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Transcript of 1 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Safe EDF for SR and MR calculations...
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
2Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
3Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
The problem is everywhere
Angular momentum projection
Zdunczuk et al, nucl-th/0610118.
Particle number projection
SIII
Sly4
Gogny force is also pathological
4Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
5Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
6Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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)
7Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
8Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
9Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
Interest of the Bloch-Messiah-Zumino techniqueSome Theorem made simple
(Onishi-Yoshida)Overlaps
Thouless Theorem
if We define
Expectationvalues
with
SR
MR
10Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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.
11Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
12Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
Particle Number Restoration
with
Trial state
EDF calculations
Connecting states
with
(II) Bender et al, First application to PNR, arXiv/0809.2045
13Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
14Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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
15Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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)