modern nuclear energy density functional methods · modern nuclear energy density functional...
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![Page 1: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/1.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Modern nuclear energy density functional methods
T. Duguet
1DSM/Irfu/SPhN, CEA Saclay, France
2National Superconducting Cyclotron Laboratory,Department of Physics and Astronomy, Michigan State University, USA
Euroschool on exotic beams, Jyvaskyla, Finland, Aug. 20th - 26th, 2011
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Lecture series
0utline
1 Basics of the nuclear energy density functional method
■ Elements of formalism
■ Form, performances and limitations of empirical energy functionals
2 Typical applications and results
Practice session
1 Deriving the (simplified) Skyrme energy density functional
2 Computing the corresponding equation of state of infinite nuclear matter
Present lectures assume that you are familiar with
1 Basics of quantum mechanics, e.g. Dirac notation, variational method, . . .
2 Second quantization formulation of quantum mechanics, i.e. ai ;a†j
3 Basics of many-body technique, e.g. Slater determinant, Hartree-Fock, . . .
EDF methods
![Page 3: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/3.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Lecture series
0utline
1 Basics of the nuclear energy density functional method
■ Elements of formalism
■ Form, performances and limitations of empirical energy functionals
2 Typical applications and results
Practice session
1 Deriving the (simplified) Skyrme energy density functional
2 Computing the corresponding equation of state of infinite nuclear matter
Present lectures assume that you are familiar with
1 Basics of quantum mechanics, e.g. Dirac notation, variational method, . . .
2 Second quantization formulation of quantum mechanics, i.e. ai ;a†j
3 Basics of many-body technique, e.g. Slater determinant, Hartree-Fock, . . .
EDF methods
![Page 4: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/4.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 5: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/5.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 6: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/6.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
What is low-energy nuclear physics interested in?
In generic terms
■ Spectrum of H |ΨAµ 〉= EA
µ |ΨAµ 〉 for all A = N + Z
■ Observables for each state, e.g. r2 ≡ 〈ΨAµ |∑A
kr2k |Ψ
Aµ 〉/A
■ Decays between |ΨAµ 〉, i.e. nuclear, electromagnetic, electro-weak
Ground state
Mass, deformation
Spectroscopy
Excitations modes
Limits
Drip-lines, halos
Reaction properties
Fusion, transfer...
Heavy elements
Fission, fusion, SHE
Astrophysics
NS, SN, r-process
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Towards a unified theoretical description of nucleonic matter
Extended matter ρ ∈ [0,∼ 3ρsat]
-50
0
50
100
150
200
250
300
350
0 0.16 0.32 0.48 0.64 0.8 0.96
E/A
ρ [fm-3
]
f3f4f5
SLY5’
[T. Lesinski et al., PRC74 (2006) 044315]
Theoretical methods
1 "Exact" (VMC, . . . )
2 Ab-initio (SCGF, BHF, . . . )
3 Effective (EDF)
Finite nuclei Z ∈ [0,118+]
Theoretical methods
1 "Exact" (GFMC, NCSM, . . . )
2 Ab-initio (CC, SCGF, IMSRG)
3 Effective (SM, EDF)
EDF methods
![Page 8: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/8.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Basic ingredients
1 Approach NOT based on {H ; |Ψ〉} ; i.e. E = 〈Ψ|H |Ψ〉 with |Ψ〉 ≈ |Ψexact〉
2 Key object is the off-diagonal energy kernel Eqq′
Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′
ij ,κqq′
ij ,κq′q∗ij ]
which is a functional of one-body transition density matrices
ρqq′
ij ≡〈Φq |a†j ai |Φ
q′〉
〈Φq|Φq′ 〉; κqq′
ij ≡〈Φq |ajai |Φ
q′〉
〈Φq|Φq′ 〉; κq′q ∗
ij ≡〈Φq|a†i a
†j |Φ
q′ 〉
〈Φq |Φq′〉
with {a†i }= arbitrary single-particle basis; e.g. i = (~r ,σ,τ )
■ |Φq〉= product (Bogoliubov) state with collective label q
■ EDF kernel re-sums bulk correlations through functional character
■ Approach relies on symmetry breaking and restoration (J2,N ,Z ,Π, . . .)
3 Empirical param. (Gogny, Skyrme,. . . ) successful but lack predictive power
EDF methods
![Page 9: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/9.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Basic ingredients
1 Approach NOT based on {H ; |Ψ〉} ; i.e. E = 〈Ψ|H |Ψ〉 with |Ψ〉 ≈ |Ψexact〉
2 Key object is the off-diagonal energy kernel Eqq′
Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′
ij ,κqq′
ij ,κq′q∗ij ]
which is a functional of one-body transition density matrices
ρqq′
ij ≡〈Φq |a†j ai |Φ
q′〉
〈Φq|Φq′ 〉; κqq′
ij ≡〈Φq |ajai |Φ
q′〉
〈Φq|Φq′ 〉; κq′q ∗
ij ≡〈Φq|a†i a
†j |Φ
q′ 〉
〈Φq |Φq′〉
with {a†i }= arbitrary single-particle basis; e.g. i = (~r ,σ,τ )
■ |Φq〉= product (Bogoliubov) state with collective label q
■ EDF kernel re-sums bulk correlations through functional character
■ Approach relies on symmetry breaking and restoration (J2,N ,Z ,Π, . . .)
3 Empirical param. (Gogny, Skyrme,. . . ) successful but lack predictive power
EDF methods
![Page 10: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/10.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Basic ingredients
1 Approach NOT based on {H ; |Ψ〉} ; i.e. E = 〈Ψ|H |Ψ〉 with |Ψ〉 ≈ |Ψexact〉
2 Key object is the off-diagonal energy kernel Eqq′
Eqq′ ≡ E [〈Φq |; |Φq′〉] = E [ρqq′
ij ,κqq′
ij ,κq′q∗ij ]
which is a functional of one-body transition density matrices
ρqq′
ij ≡〈Φq |a†j ai |Φ
q′〉
〈Φq|Φq′ 〉; κqq′
ij ≡〈Φq |ajai |Φ
q′〉
〈Φq|Φq′ 〉; κq′q ∗
ij ≡〈Φq|a†i a
†j |Φ
q′ 〉
〈Φq |Φq′〉
with {a†i }= arbitrary single-particle basis; e.g. i = (~r ,σ,τ )
■ |Φq〉= product (Bogoliubov) state with collective label q
■ EDF kernel re-sums bulk correlations through functional character
■ Approach relies on symmetry breaking and restoration (J2,N ,Z ,Π, . . .)
