1. 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, Jyvaskyla, Oct. 9th-10th 2008

2. 2 Thomas 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=9 19 39 59 79 99 Example: Particle Number Restoration with PNR (or PNP) on a mesh M=9 M=99 Divergence Jump

3. 3 Thomas 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

4. 4 Thomas 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

5. 5 Thomas 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

6. 6 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Construction of EDF for configuration mixing: the “nuclear physics strategy” EDF case Hamiltonian case Single-R,, Standard Wick Theorem,,… Practical difficulties Origin of divergences Multi-R,,… Generalized Wick Theorem Balian, Brezin, Nuovo Cimento 64 (1969)

7. 7 Thomas 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 kink Cure to MR Find an Alternative to GWT Identify critical terms Remove pathologies

8. 8 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Starting point: The Balian-Brezin (Thouless) strategy valid for Start from Work on the transformation with The Bloch-Messiah-Zumino Strategy Work directly on A and B D C Simplify the connection (valid if ) Ring and Schuck Book Donau PRC(1998), Dobaczewski PRC(2000) Methods for connecting different quasi-particle vacua (I) Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041

9. 9 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Interest of the Bloch-Messiah-Zumino technique Some Theorem made simple (Onishi-Yoshida) Overlaps Thouless Theorem ifWe define Expectation values with SR MR

10. 10 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Sources of problems: Lacroix, Duguet, Bender (2007) Correcting for the misuse of the GWT and sources of difficulties Notations with and !!!Condition!!!  term  term MR Divergences Exchange (self-interaction) Condition for SR Summary of pathologies SR- self-interaction SR-self-inter /self-pairing } } MR self-inter /self-pairing MR self-inter /self-pairing Diverg.

11. 11 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Correction of spurious contributions Identify problems and correct MR calculations without breaking the actual strategy of EDF Limited to integer power of densities (Generalization to k-body interactions) Valid also for mixing Slater Determinants Should 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

12. 12 Thomas 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

13. 13 Thomas 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

14. 14 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 It is overall small BUT of the order of the accuracy required. In all cases, correction reduces 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

15. 15 Thomas Duguet FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008 Summary and discussion Energy 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)