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Ab Initio Density FunctionalTheory for Nuclei
Dick Furnstahl
Department of PhysicsOhio State University
July, 2008
Outline
Overview: UNEDF project and Ab Initio DFT
Current Status and Possible Problems
Outlook and Additional Issues for Nuclear DFT
Extras
Outline
Overview: UNEDF project and Ab Initio DFT
Current Status and Possible Problems
Outlook and Additional Issues for Nuclear DFT
Extras
SciDAC 2 Project: Building a Universal NuclearEnergy Density Functional
Understand nuclear properties “for element formation, forproperties of stars, and for present and future energy anddefense applications”
Scope is all nuclei, with particular interest in reliablecalculations of unstable nuclei and in reactions
=⇒ DFT is method of choice
Order of magnitude improvement over present capabilities=⇒ precision calculations of masses, . . .
Connected to the best microscopic physics
Maximum predictive power with well-quantified uncertainties
[See http://www.scidacreview.org/0704/html/unedf.htmlby Bertsch, Dean, and Nazarewicz]
SciDAC 2 Project: Building a Universal NuclearEnergy Density Functional
Collaboration of physicists, applied mathematicians,and computer scientists
Funding in US but international collaborators also
Paths to a Nuclear Energy Functional (EDF)Empirical energy functional (Skyrme or RMF)Emulate Coulomb DFT: LDA based on precision calculationof uniform system E [ρ] =
∫dr E(ρ(r)) plus constrained
gradient corrections (∇ρ factors)
SLDA (Bulgac et al.)
Fayans and collaborators(e.g., nucl-th/0009034)
Ev = 23εFρ0
[av
+1−hv
1+x1/3+
1−hv2+x1/3
+
x2+
+ av−
1−hv1−x1/3
+
1−hv2−x1/3
+
x2−
]where x± = (ρn ± ρp)/2ρ0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(fm-3)
-200
204060
80100120140160
E/A
(MeV
)
neutron matter
nuclear matter
FP81WFF88FaNDF0
RG approach (J. Braun, from Polonyi and Schwenk, nucl-th/0403011)
EDF from perturbative chiral interactions + DME (Kaiser et al.)
Constructive Kohn-Sham DFT with RG-softened VχEFT’s
Two-Neutron Separation Energies
Quadrupole Deformations and B(E2)
Fission: Energy Surface from DFT
HFB Mass Formula: ∆m ∼ 1–2 MeV
Current empirical functionals hit wall at ∼ 1–2 MeV (!)
Accuracy of an ab initio functional fit to few-body data?
Jacek Dobaczewski
New-generation energy density functionals
Jacek DobaczewskiUniversity of Warsaw & University of Jyväskylä
UNEDF Annual Workshop, Pack Forest (WA)June 23-26, 2008
In collaboration with:Gillis Carlsson, Markus Kortelainen,Kazuhito Mizuyama, Jussi Toivanen
Jacek Dobaczewski
Nuclear Energy Density Functional
Jacek Dobaczewski
Fit to masses
Bertsch, Sabbey, and Uusnakki
Phys. Rev. C71, 054311 (2005)
How well can we describe masses with 12 coupling constants?
Jacek Dobaczewski
Fit tosingle-particle
energies
How well can we describe single-particle energies with 12 coupling constants?
SLy5SkPSkO'SIII
1.0
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12
RM
S de
viat
ion
ofs.
p. e
nerg
ies
(MeV
)
Number of singular valueM. Kortelainen et al., Phys. Rev. C77, 064307 (2008)
Jacek Dobaczewski
How to extend the nuclear energy density functionalbeyond the current standard form?
I. Density dependence of all the coupling constants
II. Derivatives of higher order up to N3LO:
III. Products of more than two densities, for example:
Quest for the spectroscopic-quality functional
Issues with Empirical EDF’s
Density dependencies might be too simplistic
Isovector components not well constrained
No (fully) systematic organization of terms in the EDF
Difficult to estimate theoretical uncertainties
What’s the connection to many-body forces?
Pairing part of the EDF not treated on same footing
and so on . . .
=⇒ Turn to microscopic many-body theory for guidance(although not for higher precision description!)