3 Empirical param. (Gogny, Skyrme,. . . ) successful but lack predictive power
EDF methods
![Page 11: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/11.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Two successive levels of implementation
1 Single-Reference ("mean-field")
■ Invokes single |Φq〉 and diagonal energy kernel ESR ≡Min{|Φq〉} E|q||q|
■ |Φq〉 may break symmetries and acquire finite order parameters |q|eiϕq
■ Provides first approx to BE, 〈r2ch〉, ρτ (~r), β2 and ESPE {ǫα}
2 Multi-Reference ("beyond mean-field")
■ Mixes off-diagonal energy kernels associated with chosen MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
■ Large-amplitude quantum fluctuations
■ Treats collective vibrations (|q|)■ Restores broken symmetries (ϕq)■ Includes associated G.S. correlations■ Provides associated collective excitations
■ QRPA, Bohr-Hamiltonian are approximations
EDF methods
![Page 12: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/12.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Two successive levels of implementation
1 Single-Reference ("mean-field")
■ Invokes single |Φq〉 and diagonal energy kernel ESR ≡Min{|Φq〉} E|q||q|
■ |Φq〉 may break symmetries and acquire finite order parameters |q|eiϕq
■ Provides first approx to BE, 〈r2ch〉, ρτ (~r), β2 and ESPE {ǫα}
2 Multi-Reference ("beyond mean-field")
■ Mixes off-diagonal energy kernels associated with chosen MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
■ Large-amplitude quantum fluctuations
■ Treats collective vibrations (|q|)■ Restores broken symmetries (ϕq)■ Includes associated G.S. correlations■ Provides associated collective excitations
■ QRPA, Bohr-Hamiltonian are approximations
EDF methods
![Page 13: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/13.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy density functional method in a nutshell
Two successive levels of implementation
1 Single-Reference ("mean-field")
■ Invokes single |Φq〉 and diagonal energy kernel ESR ≡Min{|Φq〉} E|q||q|
■ |Φq〉 may break symmetries and acquire finite order parameters |q|eiϕq
■ Provides first approx to BE, 〈r2ch〉, ρτ (~r), β2 and ESPE {ǫα}
2 Multi-Reference ("beyond mean-field")
■ Mixes off-diagonal energy kernels associated with chosen MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
■ Large-amplitude quantum fluctuations
■ Treats collective vibrations (|q|)■ Restores broken symmetries (ϕq)■ Includes associated G.S. correlations■ Provides associated collective excitations
■ QRPA, Bohr-Hamiltonian are approximations
EDF methods
![Page 14: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/14.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 15: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/15.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 16: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/16.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Auxiliary many-body state
Single-particle basis {|α〉}
■ Creation operators a†α|0〉 = |α〉 with
{
aα,a†β
}
= δαβ
■ Wave-functions ψα(~rστ )≡ 〈~rστ |α〉
Slater determinant
1 Standard product state
|Φ〉 ≡
N∏
µ=1
b†µ |0〉
b†µ ≡
∑
α
Uαµ a†α
2 Unitary transformation U
3 Effective "vacuum"
b†µ |Φ〉 = 0 forµ ∈ [1,N ]
bµ |Φ〉 = 0 forµ ∈ [N +1,∞]
4 Particle number N |Φ〉 = N |Φ〉
Bogoliubov state
1 Product of quasi-particle operators
|Φ〉 ≡∏
µ
βµ |0〉
βµ ≡∑
α
U∗αµ aα+ V
∗αµ a
+α
2 Unitary transformation
(
U † V †
V T U T
)
3 Vacuum of quasi-particle operators
βµ |Φ〉 = 0 ∀µ
4 Symmetry breaking N |Φ〉 6= N |Φ〉
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Auxiliary many-body state
Single-particle basis {|α〉}
■ Creation operators a†α|0〉 = |α〉 with
{
aα,a†β
}
= δαβ
■ Wave-functions ψα(~rστ )≡ 〈~rστ |α〉
Slater determinant
1 Standard product state
|Φ〉 ≡
N∏
µ=1
b†µ |0〉
b†µ ≡
∑
α
Uαµ a†α
2 Unitary transformation U
3 Effective "vacuum"
b†µ |Φ〉 = 0 forµ ∈ [1,N ]
bµ |Φ〉 = 0 forµ ∈ [N +1,∞]
4 Particle number N |Φ〉 = N |Φ〉
Bogoliubov state
1 Product of quasi-particle operators
|Φ〉 ≡∏
µ
βµ |0〉
βµ ≡∑
α
U∗αµ aα+ V
∗αµ a
+α
2 Unitary transformation
(
U † V †
V T U T
)
3 Vacuum of quasi-particle operators
βµ |Φ〉 = 0 ∀µ
4 Symmetry breaking N |Φ〉 6= N |Φ〉
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Auxiliary many-body state
Single-particle basis {|α〉}
■ Creation operators a†α|0〉 = |α〉 with
{
aα,a†β
}
= δαβ
■ Wave-functions ψα(~rστ )≡ 〈~rστ |α〉
Slater determinant
1 Standard product state
|Φ〉 ≡
N∏
µ=1
b†µ |0〉
b†µ ≡
∑
α
Uαµ a†α
2 Unitary transformation U
3 Effective "vacuum"
b†µ |Φ〉 = 0 forµ ∈ [1,N ]
bµ |Φ〉 = 0 forµ ∈ [N +1,∞]
4 Particle number N |Φ〉 = N |Φ〉
Bogoliubov state
1 Product of quasi-particle operators
|Φ〉 ≡∏
µ
βµ |0〉
βµ ≡∑
α
U∗αµ aα+ V
∗αµ a
+α
2 Unitary transformation
(
U † V †
V T U T
)
3 Vacuum of quasi-particle operators
βµ |Φ〉 = 0 ∀µ
4 Symmetry breaking N |Φ〉 6= N |Φ〉
EDF methods
![Page 19: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/19.jpg)
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Single-reference one-body density matrices
Hermitian normal density matrix
ραβ ≡〈Φ|a†βaα|Φ〉
〈Φ|Φ〉
=∑
µ
V∗αµV
Tµβ
= ρ∗βα
Antisymmetric anomalous density matrix
καβ ≡〈Φ|aβaα|Φ〉
〈Φ|Φ〉
=∑
µ
V∗αµU
Tµβ
= −κβα
EDF methods
![Page 20: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/20.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Single-reference one-body density matrices
Hermitian normal density matrix
ραβ ≡〈Φ|a†βaα|Φ〉
〈Φ|Φ〉
Antisymmetric anomalous density matrix
καβ ≡〈Φ|aβaα|Φ〉
〈Φ|Φ〉
Slater determinant
1 Fixed particle number κ= 0
2 ρ is idempotent ρ2 = ρ
3 In basis {b†µ} diagonalizing ρ
ρ=
1 0 . . . 0 . . .
0. . .
... 10 0...
. . .
with ρµµ = individual occupations
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Single-reference one-body density matrices
Hermitian normal density matrix
ραβ ≡〈Φ|a†βaα|Φ〉
〈Φ|Φ〉
Antisymmetric anomalous density matrix
καβ ≡〈Φ|aβaα|Φ〉
〈Φ|Φ〉
Slater determinant
1 Fixed particle number κ= 0
2 ρ is idempotent ρ2 = ρ
3 In basis {b†µ} diagonalizing ρ
ρ=
1 0 . . . 0 . . .
0. . .
... 10 0...
. . .
