1. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Cures and pathologies of Multi-Reference EDF methods Denis Lacroix, GANIL-Caen (MSU-USA) Overall Goal: Develop Energy Density Functional able to describe static and dynamical properties of nuclei. Coll: T. Duguet, M. Bender K. Bennaceur, T. Lesinski Dynamics : -Fission. fusion. -vibrations -… Static: -ground state -excited states -Deformation -Phase diagram -… from Bruyère le Châtel (CEA-DAM) website

2. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix EDF versus DFT Some Bridges and some differences Functional concept:  n,  n,… (LSD, GGA, MGGA…) Trial State Kohn-Sham state Single-Reference (SR) state (Slater) pairing effect DFT with pairing Quasi-particle vacuum Gross et al, PRL 60 (1988) Broken symmetries intrinsic density lab density Dynamics Small amplitude Small/large amplitude potential non-local in time adiabatic potential (local in time ) Runge, Gross, PRL52 (1984) Bonche et al, PRC (1976) Shared problems functional form? Guessed, fits… Skyrme, Gogny… H-K theorem Universality of  Effective Hamiltonian based EDF Motivation/Justification Several trial states (for Excited states) Ensemble Kohn-Sham Gross et al, PRA137 (1988) GCM based Multi-Reference (MR) pathologies self-interaction self-interaction, self-pairing, divergences Perdew et al PRB23 (1981)

3. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Single vacuum (SV) state 2- Minimize with respect to relevant degrees of freedom Technique: 1-Start from a given functional Skyrme, Gogny,… Improvements: Improve the functional itself Improve the fitting procedure Take more general trial state GCM “like”, Restoration of broken symmetries… Single-Reference EDF

4. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Multi-Reference: Generalities and difficulties Single Reference (SV) 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

5. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Phenomenology of divergences in the PNP 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, arXiv:0708.0441 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, NPA 696 (2001) Divergences occur when a level crosses the Fermi energy, with occupation 0.5 and  =  /2 at this point Tajima et al, NPA542 (1992) Early warnings

6. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Restoration of angular momentum The problem is generic (angular momentum projection) Zdunczuk et al, nucl-th/0610118. Nodal structure of the overlap Oi and Tajima, PLB606 (2005)    Low Resolution High Resolution Two options: We can restrict to purely effective “Hamiltonian” case. We find a general cure to pathologies of actual EDF

7. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Construction of EDF: Single-Ref. the “nuclear physics strategy” Energy Density Functional Case Breaking the link with the Hamiltonian -Technical issues: coulomb, exchange… Effective Hamiltonian Case (Standard Wick Theorem) - Introduction of new terms-Different interactions in ph and pp channels Starting point: Effective Hamiltonian (Skyrme, Gogny,…) EDF

8. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Construction of EDF for configuration mixing: the “nuclear physics strategy” Starting point: Effective Hamiltonian (Skyrme, Gogny,…) EDF EDF case Effective Hamiltonian case Single-R,, Standard Wick Theorem,,… Practical difficulties Origin of divergences Multi-R,,… Generalized Wick Theorem Balian, Brezin, Nuovo Cimento 64 (1969)

9. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Starting points: 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 Topology of reference state

10. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Interest of the Bloch-Messiah-Zumino technique Some Theorem made simple (Onishi-Yoshida) Overlaps Thouless Theorem ifWe define Expectation values with SR MR

11. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Sources of problems: Lacroix, Duguet, Bender (2007) Recovering the Generalized Wick Theorem and sources of difficulties in the GWT+EDF 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.

12. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Correction of divergences Lacroix, Duguet, Bender (2007)-Formal aspects This provides a way to identify problems and correct configuration mixing calculations without breaking the actual strategy of EDF. (limited to integer power of densities [Generalization to k-body interactions]) Valid also in the Slater Determinant case. Should correct divergences observed in Zdunczuk et al, nucl-th/0610118. Before Bloch-Messiah-Zumino { { After Bloch-Messiah-Zumino Instructive to understand the BMZ Theorem Indicates that major changes in numerical implementations might be needed Illustration with the Particle Number Restoration confirm the intuition Bender and Duguet, Int.J. Mod. Phys. E16 (2007) see M. Bender Talk and M.Bender et al (in preparation)

13. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Summary on correction of problem in EDF: General EDF framework Keep the full Power of EDF -Phenomenological terms -Different pp and ph interactions -Exchange -Self-interaction in SR and MR -Divergences in conf. mixing -Corrections are possible with different pp and ph interactions -Limit the type of terms Lacroix, Duguet, Bender (2007) Effective Hamiltonian Anguiano et al, NPA 696 (2001) -Same pp and ph interaction. -Exchange and coulomb fully treated Closer to an Hamiltonian approach -No Divergences -But EDF is ill defined: (shift invariance) Requires a constructive framework for EDF: 1-to construct Single-Ref EDF 2-To extend to conf. mixing 3-May require major changes in codes

14. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Summary and Discussion Energy Density Functional for conf. mixing Specific aspects and difficulties (jumps, divergences…) We have partially solved the problem (application to particle Number Restoration) We need a constructive framework for MR-EDF EDF: Density Functional Theory or not? If yes, we can eventually take advantage of DFT techniques. Should we keep the Hamiltonian based theory? Requires the proof of some theorem? Lacroix, et al, General solution to pathologies (I) Bender et al, First application PNR (II) Duguet et al, Fractional powers of density case. (III) We cannot use fractional powers of densities (no shift invariance)

15. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix

16. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix The Particle Number Restoration case Bender, Duguet, Lacroix (2007)-Applications with SIII +contact pairing with Trial state EDF calculations Connecting states with

17. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Correction of divergences The Particle Number Restoration Case  term  term Results Correction is important at and near divergences Corrected energy curves are now smooth Correction is overall small (~500keV) but of the order of the required accuracy. For the presented case, second 0 + at 6.03 MeV ?

18. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix More results of the correction The correction is systematically needed even when no divergences are seen. It is overall small BUT of the order of the UNEDF goal In all cases, correction reduces the noise Systematic study with other type of conf. mixing?

19. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Discussion on constructive Framework: from Density Matrix Expansion (DME)? r2r2 r1r1 r s DME Method (for local interactions) : with Basic Idea: where Negele, Vautherin, PRC (1972) Example of decomposition: symmetries of  Constraints Asymptotic conditions Taylor 0

20. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Discussion on constructive Framework: from Density Matrix Expansion (DME)? Example : Negele, Vautherin, PRC(1972) Taylor j1j1 j 1 + j 3 Hartree-Fock alone is not sufficient Single-Vacuum Hartree-Fock +perturbative expansion (Bogner Talk, ECT* April 2007) Multi-Vacua with Some symmetries of  are lostNew terms in the EDF

21. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix

22. Cures and Pathologies of MR-EDF – Saclay November 2007 Denis Lacroix Some remarks on the construction of Skyrme EDF