Comparing Dilute and Skyrme Functionals
Skyrme energy density functional (for N = Z )
E [ρ, τ,J] =
∫d3x
τ
2M+
38
t0ρ2 +116
(3t1 + 5t2)ρτ+164
(9t1 − 5t2)(∇ρ)2
− 34
W0ρ∇ · J +116
t3ρ2+α + · · ·
Dilute ρτJ energy density functional for ν = 4 (Vexternal = 0)
E [ρ, τ,J] =
∫d3x
τ
2M+
38
C0ρ2 +
116
(3C2 + 5C′2)ρτ+
164
(9C2 − 5C′2)(∇ρ)2
− 34
C′′2 ρ∇ · J +
c1
2MC2
0ρ7/3 +
c2
2MC3
0ρ8/3 +
116
D0ρ3 + · · ·
Same functional as dilute Fermi gas with ti ↔ Ci
Is Skyrme missing non-analytic, NNN, long-range (pion),(and so on) terms? Are proposed extensions enough?
Isn’t this a “perturbative” expansion?
Major UNEDF Research Areas
Ab initio structure — Nuclear wf’s from microscopic NN· · ·NNCSM/NCFC, CC, GFMC/AFMCAV18/ILx, chiral EFT −→ Vlow k
Ab initio energy functionals — DFT from microscopic N· · ·NCold atoms — superfluid LDA+ as prototype for nuclear DFTχEFT −→ Vlow k −→ MBPT −→ DME
DFT applications — Technology to calculate observablesSkyrme HFB+ for all nuclei (solvers)Fitting the functional to data (e.g., correlation analysis)
DFT extensions — Long-range correlations, excited statesLACM, GCM, TDDFT, QRPA, CI
Reactions – Low-energy reactions, fission, . . .
Outline
Overview: UNEDF project and Ab Initio DFT
Current Status and Possible Problems
Outlook and Additional Issues for Nuclear DFT
Extras
UNEDF Interconnections for Ab Initio Functionals
UNEDF
Outside UNEDF
Participant color key:
International collaborator
OSU (Drut, Furnstahl, Platter)MSU (Bogner, Gebremariam)
Ab Initio WF Methods
Ab Initio Functional+ Nuclear Matter
DFT Applications
CC: UT/ORNL (Dean, Hagen,Papenbrock)
UT/ORNL (Schunck, Stoitsov)
UW (Bertsch)
(also Saclay, TRIUMF)
(Epelbaum, Nogga)
TRIUMF (Schwenk)OSU, MSUVlowk/SRG
(Entem, Machleidt)
Interactions
Bonn/JulichChiral EFT
Saclay (Duguet, Lesinski, ...)
NCFC: ISU (Maris, Vary)LLNL (Navratil)
Salamanca/Idaho
UNEDF Interconnections for Ab Initio Functionals
UNEDF
Outside UNEDF
Participant color key:
International collaborator
Ab Initio densities
Systematics along isotope chains
Tests: spin−orbit splittings, time−odd terms, ...
Explicit Delta’sN3LO 3NF
Full 3NF
External potentials
Wider range of nuclei
Long−range pion contributions from NN and NNN DMENew 3NF fits
SRG 3NF evolution
Tests of DME: energies, densities with same HVary 3NF, external potential parametersCutoff dependence as diagnostic
Non−empirical pairing functional
Tests of nuclear matter: new fits, self−energies, ...Improved 3NF for DME
Generalized DMEDFT from OPM
plus fit residual Skyrme in HFB code
OSU (Drut, Furnstahl, Platter)MSU (Bogner, Gebremariam)
Ab Initio WF Methods
Ab Initio Functional+ Nuclear Matter
DFT Applications
CC: UT/ORNL (Dean, Hagen,Papenbrock)
UT/ORNL (Schunck, Stoitsov)
UW (Bertsch)
(also Saclay, TRIUMF)
(Epelbaum, Nogga)
TRIUMF (Schwenk)OSU, MSUVlowk/SRG
(Entem, Machleidt)
Interactions
Bonn/JulichChiral EFT
Saclay (Duguet, Lesinski, ...)