with ρµµ = individual occupations
Bogoliubov state
1 Indefinite particle number κ 6= 0
2 Generalized density matrix
R≡
(
ρ κ
−κ∗ 1−ρ∗
)
=R2
3 In basis {c†ν} diagonalizing ρ
|Φ〉 =∏
ν>0
(
uν + vν c+ν c
+ν
)
|0〉
BCS-like state with occupations ρνν = v2ν
EDF methods
![Page 22: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/22.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Single-reference one-body density matrices
Hermitian normal density matrix
ραβ ≡〈Φ|a†βaα|Φ〉
〈Φ|Φ〉
Antisymmetric anomalous density matrix
καβ ≡〈Φ|aβaα|Φ〉
〈Φ|Φ〉
Coordinate basis {|~rστ 〉}
■ Basis transformation
c~rστ =∑
α
ψα(~rστ ) aα
■ Density matrices are non-local functions in space
ρ~rστ~r ′σ′τ ′ =∑
αβ
ψ∗β(~r ′σ′τ )ψα(~rστ )ραβ δττ ′
κ~rστ~r ′σ′τ ′ =∑
αβ
ψβ(~r ′σ′τ )ψα(~rστ )καβ δττ ′
EDF methods
![Page 23: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/23.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Single-reference one-body density matrices
Hermitian normal density matrix
ραβ ≡〈Φ|a†βaα|Φ〉
〈Φ|Φ〉
Antisymmetric anomalous density matrix
καβ ≡〈Φ|aβaα|Φ〉
〈Φ|Φ〉
Physical content in coordinate basis
1 Value at ~r = ~r ′ ⇔ average nucleonic density distribution
ρτ (~r) ≡∑
σ
ρ~rστ~r ′στ∣
∣
~r=~r ′
2 One derivative at ~r = ~r ′ ⇔ average nucleonic current distribution
~jτ (~r) ≡i
2
(
~∇′− ~∇)
∑
σ
ρ~rστ~r ′στ∣
∣
~r=~r ′
3 Two derivatives at ~r = ~r ′ ⇔ average nucleonic kinetic distribution
ττ (~r) ≡ ~∇· ~∇′∑
σ
ρ~rστ~r ′στ∣
∣
~r=~r ′
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Ansatz
SR-EDF ansatz
■ Diagonal energy kernel postulated as a functional E ≡ E [ρ,κ,κ∗]
■ The energy results from the minimization of the diagonal kernel
ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]
within the manyfold span by product states |Φ〉
■ EDF kernel re-sums bulk correlations through functional character
Minimization and constraints
1 Independent variables are {ραβ , ρ∗αβ , καβ , κ
∗αβ for β ≤ α}
2 The constraint that R2 =R must be enforced
➨ Matrix Λ of Lagrange parameters (one per isospin τ )
3 Constrain 〈Φ|N |Φ〉 = Tr{ρ}= N as a free variation would lead to ESR→−∞
➨ Lagrange parameter λ (one per isospin τ )
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Ansatz
SR-EDF ansatz
■ Diagonal energy kernel postulated as a functional E ≡ E [ρ,κ,κ∗]
■ The energy results from the minimization of the diagonal kernel
ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]
within the manyfold span by product states |Φ〉
■ EDF kernel re-sums bulk correlations through functional character
Minimization and constraints
1 Independent variables are {ραβ , ρ∗αβ , καβ , κ
∗αβ for β ≤ α}
2 The constraint that R2 =R must be enforced
➨ Matrix Λ of Lagrange parameters (one per isospin τ )
3 Constrain 〈Φ|N |Φ〉 = Tr{ρ}= N as a free variation would lead to ESR→−∞
➨ Lagrange parameter λ (one per isospin τ )
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Equation of motion
Minimization
1 Set δR
[
E [ρ,κ,κ∗]−λTr{ρ}−Tr{Λ(R2−R)}]
= 0 leads to [H,R] = 0
H≡
(
h−λ ∆−∆∗ −h∗+λ
)
where the effective one-body fields are defined through
hαγ ≡δE [ρ,κ,κ∗]
δργα; ∆αγ ≡
δE [ρ,κ,κ∗]
δκ∗αγ
2 This is equivalent to solving the Bogoliubov-De-Gennes eigenvalue problem
H
(
U
V
)
µ
= Eµ
(
U
V
)
µ
3 {(U ,V )µ} and {Eµ} denote quasi-particle states and energies
4 The field h drives the shell structure while ∆ drives pair scattering
5 Iterative procedure H[{(U ,V )[n]µ }]→{(U ,V )
[n+1]µ } →H[{(U ,V )
[n+1]µ }]→ . . .
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Iterative procedure and self-consistency
In
1 E ≡ E [ρ,κ,κ∗]
2 N ,Z
3 |α〉(0)
4 (U ,V )(0)µ
E (n)
(ρ,κ)(n)(h,∆)(n)
(U ,V )(n+1)µ
δEDF
Bogo
SR
Out
■ ESR
■ (U ,V )µ
■ Eµ
■ ρ(~r),r2ch
■ . . .
EDF methods
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Individual excitations
HFB spectrum Eµ with ≈√
(hµµ−λ)2 + ∆µµ and without = |hµµ−λ| pairing
Pairing gap opens at low energy + mix of hole- and particle-like excitations
Slater determinant
1 Auxiliary excited state
|Φph〉 ≡ b†p bh |Φ〉
2 Spectrum (hµν ≡ ǫµδµν)
ESRph −E
SR ≈ ǫp− ǫh≥ 0
Bogoliubov state
1 Auxiliary excited state
|Φµν〉 ≡ ↵β†ν |Φ〉
2 Spectrum
ESRαβ −E
SR ≈ Eα+ Eβ ≥ 2∆F
EDF methods
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Observable
At convergence
1 Binding energy ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]
2 Density matrices (ρ,κ) from which other observables can be computed, e.g.
Local matter density ρ0(~r) ≡∑
σ,τ
ρ~rστ~rστ
Matter square radius r2mat ≡
1
A
∫
d~r r2 ρ0(~r)
SR-EDF method
1 Correlations that vary smoothly with N and Z are mocked up
1 Through rich (enough?) functional form
2 Through the fitting of its parameters
2 Not all correlations are easily resummed into E [ρ,κ,κ∗] itself
1 Symmetry breaking captures important correlations
2 Further correlations brought by restoring symmetries = MR-EDF
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Observable
At convergence
1 Binding energy ESR ≡Min{|Φ〉} E [ρ,κ,κ∗]
2 Density matrices (ρ,κ) from which other observables can be computed, e.g.
Local matter density ρ0(~r) ≡∑
σ,τ
ρ~rστ~rστ
Matter square radius r2mat ≡
1
A
∫
d~r r2 ρ0(~r)
SR-EDF method
1 Correlations that vary smoothly with N and Z are mocked up
1 Through rich (enough?) functional form
2 Through the fitting of its parameters
2 Not all correlations are easily resummed into E [ρ,κ,κ∗] itself
1 Symmetry breaking captures important correlations
2 Further correlations brought by restoring symmetries = MR-EDF
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Symmetries
Symmetries of H and quantum numbers
■ In nuclear physics [X ,H ] = 0 for {X}= {N , Z , ~P, J2, Jz , Π, T2}
■ Solutions |Ψxi 〉 are eigenstates of {X} and labeled by quantum numbers {x}
➨ (N ,Z) ∈ N2
➨ ~P ∈ R3
➨ 2J ∈ N and 2M ∈ Z such that −J ≤M ≤+J
➨ Π =±1➨ T 2 = (−1)N (−1)Z
■ The nuclear part of H also nearly commutes with isospin operators (T2, Tz)
In which space do we actually vary |Φ〉/{ρ,κ} when minimizing E [ρ,κ,κ∗]?
■ Natural to vary such that |Φ〉 carries the same quantum numbers {x} as |Ψxi 〉
➨ Too constraining for simple product states, e.g. Slater determinants
■ Example: ~P|Φ〉 = ~P|Φ〉 ⇒ ϕα(~r) = ϕ~k(~r) = ei~k·~r
■ Miss correlations that induce spatial localization of the internal motion■ A similar situation holds for all symmetries
➨ Need to expand our horizon. . .
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Symmetries
Symmetries of H and quantum numbers
■ In nuclear physics [X ,H ] = 0 for {X}= {N , Z , ~P, J2, Jz , Π, T2}
■ Solutions |Ψxi 〉 are eigenstates of {X} and labeled by quantum numbers {x}
➨ (N ,Z) ∈ N2
➨ ~P ∈ R3
➨ 2J ∈ N and 2M ∈ Z such that −J ≤M ≤+J
➨ Π =±1➨ T 2 = (−1)N (−1)Z
■ The nuclear part of H also nearly commutes with isospin operators (T2, Tz)
In which space do we actually vary |Φ〉/{ρ,κ} when minimizing E [ρ,κ,κ∗]?
■ Natural to vary such that |Φ〉 carries the same quantum numbers {x} as |Ψxi 〉
➨ Too constraining for simple product states, e.g. Slater determinants
■ Example: ~P|Φ〉 = ~P|Φ〉 ⇒ ϕα(~r) = ϕ~k(~r) = ei~k·~r
■ Miss correlations that induce spatial localization of the internal motion■ A similar situation holds for all symmetries
➨ Need to expand our horizon. . .
