Safe EDF for SR and MR calculations

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

2. Multi-Reference: generalities and difficulties Multi- Ref. (MR) Single Reference (SR) Example: Particle Number Restoration M=99 with M=9 19 39 M=9 PNR (or PNP) on a mesh 59 79 99 Bender, Duguet, Int. J. Mod. Phys. E16 (2007) Divergence Jump

3. The problem is everywhere SIII Gogny force is also pathological Angular momentum projection Particle number projection Sly4 Zdunczuk et al, nucl-th/0610118.

4. Early warnings in the PNR case Anguiano et al, NPA696 (2001) Tajima et al, NPA542 (1992) Divergences occur in PNR when a level crosses the Fermi energy, with occupation 0.5 and j=p/2 at this point Early warnings 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) Analysis in the complex plane for the PNP case 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. Dobaczewski et al, PRC76 (2007) Bender and Duguet, Int. J. Mod. Phys. E16 (2007)

5. Construction of the EDF: Single-Ref. The “nuclear physics strategy” - Introduction of new terms • Different interactions in ph and pp channels -Technical issues: coulomb, exchange… Energy Density Functional case Hamiltonian case Standard Wick Theorem Breaking the link with the Hamiltonian

6. Construction of EDF for configuration mixing: the “nuclear physics strategy” EDF case Hamiltonian case Standard Wick Theorem Single-R , , Generalized Wick Theorem Multi-R , , ,… ,… Balian, Brezin, Nuovo Cimento 64 (1969) Practical difficulties Origin of divergences

7. Identification and cure to pathologies SP (I) Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041 SI Sources of pathologies • Self-Interaction (SR+MR) • Not dramatic a priori • Need to be characterized • Self-Pairing (SR+MR) • Not dramatic a priori • Need to be characterized • 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. Methods for connecting different quasi-particle vacua Starting point: The Bloch-Messiah-Zumino Strategy The Balian-Brezin (Thouless) strategy Work directly on A and B D Start from C valid for Work on the transformation Simplify the connection (valid if ) Donau PRC(1998), Dobaczewski PRC(2000) (I) Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041 Ring and Schuck Book with

9. Interest of the Bloch-Messiah-Zumino technique Some Theorem made simple Overlaps (Onishi-Yoshida) if We define Thouless Theorem with Expectation values MR SR

10. Correcting for the misuse of the GWT and sources of difficulties Notations with and Summary of pathologies rr term SR SR- self-interaction } MR MR self-inter /self-pairing !!!Condition!!! Diverg. SR SR-self-inter /self-pairing kk term Sources of problems: } Condition for MR MR self-inter /self-pairing Divergences Exchange (self-interaction) Diverg. Lacroix, Duguet, Bender (2007)

11. 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. Particle Number Restoration Trial state with EDF calculations Connecting states with (II) Bender et al, First application to PNR, arXiv/0809.2045

13. Particle Number Restoration rr term kk term Results Correction is important at but also away from steps Correction is overall small (~1MeV) but of the order of the required mass accuracy or spectroscopy of interest Corrected energy curve is smooth (II) Bender et al, First application to PNR, arXiv/0809.2045

14. Particle Number Restoration After correction the Functional is independent on the mesh The correction is of the 500 keV/1 MeV 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 (II) Bender et al, First application to PNR, arXiv/0809.2045

15. Summary and discussion Specific difficulties to be considered seriously Regularization method valid for any MR calculation Application to Particle Number Restoration (I) Lacroix, et al, General solution to pathologies, arXiv/0809.2041 (II) Bender et al, First application to PNR, arXiv/0809.2045 Specific case of fractional power of the density: rg (III) Duguet et al, Non-integer powers of density matrices, arXiv/0809.2049 Need to build (i) Correctable EDF for MR calculations (ii) SI- and SP-free EDF Need a constructive framework for MR-EDF Energy Density Functional methods Dobaczewski et al, Identification of problem in PNR, PRC76, 054315 (2007)