NCFC: ISU (Maris, Vary)LLNL (Navratil)
Salamanca/Idaho
Microscopic Nuclear Structure MethodsWave function methods (GFMC/AFMC, NCSM/FCI, CC, . . . )
many-body wave functions (in approximate form!)Ψ(x1, · · · , xA) =⇒ everything (if operators known)limited to A < 100 (??)
Green’s functions (see W. Dickhoff and D. Van Neck text)response of ground state to removing/adding particlessingle-particle Green’s function =⇒ expectation value
of one-body operators, Hamiltonianenergy, densities, single-particle excitations, . . .
DFT (see C. Fiolhais et al., A Primer in Density Functional Theory)response of energy to perturbations of the density J(x)ψ†ψ
natural framework is effective actions for composite operatorsΓ[ρ] = Γ0[ρ] + Γint[ρ] (e.g., for EFT/DFT) but also considerquantum chemistry MBPT+ approach (Bartlett et al.)energy functional =⇒ plug in candidate density, get out
trial energy, minimize (variational?)energy and densities (TDFT =⇒ excitations)
Orbital Dependent DFT (OEP, OPM, . . . ) [J. Drut]
Construct expansion for Γint[ρ, τ,J, . . .]; densities are sumsover orbitals solving from Kohn-Sham S-eqn with J0(r), . . .Self-consistency from J(r) = 0 =⇒ J0(r) = δΓint[ρ, . . .]/δρ(r)
i.e., Kohn-Sham potential is functional derivative of interactingenergy functional (or Exc) wrt densitiesHow do we calculate this functional derivative?
Approximations with explicit ρ(r) dependence: LDA, DME, . . .
Orbital-dependent DFT =⇒ full derivative via chain rule:
J0(r) =δΓint[φα, εα]
δρ(r)=
∫dr′
δJ0(r′)δρ(r)
∑α
∫dr′′
[δφ†α(r′′)δJ0(r′)
δΓint
δφ†α(r′′)+ c.c.
]+
δεα
δJ0(r′)∂Γint
∂εα
Solve the OPM equation for J0 using χs(r, r′) = δρ(r)/δJ0(r′)∫
d3r ′ χs(r, r′) J0(r′) = Λxc(r)
Λxc(r) is functional of the orbitals φα, eigenvalues εα, and G0KS
Orbital Dependent DFT (OEP, OPM, . . . ) [J. Drut]
Construct expansion for Γint[ρ, τ,J, . . .]; densities are sumsover orbitals solving from Kohn-Sham S-eqn with J0(r), . . .Self-consistency from J(r) = 0 =⇒ J0(r) = δΓint[ρ, . . .]/δρ(r)
i.e., Kohn-Sham potential is functional derivative of interactingenergy functional (or Exc) wrt densitiesHow do we calculate this functional derivative?
Approximations with explicit ρ(r) dependence: LDA, DME, . . .Orbital-dependent DFT =⇒ full derivative via chain rule:
J0(r) =δΓint[φα, εα]
δρ(r)=
∫dr′
δJ0(r′)δρ(r)
∑α
∫dr′′
[δφ†α(r′′)δJ0(r′)
δΓint
δφ†α(r′′)+ c.c.
]+
δεα
δJ0(r′)∂Γint
∂εα
Solve the OPM equation for J0 using χs(r, r′) = δρ(r)/δJ0(r′)∫
d3r ′ χs(r, r′) J0(r′) = Λxc(r)
Λxc(r) is functional of the orbitals φα, eigenvalues εα, and G0KS
Microscopic EDF’s from the DME
• density matrices and s.p. propagators• finite range and non-local resummed vertices K
• Dominant MBPT contributions to bulk properties take the form
non-local functionalsof ρ
• DME => expand DM in local operators w/factorized non-locality
• Original DME => calculate Πn from expanding about infinite NM
Maps <V> into a extended Skyrme-like EDF!