EDF methods
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Spontaneous symmetry breaking
Variation allowing for spontaneous symmetry breaking
■ Let |Φ〉 span several irreducible representations of the symmetry group G
|Φ〉 ≡∑
x
cx |Θx〉
✔ ESR lower than for a symmetry-conserving variation
■ Specific long-range correlations are included in this way
■ SM method grasps such correlations through |Ψx〉 ≈∑
jaj |Φ
xj 〉
✘ ESR does not access the ground-state energy but the one of a wave-packet
ESR =
∑
x
|cx |2
Ex
Symmetries have to be restored eventually ⇒ MR-EDF method
■ Additional ground-state correlation energy by extracting Ex
■ Compute reliably quantities dictated by selection rules, e.g. B(E2)
■ Particularly important for weakly-broken symmetries
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Spontaneous symmetry breaking
Variation allowing for spontaneous symmetry breaking
■ Let |Φ〉 span several irreducible representations of the symmetry group G
|Φ〉 ≡∑
x
cx |Θx〉
✔ ESR lower than for a symmetry-conserving variation
■ Specific long-range correlations are included in this way
■ SM method grasps such correlations through |Ψx〉 ≈∑
jaj |Φ
xj 〉
✘ ESR does not access the ground-state energy but the one of a wave-packet
ESR =
∑
x
|cx |2
Ex
Symmetries have to be restored eventually ⇒ MR-EDF method
■ Additional ground-state correlation energy by extracting Ex
■ Compute reliably quantities dictated by selection rules, e.g. B(E2)
■ Particularly important for weakly-broken symmetries
EDF methods
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Physical content and patterns
Origin and manifestation of spontaneous symmetry breaking
Group Correlations
Label Casimir V NN Internal motion
Translation ~a T (3) P Short range Spatial localization
Rotation ϕ U (1) N S-wave virtual state Pairing
Rotation α,β,γ SO(3) J2 Quadrup.-quadrup. Angular localization
Nuclei and spontaneous symmetry breaking
Nuclei Excitation pattern
Translation ~a All of them Surface vibrationsRotation ϕ All but doubly-magic ones Energy gapRotation α,β,γ All but singly-magic ones Rotational bands
■ Symmetries are enforced or relaxed according to which nucleus is studied
■ Enforcing symmetries leads to a great gain in CPU time
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Physical content and patterns
Origin and manifestation of spontaneous symmetry breaking
Group Correlations
Label Casimir V NN Internal motion
Translation ~a T (3) P Short range Spatial localization
Rotation ϕ U (1) N S-wave virtual state Pairing
Rotation α,β,γ SO(3) J2 Quadrup.-quadrup. Angular localization
Nuclei and spontaneous symmetry breaking
Nuclei Excitation pattern
Translation ~a All of them Surface vibrationsRotation ϕ All but doubly-magic ones Energy gapRotation α,β,γ All but singly-magic ones Rotational bands
■ Symmetries are enforced or relaxed according to which nucleus is studied
■ Enforcing symmetries leads to a great gain in CPU time
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Symmetry breaking and densities
Symmetry, densities and collective deformation
■ Quantum numbers ~P,(L,M ),(N ,Z) . . . translate into properties of {ρ,κ}
1 Local density ρ(~r) is constant over space for state with any good ~P
2 Local density ρ(~r) is spherically symmetric for state with L = 0
3 Anomalous density καβ is a null matrix for state with fixed N ,Z
■ Symmetry breaking translates into a violation of such properties
➨ Deformation with respect to a collective variable
~P vs density localization L = 0 vs density deformation
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Symmetry breaking and densities
Symmetry, densities and collective deformation
■ Quantum numbers ~P,(L,M ),(N ,Z) . . . translate into properties of {ρ,κ}
■ Symmetry breaking translates into a violation of such properties
■ Deformation with respect to a collective variable
➨ Characterized by a non-zero order parameter q ≡ |q|eiArg(q)
Group Order parameter"Phase" Arg(q) Magnitude |q|
SO(3) Euler angles (α,β,γ) ρλµ ≡ 〈rλYµλ 〉
{
odd λ≤ 2L and any µeven λ≤ 2L and µ 6= 0any λ > 2L and any µ
U (1) Gauge angle ϕ ‖κ‖ such that καβ ≡ 〈Φ|aβaα|Φ〉
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Minimization and correlation energy
Order parameter q ≡ |q|eiArg(q) and energy minimization
■ ESR is independent of Arg(q) as Eqq = E|q||q|
■ The magnitude |q| may vary when minimizing Eqq
1 Variation at fixed |q|= 0 provides symmetry-conserving solution2 Free variation (|q| 6= 0) may provide a lower solution in the larger space3 Constrained variation via Lagrange (−λq 〈Φq |Q|Φq〉) to access E|q||q|
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Minimization and correlation energy
Order parameter ”|q|eiArg(q)” and energy minimization
■ ESR is independent of Arg(q) as Eqq = E|q||q|
■ The magnitude |q| may vary when minimizing Eqq
1 Variation at fixed |q|= 0 provides symmetry-conserving solution2 Free variation (|q| 6= 0) may provide a lower solution in the larger space3 Constrained variation via Lagrange (−λq 〈Φq |Q|Φq〉) to access E|q||q|
Quadrupole (J/ = 0) correlation ∆Eρ20
[M. Bender, private communication]
Octupole (Jπ/ = 0+) correlation ∆Eρ30
[M. Bender, private communication]
EDF methods
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Spatial symmetry and quantum numbers
Symmetries of quasi-particle wave-functions (U ,V )µ
■ Quantum numbers µ reflects symmetries of (h,∆) and thus of (ρ,κ)
Sym. ρ Pref. coord. Broken sym. Quant. numb. Degeneracy
ρ (x,y,z) none (kxkykzσq) ρ(E)
ρ00 (r , θ,φ) ~p (nljmq) 2j + 1
ρ2l0 (ρ,θ,z) ~p, j2 (nπmq) 2
ρ2lµ none ~p, j2, jz (nπζq) 2
ρλµ none ~p, j2, jz ,π (nζq) 2
ρλµ none ~p, j2, jz ,π,T (nq) 0
➨ Degeneracies of Eµ reflect symmetries of (h,∆)/(ρ,κ)
Practical consequences
■ Symmetries are enforced or relaxed according to which nucleus is studied
■ Enforcing symmetries leads to a great gain in CPU time
EDF methods
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Deformation and Nilsson diagram
Calculation of 250Fm (N = 150,Z = 100) as a function of β2 ∝ ρ20
■ Spectrum of hϕα = ǫαϕα as a function of β2 = Nilsson diagram
[M. Bender, P.-H. Heenen, unpublished]
■ β2,min is a compromise between N = 150 and Z ≈ 100 deformed gaps
EDF methods
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Manifestations of pairing correlations in nuclei
Fingerprint at finite density of n-n virtual state in vacuum
■ Additional ground-state correlations
■ Odd-even mass staggering (OEMS)
■ Gap opens up in individual excitation spectrum
■ Moment of inertia ր with J ⇐⇒ Meissner effect in superconductors
Pairing (N/ = 70) correlation ∆E‖κ‖
Data from [M. Bender, T.D., IJME16 (2007) 222]
Pairing correlations
■ Weak symmetry breaking
■ |∆E‖κ‖| ≪∆Eρ20
■ |∆E‖κ‖| ≥ needed accuracy
■ Sharp collapse for ‖κ‖ ≤ ‖κ‖crit.
