Safe EDF for SR and MR calculations
M. Bender, T. Duguet, D. Lacroix
FIDIPRO Workshop, Jyvaskyla, Oct. 9th-10th 2008
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
1
Multi-Reference: generalities and difficulties
Multi- Ref. (MR)
Single Reference (SR)
Example: Particle Number Restoration
Divergence
Jump
M=99
with
PNR (or PNP) on a mesh
Thomas Duguet
M=9
19
39
59
79
99
M=9
Bender, Duguet,
Int. J.
Mod.
Phys.2008
E16 (2007)
FIDIPRO Workshop - Jyvaskyla,
Oct
9th-10th
2
The problem is everywhere
Particle number projection
Angular momentum projection
Sly4
SIII
Zdunczuk et al, nucl-th/0610118.
Gogny force is also pathological
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
3
Early warnings in the PNR case
Tajima et al, NPA542 (1992)
Early warnings
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
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.
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
4
Construction of the EDF: Single-Ref.
The “nuclear physics strategy”
Hamiltonian case
Standard Wick Theorem
Breaking the link with the Hamiltonian
- Introduction of new terms
-Different interactions in ph and pp channels
-Technical issues: coulomb, exchange…
Energy Density Functional case
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
5
Construction of EDF for configuration mixing:
the “nuclear physics strategy”
EDF case
Hamiltonian case
Single-R
Standard Wick Theorem
,
,
Multi-R
Generalized Wick Theorem
,
,…
Balian, Brezin, Nuovo Cimento 64 (1969)
Thomas Duguet
,
,…
Practical difficulties
Origin of divergences
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
6
Identification and cure to pathologies
(I)
Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041
SP
Sources of pathologies
SI
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
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
7
Methods for connecting different quasi-particle vacua
(I)
Lacroix, et al, General solution to MR pathologies,arXiv/0809.2041
Starting point:
The Balian-Brezin (Thouless)
strategy
The Bloch-Messiah-Zumino
Strategy
Work directly on A and B
Start from
D
C
valid for
Work on the transformation
Thomas Duguet
Simplify the connection (valid if
Donau PRC(1998),
2008
8
Dobaczewski PRC(2000) FIDIPRO Workshop - Jyvaskyla, Oct 9th-10thRing
and Schuck Book
)
Interest of the Bloch-Messiah-Zumino technique
Some Theorem made simple
Overlaps
Thouless
Theorem
(Onishi-Yoshida)
if
We define
with
Expectation
values
MR
SR
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
9
Correcting for the misuse of the GWT
and sources of difficulties
Notations
with
and
Summary of pathologies
SR
MR
SR
MR
Thomas Duguet
rr term
SR- self-interaction
}
MR self-inter
/self-pairing
!!!Condition!!!
Diverg.
SR-self-inter
kk term
/self-pairing
Sources
of problems:
}
Condition for
MR
self-inter
Divergences
/self-pairing
Exchange (self-interaction)
Diverg.
FIDIPRO Workshop - Jyvaskyla,
Oct 9th-10th
Lacroix,
Duguet,2008
Bender (2007)
10
Correction of spurious contributions
(I)
Lacroix, et al, General solution to MR pathologies, arXiv/0809.2041
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)
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
11
Particle Number Restoration
(II) Bender et al, First application to PNR, arXiv/0809.2045
Trial state
with
EDF calculations
Connecting states
with
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
12
Particle Number Restoration
(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
Thomas Duguet
Correction is overall small (~1MeV) but
of the order of the required mass
accuracy or spectroscopy of interest
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
13
Particle Number Restoration
(II) Bender et al, First application to PNR, arXiv/0809.2045
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
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
14
Summary and discussion
Energy Density Functional methods
Specific difficulties to be considered seriously
Dobaczewski et al, Identification of problem in PNR, PRC76, 054315 (2007)
Regularization method valid for any MR calculation
(I)
Lacroix, et al, General solution to pathologies, arXiv/0809.2041
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
Need to build
(i) Correctable EDF for MR calculations
(ii) SI- and SP-free EDF
Need a constructive framework for MR-EDF
Thomas Duguet
FIDIPRO Workshop - Jyvaskyla, Oct 9th-10th 2008
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