• Optimized DME => Fit Πn, constrain by symmetries and sum rules
• density dependencies, isovector, time-odd,… missing in Skyrme
Density Matrix Expansion Revisited [Negele/Vautherin]
DME: Write one-particle density matrix in Kohn-Sham basis
ρ(r1, r2) =∑
εα≤εF
ψ†α(r1)ψα(r2)
r1r2
R-r/2 +r/2
ρ(r1, r2) falls off with |r1 − r2| =⇒ expand in r (and resum)
Fall off is well approximated by nuclear matter=⇒ expand so that first term exact in uniform system
Change to R = 12(r1 + r2) and r = r1 − r2 and resum in r
ρ(R + r/2,R− r/2) = er·(∇1−∇2)/2 ρ(r1, r2)|r=0
=⇒ 3j1(rkF)
rkFρ(R) +
35j3(rkF)
2rk3F
(14∇2ρ(R)− τ(R) +
35
k2Fρ(R) + · · ·
)In terms of local densities ρ(R), τ(R), . . . =⇒ DFT with these
Extend to k space, NNN [Bogner, rjf, Platter]
Density Matrix Expansion Revisited [Negele/Vautherin]
DME: Write one-particle density matrix in Kohn-Sham basis
ρ(r1, r2) =∑
εα≤εF
ψ†α(r1)ψα(r2)
r1r2
R-r/2 +r/2
ρ(r1, r2) falls off with |r1 − r2| =⇒ expand in r (and resum)
Fall off is well approximated by nuclear matter=⇒ expand so that first term exact in uniform system
Change to R = 12(r1 + r2) and r = r1 − r2 and resum in r
ρ(R + r/2,R− r/2) = er·(∇1−∇2)/2 ρ(r1, r2)|r=0
=⇒ 3j1(rkF)
rkFρ(R) +
35j3(rkF)
2rk3F
(14∇2ρ(R)− τ(R) +
35
k2Fρ(R) + · · ·
)In terms of local densities ρ(R), τ(R), . . . =⇒ DFT with these
Extend to k space, NNN [Bogner, rjf, Platter]
Physics of the DME [Negele et al.]
Local rather than global properties of density matrixNot a short-distance expansion; keeps long-range effects
Expanding the difference between exact and nuclear matterresults in powers of s (nuclear matter kF)
Exact neutron densitymatrix squared in 208Pbcompared with DME
Improvements . . . [B. Gebremariam, S. Bogner, T. Duguet]
DME for Low-momentum Interactions (HF/NN only)
Test with HO model; cf. schematic V ’s (1970’s)
Λ–independent errors ≈ +5 MeV (NLO), ≈ −10 MeV (N3LO)
1.5 2 2.5 3 3.5 4Λ [fm−1]
−40
−20
0
20
40
⟨V⟩ D
ME
−⟨V
⟩ exac
t [MeV
]
Negele-Vautherin G-matrixBrink-Boeker (Sprung et. al.)Vlow k (NLO)
Vlow k (N3LO)
HO Model: ⟨V⟩DME − ⟨V⟩exact [MeV]
40Ca
−500
−400
−300
−200
−100
0
100
⟨V⟩ [
MeV
]
Vlow k Λ = 2.0 fm−1 (N3LO)
A
B C
DME Exacttotal
16O
1π onlyNLO 2π only
Adaptation to Skyrme HFB Implementations
ESkyrme =τ
2M+
38
t0ρ2 +1
16t3ρ2+α +
116
(3t1 + 5t2)ρτ+164
(9t1 − 5t2)|∇ρ|2 + · · ·
=⇒ EDME =τ
2M+ A[ρ] + B[ρ]τ + C[ρ]|∇ρ|2 + · · ·
Orbitals and Occupation #’s
Kohn−Sham Potentials
t , t0 1 , ..., t2
Skyrmeenergy
functionalHFB
solver
J0(r) =δEint[ρ]
δρ(r)⇐⇒ [−∇2
2m−J0(x)]ψα = εαψα =⇒ ρ(x) =
∑α
nα|ψα(x)|2
Adaptation to Skyrme HFB Implementations
ESkyrme =τ
2M+
38
t0ρ2 +1
16t3ρ2+α +
116
(3t1 + 5t2)ρτ+164
(9t1 − 5t2)|∇ρ|2 + · · ·
=⇒ EDME =τ
2M+ A[ρ] + B[ρ]τ + C[ρ]|∇ρ|2 + · · ·
Orbitals and Occupation #’s
Kohn−Sham Potentials
energyfunctional
HFBsolver
DME
ρρA[ ], B[ ], ...