EDF methods
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Manifestations of pairing correlations in nuclei
Fingerprint at finite density of n-n virtual state in vacuum
■ Additional ground-state correlations
■ Odd-even mass staggering (OEMS)
■ Gap opens up in individual excitation spectrum
■ Moment of inertia ր with J ⇐⇒ Meissner effect in superconductors
Odd-even mass staggering
From [S. Baroni, private communication]
Gap extracted from the OEMS
Data from [T. D., T. Lesinski, arXiv:0907.1043]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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SR-EDF numerical codes
Some published SR-EDF (Skyrme) codes
1 1D code with spherical symmetry HFBRAD[K. Bennaceur, J. Dobaczewski, Comp. Phys. Comm. 168 (2005) 96]
http://www.sciencedirect.com/science/article/pii/S0010465505002304
2 2D code with axial symmetry HFBTHO[M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comp. Phys. Comm. 167 (2005) 43]
http://www.sciencedirect.com/science/article/pii/S0010465505000305
3 3D code with 3 symmetry planes ev8[P. Bonche, H. Flocard, P.-H. Heenen, Comp. Phys. Comm. 171 (2005) 49]
http://www.sciencedirect.com/science/article/pii/S0010465505002821
4 3D code with no symmetry HFODD[J. Dobaczewski et al., Comp. Phys. Comm. 180 (2009) 2361]
http://www.sciencedirect.com/science/article/pii/S0010465509002598
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of the single-reference method
Using a single reference state |Φq〉
■ Casimir operator A, i.e. J2, of G
A |Φqmin〉 6= a|Φqmin〉 with a≡ Irrep(a)
➨ Zero-energy mode at any |q| 6= 0
Collective landscape
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of the single-reference method
Using a single reference state |Φq〉
■ Casimir operator A, i.e. J2, of G
A |Φqmin〉 6= a|Φqmin〉 with a≡ Irrep(a)
➨ Zero-energy mode at any |q| 6= 0
■ What about using a fixed |q|min?
Different categories of nuclei (1) Stiff
[M. Bender, private communication]
Collective landscape
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of the single-reference method
Using a single reference state |Φq〉
■ Casimir operator A, i.e. J2, of G
A |Φqmin〉 6= a|Φqmin〉 with a≡ Irrep(a)
➨ Zero-energy mode at any |q| 6= 0
■ What about using a fixed |q|min?
➨ Soft mode as a function of |q|?
Different categories of nuclei (2) Soft
[M. Bender, private communication]
Collective landscape
EDF methods
![Page 53: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/53.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of the single-reference method
Using a single reference state |Φq〉
■ Casimir operator A, i.e. J2, of G
A |Φqmin〉 6= a|Φqmin〉 with a≡ Irrep(a)
➨ Zero-energy mode at any |q| 6= 0
■ What about using a fixed |q|min?
➨ Soft mode as a function of |q|?
Collective landscape
How to improve on that consistently?
■ Go beyond |Φq〉 at fixed q
➨ Mixing in |q| and Arg(q)
➨ Add dynamical correlations
➨ Provide excitations
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Multi-reference energy density functional method in a nutshell
Second level of implementation
■ Mixes off-diagonal energy kernels associated with MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
➨ Treats collective vibrations = mixing along |q|
➨ {f kq } obtained via minimization
➨ Restores broken symmetries = mixing along ϕq
➨ {f kq } fixed by symmetry-group structure
■ Large-amplitude quantum fluctuations
1 Includes G.S. correlations
2 Provides associated collective excitations
■ Common approximation schemes
1 QRPA = harmonic fluctuations
2 Bohr-Hamiltonian = GOA
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Multi-reference energy density functional method in a nutshell
Second level of implementation
■ Mixes off-diagonal energy kernels associated with MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
➨ Treats collective vibrations = mixing along |q|
➨ {f kq } obtained via minimization
➨ Restores broken symmetries = mixing along ϕq
➨ {f kq } fixed by symmetry-group structure
■ Large-amplitude quantum fluctuations
1 Includes G.S. correlations
2 Provides associated collective excitations
■ Common approximation schemes
1 QRPA = harmonic fluctuations
2 Bohr-Hamiltonian = GOA
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Multi-reference energy density functional method in a nutshell
Second level of implementation
■ Mixes off-diagonal energy kernels associated with MR set {|Φq〉}
EMRk ≡ Min{f k
q }
∑
q,q′∈MR f k ∗q f k
q′ Eqq′ 〈Φq|Φq′ 〉
∑
q,q′∈MR f k ∗q f k
q′〈Φq|Φq′ 〉
➨ Treats collective vibrations = mixing along |q|
➨ {f kq } obtained via minimization
➨ Restores broken symmetries = mixing along ϕq
➨ {f kq } fixed by symmetry-group structure
■ Large-amplitude quantum fluctuations
1 Includes G.S. correlations
2 Provides associated collective excitations
■ Common approximation schemes
1 QRPA = harmonic fluctuations
2 Bohr-Hamiltonian = GOA
EDF methods
![Page 57: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/57.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 58: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/58.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Building Eqq′ = E [〈Φq |; |Φq′〉] = E [ρqq′ ,κqq′ ,κq′q ∗]
The mathematical form
1 It can be as elaborate and as complicated as one finds empirically necessary
➨ It can be a very non-local functional in space, i.e.
Eqq′ =∑
σ1...σn ,τ1...τn
∫
· · ·
∫
d~r1 . . .d~rn E(ρqq′
~r1σ1τ1~r2σ2τ2, . . . ,κqq′
~rn−1σn−1τn−1~rnσnτn)
➨ One tries to limit the complexity and the number of free parameters
2 The form is constrained as Eqq′ (implicitly) relates to the scalar operator H
E [〈Φq|R†(g);R(g)|Φq′〉] = E [〈Φq|; |Φq′ 〉] for any R(g) ∈ G
Fitting its free parameters (so far done at the SR level)
1 Use empirical knowledge of Infinite Nuclear Matter equation of state
➨ esat, ρsat,K∞,m∗0 , asym + various ρinst⇒see practice session
2 Use a selected set of finite nuclei properties
➨ BE , r2ch,∆ǫj=l∓1/2,∆Efission
isomer, . . .
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Building Eqq′ = E [〈Φq |; |Φq′〉] = E [ρqq′ ,κqq′ ,κq′q ∗]
The mathematical form
1 It can be as elaborate and as complicated as one finds empirically necessary
➨ It can be a very non-local functional in space, i.e.
Eqq′ =∑
σ1...σn ,τ1...τn
∫
· · ·
∫
d~r1 . . .d~rn E(ρqq′
~r1σ1τ1~r2σ2τ2, . . . ,κqq′
~rn−1σn−1τn−1~rnσnτn)
➨ One tries to limit the complexity and the number of free parameters
2 The form is constrained as Eqq′ (implicitly) relates to the scalar operator H
E [〈Φq|R†(g);R(g)|Φq′〉] = E [〈Φq|; |Φq′ 〉] for any R(g) ∈ G
Fitting its free parameters (so far done at the SR level)
1 Use empirical knowledge of Infinite Nuclear Matter equation of state
➨ esat, ρsat,K∞,m∗0 , asym + various ρinst⇒see practice session
2 Use a selected set of finite nuclei properties
➨ BE , r2ch,∆ǫj=l∓1/2,∆Efission
isomer, . . .
