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.
Static:
-ground state
-excited states
-Deformation
-Phase diagram
-…
from Bruyère le Châtel (CEA-DAM) website
Dynamics :
-Fission. fusion.
-vibrations
-…
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Cures and pathologies of Multi-Reference EDF methods
Some Bridges and some differences
r,k,r,t,...
intrinsic density
Universality of r,
Effective Hamiltonian
based EDF
Functional concept:
n,n,… (LSD, GGA,
MGGA…)
lab density
Motivation/Justification
H-K theorem
Single-Reference (SR)
state (Slater)
Trial State
Kohn-Sham state
Quasi-particle vacuum
Broken symmetries
pairing effect
GCM based
Multi-Reference (MR)
Several trial states
(for Excited states)
Small/large amplitude
Dynamics
Bonche et al, PRC (1976)
DFT with pairing
Gross et al, PRL 60 (1988)
Ensemble Kohn-Sham
Gross et al, PRA137 (1988)
Small amplitude
Runge, Gross, PRL52 (1984)
adiabatic potential (local
in time )
Shared problems
Skyrme, Gogny…
functional form?
self-interaction,
self-pairing, divergences
pathologies
potential non-local in time
Guessed, fits…
self-interaction
Perdew et al PRB23 (1981)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
EDF versus DFT
Technique:
1-Start from a given functional
Single
vacuum (SV)
state
Skyrme, Gogny,…
2- Minimize with respect to relevant degrees of freedom
Improvements:
Improve the fitting procedure
Improve the functional itself
Take more general trial state
GCM “like”, Restoration of broken symmetries…
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Single-Reference EDF
Single Reference (SV)
Multi- Ref. (MR)
Example: Particle Number Restoration
M=99
with
PNR (or PNP) on a mesh
M=9
19
39
59
79
99
M=9
Bender, Duguet,
Int. J. Mod. Phys. E16 (2007)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Multi-Reference: Generalities and difficulties
Tajima et al, NPA542 (1992)
Early warnings
Anguiano et al, NPA 696 (2001)
Divergences occur 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, 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.
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Phenomenology of divergences in the PNP case
The problem is generic
(angular momentum projection)
Nodal structure of the overlap
Oi and Tajima, PLB606 (2005)
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
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Restoration of angular momentum
the “nuclear physics strategy”
Starting point: Effective Hamiltonian (Skyrme, Gogny,…)
EDF
Effective 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
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Construction of EDF: Single-Ref.
the “nuclear physics strategy”
Starting point: Effective Hamiltonian (Skyrme, Gogny,…)
EDF
Effective Hamiltonian case
EDF case
Single-R
Standard Wick Theorem
,
,
,
,…
Balian, Brezin, Nuovo Cimento 64 (1969)
,
Practical difficulties
Origin of divergences
,…
Denis Lacroix
Generalized Wick Theorem
Multi-R
Cures and Pathologies of MR-EDF – Saclay November 2007
Construction of EDF for configuration mixing:
Topology of reference state
Starting points:
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
Donau PRC(1998),
Dobaczewski PRC(2000)
Simplify the connection (valid if
Ring and Schuck Book
)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Methods for connecting different quasi-particle vacua
Some Theorem made simple
Overlaps
Thouless
Theorem
(Onishi-Yoshida)
if
We define
with
Expectation
values
MR
SR
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Interest of the Bloch-Messiah-Zumino technique
and sources of difficulties in the GWT+EDF
Notations
with
and
Summary of pathologies
SR
MR
SR
MR
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.
Lacroix, Duguet, Bender (2007)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Recovering the Generalized Wick Theorem
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)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Correction of divergences
General EDF framework
Effective Hamiltonian
Keep the full Power of EDF
-Phenomenological terms
-Different pp and ph
interactions
-Exchange
-Same pp and ph interaction.
-Exchange and coulomb
fully treated
Closer to an Hamiltonian
approach
Anguiano et al,
NPA 696 (2001)
-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)
-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
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Summary on correction of problem in EDF:
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?
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Summary and Discussion
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Bender, Duguet, Lacroix (2007)-Applications with SIII +contact pairing
Trial state
with
EDF calculations
Connecting states
with
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
The Particle Number Restoration case
The Particle Number Restoration Case
rr
term
kk
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 ?
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Correction of divergences
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?
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
More results of the correction
from Density Matrix Expansion (DME)?
r1
r s
r2
DME Method (for
local interactions)
:
with
where
Basic Idea:
symmetries of r
Constraints
Asymptotic
conditions
Example of decomposition:
Taylor
0
Negele, Vautherin, PRC (1972)
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Discussion on constructive Framework:
Example :
from Density Matrix Expansion (DME)?
Single-Vacuum
Negele, Vautherin, PRC(1972)
Hartree-Fock
alone is not
sufficient
Hartree-Fock
+perturbative
expansion
(Bogner Talk, ECT*
April 2007)
j1
j1+ j3
Multi-Vacua
Taylor
with
Some symmetries of r are lost
New terms in the EDF
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Discussion on constructive Framework:
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Denis Lacroix
Cures and Pathologies of MR-EDF – Saclay November 2007
Some remarks on the construction of Skyrme EDF