Safe EDF for SR and MR calculations M. Bender, T. Duguet, D. Lacroix 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
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
The problem is everywhere Angular momentum projection Zdunczuk et al, nucl-th/0610118. Particle number projection SIII Sly4 Gogny force is also pathological
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
j=p/2 at this point Tajima et al, NPA542 (1992) Early warnings
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
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)
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
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
Interest of the Bloch-Messiah-Zumino technique
Some Theorem made simple (Onishi-Yoshida) Overlaps Thouless
Theorem if We define Expectation
values with SR MR
Sources of problems: Lacroix, Duguet, Bender (2007) Correcting for the misuse of the GWT
and sources of difficulties Notations with and !!!Condition!!! kk term rr 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.
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
Particle Number Restoration
with Trial state EDF calculations Connecting states with (II) Bender et al, First application to PNR, arXiv/0809.2045
rr
term kk
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
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
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: rg (III) Duguet et al, Non-integer powers of density matrices, arXiv/0809.2049 Dobaczewski et al, Identification of problem in PNR, PRC76, 054315 (2007)
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