EDF methods
![Page 61: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/61.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of E = E [ρ,κ,κ∗]
Strategy followed below
1 Diagonal kernel Eqq ≡ E [ρ,κ,κ∗]
2 Focus on the ρ dependence
Scalar and vector non-local densities
ρτ (~r ,~r′) ≡
∑
σσ′
ρ~rστ~r ′σ′τ 1σ′σ
sτ,ν(~r ,~r′) ≡
∑
σσ′
ρ~rστ~r ′σ′τ σσ′σν
Time-even local densities
ρτ (~r) ≡ ρτ (~r ,~r′)|~r=~r ′ Matter density
ττ (~r) ≡∑
µ
∇µ∇′µ ρτ (~r ,~r
′)|~r=~r ′ Kinetic density
Jτ,µν(~r) ≡i
2
(
∇′µ−∇µ)
sτ,ν(~r ,~r′)|~r=~r ′ Spin-current tensor density
Jτ,κ(~r) ≡
z∑
µ,ν=x
ǫκµν Jτ,µν(~r) Spin-orbit density
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of E = E [ρ,κ,κ∗]
Strategy followed below
1 Diagonal kernel Eqq ≡ E [ρ,κ,κ∗]
2 Focus on the ρ dependence
Scalar and vector non-local densities
ρτ (~r ,~r′) ≡
∑
σσ′
ρ~rστ~r ′σ′τ 1σ′σ
sτ,ν(~r ,~r′) ≡
∑
σσ′
ρ~rστ~r ′σ′τ σσ′σν
Time-odd local densities
sτ,ν(~r) ≡ sτ,ν(~r ,~r′)|~r=~r ′ Spin density
Tτ,ν(~r) ≡∑
µ
∇µ∇′µ sτ,ν(~r ,~r
′)|~r=~r ′ Spin-kinetic density
Fτ,µ(~r) ≡1
2
∑
ν
(
∇µ∇′ν +∇ν∇
′µ
)
sτ,ν(~r ,~r′)|~r=~r ′ Tensor-kinetic density
jτ,µ(~r) ≡i
2
(
∇′µ−∇µ)
ρτ (~r ,~r′)|~r=~r ′ Current density
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Building procedure
1 Local bilinear form up to two gradients and two Pauli matrices
2 Form scalars
3 Couplings Cff ′
ττ ′ may further depend on ~r ≡ higher-order terms
Skyrme energy density functional
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτ ρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′−~jτ ·~jτ ′)
+Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Building procedure
1 Local bilinear form up to two gradients and two Pauli matrices
2 Form scalars with respect to (i) Galilean transformation
3 Couplings Cff ′
ττ ′ may further depend on ~r ≡ higher-order terms
Skyrme energy density functional
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτ ρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′−~jτ ·~jτ ′)
+Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Building procedure
1 Local bilinear form up to two gradients and two Pauli matrices
2 Form scalars with respect to (ii) space rotation
3 Couplings Cff ′
ττ ′ may further depend on ~r ≡ higher-order terms
Skyrme energy density functional
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτ ρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′−~jτ ·~jτ ′)
+Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Building procedure
1 Local bilinear form up to two gradients and two Pauli matrices
2 Form scalars with respect to (iii) time reversal
3 Couplings Cff ′
ττ ′ may further depend on ~r ≡ higher-order terms
Skyrme energy density functional
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτ ρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′−~jτ ·~jτ ′)
+Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Building procedure
1 Local bilinear form up to two gradients and two Pauli matrices
2 Form scalars
3 Couplings Cff ′
ττ ′ may further depend on ~r ≡ higher-order terms
Skyrme energy density functional
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτ ρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′−~jτ ·~jτ ′)
+Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+C∇s∇sττ ′ (∇·~sτ ) (∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 Quasi zero-range two-body vertex with 9 parameters
Vskyrme ≡ Vcent + Vls + Vtens
Vcent = t0 (1 + x0Pσ)δ(~r)
+t1
2(1 + x1Pσ)
[
δ(~r)−→k
2 +←−k′2 δ(~r)
]
+ t2 (1 + x2Pσ)←−k′ · δ(~r)
−→k
Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)
−→k
Vtens =te
2
{
[
3(~σ1 ·←−k′ ) (~σ2 ·
←−k′ )− (~σ1 ·~σ2)
←−k′2]
δ(~r)
+δ(~r)[
3(~σ1 ·−→k ) (~σ2 ·
−→k )− (~σ1 ·~σ2)
−→k
2]}
+ to
{
3(~σ1 ·←−k′ )δ(~r) (~σ2 ·
−→k )− (~σ1 ·~σ2)
←−k′ · δ(~r)
−→k
}
2 E [ρ,κ,κ∗]≡ 〈Φ|T + Vskyrme|Φ〉 ⇐⇒ Effective Hartree-Fock
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 Quasi zero-range two-body vertex with 9 parameters
Vskyrme ≡ Vcent + Vls + Vtens
Vcent = t0 (1 + x0Pσ)δ(~r)
+t1
2(1 + x1Pσ)
[
δ(~r)−→k
2 +←−k′2 δ(~r)
]
+ t2 (1 + x2Pσ)←−k′ · δ(~r)
−→k
Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)
−→k
Vtens =te
2
{
[
3(~σ1 ·←−k′ ) (~σ2 ·
←−k′ )− (~σ1 ·~σ2)
←−k′2]
δ(~r)
+δ(~r)[
3(~σ1 ·−→k ) (~σ2 ·
−→k )− (~σ1 ·~σ2)
−→k
2]}
+ to
{
3(~σ1 ·←−k′ )δ(~r) (~σ2 ·
−→k )− (~σ1 ·~σ2)
←−k′ · δ(~r)
−→k
}
2 E [ρ,κ,κ∗]≡ 〈Φ|T + Vskyrme|Φ〉 ⇐⇒ Effective Hartree-Fock
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 Quasi zero-range two-body vertex with 9 parameters
Vskyrme ≡ Vcent + Vls + Vtens
Vcent = t0 (1 + x0Pσ)δ(~r)
+t1
2(1 + x1Pσ)
[
δ(~r)−→k
2 +←−k′2 δ(~r)
]
+ t2 (1 + x2Pσ)←−k′ · δ(~r)
−→k
Vls = iW0 ( ~σ1 + ~σ2)←−k′ ∧ δ(~r)
−→k
Vtens =te
2
{
[
3(~σ1 ·←−k′ ) (~σ2 ·
←−k′ )− (~σ1 ·~σ2)
←−k′2]
δ(~r)
+δ(~r)[
3(~σ1 ·−→k ) (~σ2 ·
−→k )− (~σ1 ·~σ2)
−→k
2]}
+ to
{
3(~σ1 ·←−k′ )δ(~r) (~σ2 ·
−→k )− (~σ1 ·~σ2)
←−k′ · δ(~r)
−→k
}
2 E [ρ,κ,κ∗]≡ 〈Φ|T + Vskyrme|Φ〉 ⇐⇒ Effective Hartree-Fock
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 〈Φ|T + Vskyrme|Φ〉 = purely bilinear Skyrme-type EDF ⇒see practice session
2 The 9 vertex parameters induce correlations among the 18 EDF couplings
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′ −~jτ ·~jτ ′)
+ Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+ CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
➨ Not enough (i) powers of the densities and (ii) freedom to fix the Cff ′
ττ ′
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 〈Φ|T + Vskyrme|Φ〉 = purely bilinear Skyrme-type EDF ⇒see practice session
2 The 9 vertex parameters induce correlations among the 18 EDF couplings
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′ −~jτ ·~jτ ′)
+ Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+ CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
➨ Not enough (i) powers of the densities and (ii) freedom to fix the Cff ′
ττ ′
EDF methods
![Page 74: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/74.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from an effective pseudo-potential
1 〈Φ|T + Vskyrme|Φ〉 = purely bilinear Skyrme-type EDF ⇒see practice session
2 The 9 vertex parameters induce correlations among the 18 EDF couplings
E [ρ,κ,κ∗] =∑
τ
∫
d~rh2
2mττ
+∑
ττ ′
∫
d~r
[
Cρρττ ′ ρτρτ ′ + C
ρ∆ρττ ′ ρτ∆ρτ ′ + C
ρτττ ′
(
ρτ ττ ′ −~jτ ·~jτ ′)
+ Cssττ ′ ~sτ ·~sτ ′ + C
s∆sττ ′ ~sτ ·∆~sτ ′ + C
ρ∇Jττ ′
(
ρτ ~∇·~Jτ ′ +~jτ · ~∇×~sτ ′)
+ C∇s∇sττ ′ (∇·~sτ )(∇·~sτ ′) + C
JJττ ′
(
∑
µν
Jτ,µνJτ ′,µν −~sτ · ~Tτ ′)
+ CJJττ ′
(
∑
µν
Jτ,µµ Jτ ′,νν + Jτ,µν Jτ ′,νµ−2~sτ · ~Fτ ′)
]
+ terms in [κ,κ∗]
➨ Not enough (i) powers of the densities and (ii) freedom to fix the Cff ′
ττ ′
EDF methods
![Page 75: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/75.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from a density-dependent effective pseudo-potential
1 Step towards EDF philosophy by adding a density dependence to Vskyrme
”Vcent” = t0 (1 + x0Pσ)δ(~r)
+1
2t1 (1 + x1Pσ)
[
δ(~r)−→k
2 +←−k′ 2 δ(~r)
]
+ t2 (1 + x2Pσ)←−k′ · δ(~r)
−→k
+1
6t3 (1 + x3Pσ)ρα0 (~r)δ(~r)
2 Compute E [ρ,κ,κ∗]≡ 〈Φ|T + ”Vskyrme”|Φ〉
➨ Only Cρρττ ′ further depend on ~r via ρα0 (~r)
3 Modern EDF approach = exploit full freedom offered by EDF kernel
■ Bypass ”Vskyrme” entirely to parameterize E [ρ,κ,κ∗] directly
■ Freedom to use any functional form; e.g. Padé, exponential. . .
