Cures and pathologies of Multi-Reference EDF methods

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

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

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

4. Multi-Reference: Generalities and difficulties Multi- Ref. (MR) Single Reference (SV) 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)

5. Phenomenology of divergences in the PNP case Anguiano et al, NPA 696 (2001) Tajima et al, NPA542 (1992) Divergences occur 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, arXiv:0708.0441 Bender and Duguet, Int.J. Mod. Phys. E16 (2007)

6. Restoration of angular momentum Nodal structure of the overlap The problem is generic (angular momentum projection) Oi and Tajima, PLB606 (2005) b Low Resolution High Resolution g a Zdunczuk et al, nucl-th/0610118. Two options: We can restrict to purely effective “Hamiltonian” case. We find a general cure to pathologies of actual EDF

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

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

9. Methods for connecting different quasi-particle vacua Topology of reference state Starting points: 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) Ring and Schuck Book with

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

11. Recovering the Generalized Wick Theorem and sources of difficulties in the GWT+EDF 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)

12. Correction of divergences 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) Lacroix, Duguet, Bender (2007)-Formal aspects

13. Summary on correction of problem in EDF: Effective Hamiltonian General EDF framework -Same pp and ph interaction. -Exchange and coulomb fully treated Keep the full Power of EDF -Phenomenological terms -Different pp and ph interactions -Exchange Closer to an Hamiltonian approach Anguiano et al, NPA 696 (2001) -Self-interaction in SR and MR -Divergences in conf. mixing -No Divergences -But EDF is ill defined: (shift invariance) -Corrections are possible with different pp and ph interactions -Limit the type of terms Lacroix, Duguet, Bender (2007) 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. 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) 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) 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?

16. The Particle Number Restoration case Trial state with EDF calculations Connecting states with Bender, Duguet, Lacroix (2007)-Applications with SIII +contact pairing

17. Correction of divergences The Particle Number Restoration Case rr term kk term Results Correction is important at and near divergences Correction is overall small (~500keV) but of the order of the required accuracy. For the presented case, second 0+ at 6.03 MeV ? Corrected energy curves are now smooth

18. 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. Discussion on constructive Framework: from Density Matrix Expansion (DME)? r1 DME Method (for local interactions): s r r2 with where Basic Idea: symmetries of r Constraints Taylor Asymptotic conditions 0 Example of decomposition: Negele, Vautherin, PRC (1972)

20. Discussion on constructive Framework: from Density Matrix Expansion (DME)? Single-Vacuum Hartree-Fock alone is not sufficient Hartree-Fock +perturbative expansion (Bogner Talk, ECT* April 2007) Multi-Vacua with Some symmetries of r are lost New terms in the EDF Example : Negele, Vautherin, PRC(1972) j1 j1+ j3 Taylor

22. Some remarks on the construction of Skyrme EDF