cures and pathologies of mr-edf – saclay november 2007 denis lacroix cures and pathologies of...
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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
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EDF versus DFT
Some Bridges and some differences
Functional concept: n,n,…(LSD, GGA,MGGA…)
Trial State Kohn-Sham stateSingle-Reference (SR)
state (Slater)
pairing effect DFT with pairingQuasi-particle vacuumGross et al, PRL 60 (1988)Broken symmetries
intrinsic density lab density
Dynamics Small amplitudeSmall/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 theoremUniversality of
Effective Hamiltonianbased EDF
Motivation/Justification
Several trial states (for Excited states)
Ensemble Kohn-ShamGross et al, PRA137 (1988)
GCM basedMulti-Reference (MR)
pathologies self-interactionself-interaction, self-pairing, divergences Perdew et al PRB23 (1981)
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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 itselfImprove the fitting procedure
Take more general trial stateGCM “like”, Restoration of broken symmetries…
Single-Reference EDF
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Multi-Reference: Generalities and difficulties
Single Reference (SV) Multi- Ref. (MR)
Bender, Duguet, Int. J. Mod. Phys. E16 (2007)
M=91939597999
Example: Particle Number Restoration
with
PNR (or PNP) on a mesh M=9
M=99
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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 =/2at this point
Tajima et al, NPA542 (1992)Early warnings
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Restoration of angular momentum
The problem is generic (angular momentum projection)
Zdunczuk et al, nucl-th/0610118.
Nodal structure of the overlapOi and Tajima, PLB606 (2005)
Low
Resolutio
nHigh
Resolutio
n
Two options:
We can restrict to purely effective “Hamiltonian” case.
We find a general cure to pathologies of actual EDF
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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
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Construction of EDF for configuration mixing:the “nuclear physics strategy”
Starting point: Effective Hamiltonian (Skyrme, Gogny,…)
EDF
EDF caseEffective Hamiltonian case
Sin
gle
-R
, ,
Standard Wick Theorem
,,…
Practical difficulties
Origin of divergences
Mu
lti-
R
,,…
Generalized Wick Theorem
Balian, Brezin, Nuovo Cimento 64 (1969)
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Starting points:
The Balian-Brezin (Thouless) strategy
valid for
Start from
Work on the transformation
The Bloch-Messiah-Zumino Strategy
Work directly on A and B D
C
Simplify the connection (valid if )
Ring and Schuck BookDonau PRC(1998), Dobaczewski PRC(2000)
Methods for connecting different quasi-particle vacuaTopology of reference state
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Interest of the Bloch-Messiah-Zumino techniqueSome Theorem made simple
(Onishi-Yoshida)Overlaps
Thouless Theorem
if We define
Expectationvalues
with
SR
MR
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Sources of problems:
Lacroix, Duguet, Bender (2007)
Recovering the Generalized Wick Theoremand sources of difficulties in the GWT+EDF
Notations
with and
!!!Condition!!!
term
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.
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Correction of divergencesLacroix, 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)
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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 EDF2-To extend to conf. mixing3-May require major changes in codes
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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)
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The Particle Number Restoration case
Bender, Duguet, Lacroix (2007)-Applications with SIII +contact pairing
with
Trial state
EDF calculations
Connecting states
with
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Correction of divergencesThe 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 ?
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More results of the correction
The correction is systematically neededeven when no divergences are seen.
It is overall smallBUTof the order of theUNEDF goal
In all cases, correctionreduces the noise
Systematic study with other type of conf. mixing?
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Discussion on constructive Framework: from Density Matrix Expansion (DME)?
r2
r1
r sDME Method (for local interactions) :
with
Basic Idea: where
Negele, Vautherin, PRC (1972)Example of decomposition:
symmetries of ConstraintsAsymptoticconditions
Taylor
0
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Discussion on constructive Framework: from Density Matrix Expansion (DME)?
Example : Negele, Vautherin, PRC(1972)
Taylor
j1
j1+ j3
Hartree-Fock alone is not sufficient
Single-Vacuum
Hartree-Fock+perturbative expansion (Bogner Talk, ECT* April 2007)
Multi-Vacua
with
Some symmetries of are lost New terms in the EDF
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Some remarks on the construction of Skyrme EDF