■ Relax constraints between EDF couplings
➨ Responsible for serious problems that cannot be discussed here. . .
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Quasi-local form of Eqq′ = E [ρqq′ ,κqq′ ,κq′q ∗]
Historically: kernel derived from a density-dependent effective pseudo-potential
1 Step towards EDF philosophy by adding a density dependence to Vskyrme
”Vcent” = t0 (1 + x0Pσ)δ(~r)
+1
2t1 (1 + x1Pσ)
[
δ(~r)−→k
2 +←−k′ 2 δ(~r)
]
+ t2 (1 + x2Pσ)←−k′ · δ(~r)
−→k
+1
6t3 (1 + x3Pσ)ρα0 (~r)δ(~r)
2 Compute E [ρ,κ,κ∗]≡ 〈Φ|T + ”Vskyrme”|Φ〉
➨ Only Cρρττ ′ further depend on ~r via ρα0 (~r)
3 Modern EDF approach = exploit full freedom offered by EDF kernel
■ Bypass ”Vskyrme” entirely to parameterize E [ρ,κ,κ∗] directly
■ Freedom to use any functional form; e.g. Padé, exponential. . .
■ Relax constraints between EDF couplings
➨ Responsible for serious problems that cannot be discussed here. . .
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Single-particle field h
Local h derived from the Skyrme EDF ⇒see practice session
hij ≡δE
δρji=
∫
d~r ϕ†i (~r)hτ (~r)ϕj(~r)
hτ (~r) ≡ −~∇·Bτ (~r)~∇+ Uτ (~r)−i
2
z∑
µ,ν=x
[
Wτ,µν(~r)∇µ+∇µWτ,µν(~r)]
σν
−~∇·[
~Cτ (~r) ·~σ]
~∇+~Sτ (~r) ·~σ−1
2
[
~∇· ~Dτ (~r)~σ · ~∇+~σ · ~∇ ~Dτ (~r) · ~∇]
−i
2
[
~Aτ (~r) · ~∇+ ~∇· ~Aτ (~r)]
with multiplicative potentials defined through
Bτ (~r) ≡δE
δττ (~r); Uτ (~r)≡
δE
δρτ (~r); Wτ,µν(~r)≡
δE
δJτ,µν(~r)
Cτ,µ(~r) ≡δE
δTτ,µ(~r); Sτ,µ(~r)≡
δE
δsτ,µ(~r); Dτ,µ(~r)≡
δE
δFτ,µ(~r)
Aτ,µ(~r) ≡δE
δjτ,µ(~r)
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Nuclear shell structure
SR-EDF h drives the s.p. motion
1 Observable fragmented shell structure
E+µ ≡E
A+1µ −E
A0 with spectroscopic factor S
+µ
E−µ ≡E
A0 −E
A-1µ with spectroscopic factor S
−µ
2 h provides the centroid shell structure
hτ (~r)ϕα(~r) ≡ ǫαϕα(~r)
[T. D., J. Sadoudi, (2011) unpublished]
➨ Add dynamical correlations to access E±µ
➨ Configuration mixing via MR-EDF
1 Symmetry restorations
2 Dynamical particle-vibrations coupling
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
![Page 81: Modern nuclear energy density functional methods · Modern nuclear energy density functional methods T. Duguet 1DSM/Irfu/SPhN, CEA Saclay, France 2National Superconducting Cyclotron](https://reader034.vdocuments.net/reader034/viewer/2022050611/5fb2b57ed606a858bc20d044/html5/thumbnails/81.jpg)
Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Ground-state properties over the nuclear chart
■ Large-scale deformed calculations in one day
■ Masses, deformation, radii, s.p.e energies. . .
■ Tables for experimentalists/astrophysics
■ Ex: http://www-phynu.cea.fr/HFB-Gogny.htm
[S. Hilaire and M. Girod, Eur. Phys. J. A33 (2007) 237]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Neutron shell-structure in Sn and Pb isotopes
■ From SR-EDF calculation hϕα = ǫαϕα
■ ǫnlj ≡ centroid of one-nucleon separation energies
[T. D., J. Sadoudi, (2011) unpublished]
■ Too spread out to anticipate correlations
■ Current interest focuses on N−Z behavior
[M. Bender, G. F. Bertsch, P.-H. Heenen, PRC73 (2006) 034322]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Z = 50 magic gaps in Sn isotopes
■ From difference of −S2p(Z) = EZ0 −EZ+2
0
■ Importance of correlations to reproduce data
■ Current focus on evolution with N−Z
[M. Bender, G. F. Bertsch, P.-H. Heenen, PRC78 (2008) 054312]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Systematic of S2q and magic gaps
■ Visible (e.g. N = 50) shell quenching
■ Collective correlations are essential
■ Shell effects too pronounced in heavy nuclei
[M. Bender, G. F. Bertsch, P.-H. Heenen, PRC78 (2008) 054312]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Charge density distribution in 48Ca
■ ρch(~r)≈ ρp(~r)
■ Saturation density ρsat = 2ρp(0)≈ 0.16 fm−3
■ Stable nuclei = good reproduction of data
■ No measure yet in unstable nuclei[V. Rotival and T. D., unpublished]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Charge density distribution in 208Pb
■ ρch(~r)
■ Saturation density ρsat ≈ 2ρp(0)≈ 0.16 fm−3
■ Sensitive to shell effects
■ Sensitive to correlations[V. Rotival and T. D., unpublished]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Charge radii in Xe and Ba isotopes
■ Good reproduction of data
■ Correct N−Z dependence
■ Importance of correlations
[M. Bender, G. F. Bertsch, P.-H. Heenen, PRC73 (2006) 034322]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Systematic of charge radii
■ Good agreement (≤ 3%)
■ More difficult to extract r2n experimentally
■ Neutron skin√
r2n−√
r2p of importance
■ New interest triggered by JLAB exp. on 208Pb
[J. P. Delaroche et al., unpublished]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Halos in medium-mass nuclei
0 4 8 12r [fm]
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
n(r
) [f
m-3
]
54Cr
80Cr
■ Nuclei with anomalous extensions
■ Predictions done over the nuclear chart
■ Can only exist very close to neutron drip-line
■ Best candidates are Cr and Fe isotopes
[V. Rotival, T. D., PRC79 (2009) 054308]
[V. Rotival, K. Bennaceur, T. D., PRC79 (2009) 054309]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Halos in medium-mass nuclei
■ Nuclei with anomalous extensions
■ Predictions done over the nuclear chart
■ Can only exist very close to neutron drip-line
■ Best candidates are Cr and Fe isotopes
[V. Rotival, T. D., PRC79 (2009) 054308]
[V. Rotival, K. Bennaceur, T. D., PRC79 (2009) 054309]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Systematic of pairing gap in semi-magic nuclei
■ The nucleus is superfluid
■ Odd-even mass staggering is a measure of ∆τ
■ First microscopic calculations
■ All low-energy nuclear properties impacted
[T. Lesinski, K. Hebeler, T. D., A. Schwenk, JPG (2011) in press]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Isoscalar giant monopole resonance in 120Sn
■ QRPA = harmonic approximation of MR-EDF
■ Strength distribution
Sλ(E) =∑
i
∣
∣
∣〈ΨA
0 |Fλ|ΨAi 〉
∣
∣
∣
2
δ(EAi −EA
0 )
■ Resonances for all λ= Lπ and T = 0,1 with L≤ 3
■ Test features of EDF (K∞, m∗τ , asym, pairing. . . )
[J. Engel, unpublished]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Low-lying collective spectroscopy of 186Pb
■ Complex nucleus displaying shape coexistence
■ Good overall picture: bands, in/out B(E2)s
■ Too spread out spectra and too large E0
■ Extensions needed
[M. Bender et al., PRC69 (2004) 064303]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Nuclear fission of 238Pu
80 100 120 140 160
Fragment Mass
10-5
10-4
10-3
10-2
10-1
Yie
ld
■ Fission fragments properties: A, Ekin distrib.
