/0610118

Before Bloch-Messiah-Zumino

{{

After Bloch-Messiah-

Zumino Instructive to understand BMZ Theorem

Changes in numerical implementations are needed

Illustration for Particle Number Restoration Confirm the intuition Bender and Duguet, Int.J. Mod. Phys. E16 (2007)

(I) Lacroix, et al, General solution to MR pathologies, arXiv/0809.2041


Particle Number Restoration

with

Trial state

EDF calculations

Connecting states

with

(II) Bender et al, First application to PNR, arXiv/0809.2045


term

term

Results

Correction is important at but also away from steps

Corrected energy curve is smooth

Correction is overall small (~1MeV) but of the order of the required mass

accuracy or spectroscopy of interest

Particle Number Restoration(II) Bender et al, First application to PNR,

arXiv/0809.2045


It is overall smallBUTof the order of theaccuracy required.

In all cases, correctionreduces the noise

Need study with other type of conf. mixing

After correction the Functional is independent on the mesh

The correction is of the 500 keV/1 MeV

(II) Bender et al, First application to PNR, arXiv/0809.2045

Particle Number Restoration


Summary and discussionEnergy Density Functional methods

Specific difficulties to be considered seriously

Need a constructive framework for MR-EDF

Regularization method valid for any MR calculation(I) Lacroix, et al, General solution to pathologies, arXiv/0809.2041

Need to build

(i) Correctable EDF for MR calculations(ii) SI- and SP-free EDF

Application to Particle Number Restoration(II) Bender et al, First application to PNR, arXiv/0809.2045

Specific case of fractional power of the density:

(III) Duguet et al, Non-integer powers of density matrices, arXiv/0809.2049

Dobaczewski et al, Identification of problem in PNR, PRC76, 054315 (2007)

Documents

1 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Safe EDF for SR and MR calculations M. Bender, T. Duguet, D. Lacroix FIDIPRO Workshop,