■ SR-EDF ⇒ static paths for asymmetric fission
■ Bohr Hamiltonian ⇒ fission dynamics
■ Need exp. Z distrib., T1/2, n/γ with N −Z
[H. Goutte et al., PRC71 (2005) 024316]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Typical results from EDF calculations
Calculations
1 Systematics
2 Shells
3 Densities
4 Radii
5 Halos
6 Superfluidity
7 Resonances
8 Spectroscopy
9 Fission
10 Reactions
Reactions from time-dependent EDF calculations
■ Ex: fusion at the Coulomb barrier (16O+208Pb)
■ Pre-transfer, barrier distribution. . .
■ Note: full EDF essential to solve old problems
[C. Simenel and B. Avez, arXiv:0711.0934]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
Systematics of Eth−Eexp (σmass2135 ≈ 1.5 MeV from SR)
■ Ravines at magic numbers at SR level
■ Improved by dynamical correlations
■ Miss MR correlations, terms in EDF, fit?
[F. Chappert et al., unpublished]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
(S ,T ) content of nuclear EOS
-20
-10
0
10
20
30
40
50
60
0 0.08 0.16 0.24 0.32 0.4E
/A [
MeV
]ρ [fm-3]
f-f0f+
SLy5LNS
■ Good reproduction of SNM and PNM EOS
■ Wrong splitting in (S ,T ) channels
■ Current EDF are overconstrained
[T. Lesinski et al., PRC74 (2006) 044315]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
(S ,T ) content of nuclear EOS
-25-20-15-10-5 0 5
10
0 0.1 0.2 0.3
E/A
S,T
=0,
0 [M
eV]
ρ [fm-3] 0 0.1 0.2 0.3 0.4
-30-25-20-15-10-5
E/A
S,T
=0,
1 [M
eV]
ρ [fm-3]
SLy5
-30-20-10
0 10 20
E/A
S,T
=1,
0 [M
eV]
-10
-5
0
5
E/A
S,T
=1,
1 [M
eV]
f-f0f+
■ Good reproduction of SNM and PNM EOS
■ Wrong splitting in (S ,T ) channels
■ Current EDFs are overconstrained
[T. Lesinski et al., PRC74 (2006) 044315]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
Spin/isospin instabilities of nuclear matter
0 0.16 0.32 0.48 0.64
0 1 2 3s.-i
sos.
ρc
[fm
-3]
q [fm-1]
f-f0f+
SLy5
0 1 2 3 4 0 0.16 0.32 0.48 0.64
v.-i
sos.
ρc
[fm
-3]
q [fm-1]
0 0.16 0.32 0.48 0.64 0.8
s.-i
sov.
ρc
[fm
-3]
0 0.16 0.32 0.48 0.64 0.8
v.-i
sov.
ρc
[fm
-3]
■ ∞/finite size instabilities of INM
■ Potentially spurious predictions in nuclei
■ Need control but EDFs are overconstrained
[T. Lesinski et al., PRC74 (2006) 044315]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
∆ǫi = ǫi− ǫexpi in doubly-magic nuclei
0
5
10
15
20
-4 -2 0 2 4
SkO' (fit)
εε εε i - εε εε
iexp
(MeV)
0
5
10
15
20SkO'
εε εε i - εε εε
iexp
-4 -2 0 2 4
SkP (fit)
(MeV)
SkP
-4 -2 0 2 4
SLy5 (fit)
(MeV)
SLy5
[M. Kortelainen et al., PRC77 (2008) 064307]
■ Refit on 58 data points
■ ∆ǫrms ≥ 1.1 MeV = poor
■ Wrong ℓ-dependence of centroids/s.o. splittings
■ Wrong evolution with N−Z
■ Enriched EDF needed
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Limitations of current EDF methods
Limitations
1 Masses
2 Nuclear EOS
3 Instabilities
4 Shells
5 Predictability
Extrapolations towards the drip-line: masses
■ Agreement within models where fitted on data
■ Divergence "in the next major shell"
■ Strong limitations of empirical EDFs
[T. Lesinski et al., PRC74 (2006) 044315]
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy Density Functional method: single-reference implementation
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Energy Density Functional method: multi-reference implementation
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Towards non-empirical EDF parameterization and method
Design non-empirical Energy Density Functionals
■ Bridge with ab-initio many-body techniques
■ Calculate properties of heavy/complex nuclei from NN+NNN
Empirical
Predictive?
Finite nuclei and extended nuclear matter
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Towards non-empirical EDF parameterization and method
Design non-empirical Energy Density Functionals
■ Bridge with ab-initio many-body techniques
■ Calculate properties of heavy/complex nuclei from NN+NNN
Empirical
Predictive?
Non-empirical
Predictive...
Low-k NN+NNN QCD / χ-EFT
Finite nuclei and extended nuclear matter
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Outline
1 The energy density functional method in a nutshell
2 Single-reference EDF methodInputsEquation of motionBreaking symmetriesNumerical codes
3 Multi-reference EDF method
4 Empirical parametrization of the EDF kernelGeneral strategySkyrme family
5 ApplicationsTypical results from EDF calculationsLimitations of current EDF methods
6 Summary and outlook
7 Bibliography
EDF methods
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Introduction SR-EDF MR-EDF Parametrization Applications Summary Bibliography
Selected bibliography
P. Ring, P. Schuck,
The nuclear many-body problem, 1980, Springer-Verlag, Berlin
M. Bender, P.-H. Heenen, P.-G. Reinhard,
Rev. Mod. Phys. 75 (2003) 121
T. Duguet, K. Bennaceur, T. Lesinski, J. Meyer,
Opportunities with Exotic Beams, 2007, World Scientific, p. 21
K. Bennaceur, J. Dobaczewski,
Comp. Phys. Comm. 168 (2005) 96
M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring,
Comp. Phys. Comm. 167 (2005) 43
P. Bonche, H. Flocard, P.-H. Heenen,
Comp. Phys. Comm. 171 (2005) 49
J. Dobaczewski et al.,
Comp. Phys. Comm. 180 (2009) 2361
EDF methods