J0(r) =δEint[ρ]
δρ(r)⇐⇒ [−∇2
2m−J0(x)]ψα = εαψα =⇒ ρ(x) =
∑α
nα|ψα(x)|2
Adaptation to Skyrme HFB Implementations
ESkyrme =τ
2M+
38
t0ρ2 +1
16t3ρ2+α +
116
(3t1 + 5t2)ρτ+164
(9t1 − 5t2)|∇ρ|2 + · · ·
=⇒ EDME =τ
2M+ A[ρ] + B[ρ]τ + C[ρ]|∇ρ|2 + · · ·
Orbitals and Occupation #’s
Kohn−Sham Potentials
functionalHFB
solver
Orbitaldependent
Solve OPM
J0(r) =δEint[ρ]
δρ(r)⇐⇒ [−∇2
2m−J0(x)]ψα = εαψα =⇒ ρ(x) =
∑α
nα|ψα(x)|2
Validation of HFBRAD DME ImplementationReproduces Skyrme results (with good accuracy)Possible issues: Sprung et al. test; dC[ρ]/dρ termsTry fine-tuned nuclear matter with low-momentum NN/NNN
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
200
400
600Vsrg λ = 2.0 fm−1 (N3LO)
A
B
C
DME Sly4total
40Ca
<V> <V>
DMEtotal
E
Sly4E
NN + NNN scaled to "fit" NM
HFBRAD
0 1 2 3 4 5 6r [fm]
0
0.02
0.04
0.06
0.08
0.1
dens
ity [f
m−3
]
Skyrme SLy4Vsrg DME
16O40Ca
HFBRAD
λ = 2 fm−1 ("fit")
Do densities look like nuclei from Skyrme EDF’s? (Yes)
New low-momentum NNN fits and Nuclear Matter
self-bound w/ saturation
N2LO 3NF fit to A =3,4B.E. and 4He radii
Loop expansion (perturbative)about HF becomes sensible
Smooth cutoff Vlow k fromN3LO(500)
[A. Nogga]
New low-momentum NNN fits and Nuclear Matter
Λ-dependence => theoretical error bands (lower limit)
Assess the impact of large uncertainties in the ci’s appearing in 2- and 3-body TPEP
Knobs to estimateTheoretical error bars:
Vary the order of the underlying EFT
Sensitivity to many-body approximations
New low-momentum NNN fits and Nuclear Matter
Excellent saturation w/outfine-tuning to nuclear matter
1) VNNN => V2N(r)2) HF propagators 3) Beyond 2-hole lines?4) Angle-averaging 5) Particle-hole channel6) …
But…
Ladder sum » 2nd-order
Coupled-cluster calculationsof nuclear matter,16O and 40Ca would be a huge help! To do: asymmetric matter
Guidance from NM for fixing EFT couplings
Supports suggestion of Navratil et al. to use 4He radii to constrain fits of 3NF couplings (cE and cD)
Different Λ-dependence for the 2 ways of fitting the 3NF lec’s
Large uncertainties in extracting c3, c4 from πN and NN => use NM to constrain (sensitivity at the 2-3 MeV level)
Sample Observations From HFBRAD DMEUntuned new 3NF fits to Vlow k and SRG [N3LO (500 MeV)]
best nuclear matter =⇒ 40Ca underbound by ≈ 1 MeV/A
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
200
400
600Vlow k Λ = 2.0 fm−1 (N3LO)
A
BC
DME Sly4total
40Ca
<V> <V>
B C
ANN + fit NNN (Λ = 2.5 fm−1)
HFBRAD
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
200
400
600VSRG λ = 2.0 fm−1 (N3LO)
A
B
C
DME Sly4total
40Ca
<V> <V>
B C
ANN + fit NNN (Λ = 2.5 fm−1)
HFBRAD
B large =⇒ M∗0 low compared to Sly4. Problem?
Does power counting support gradient expansion?
DFT Validation Against Ab Initio Calculations“Coester Lines”
Compare systematics, e.g.,by varying 3NF coupling inHamiltonian
−2.0 −1.5 −1.0cE [strength of 3NF contact]
−120
−115
−110
−105
−100
−95
−90
−85
−80
−75
Bind
ing
Ener
gy [M
eV]
NN + <3NF>NN + 0,1,2-body 3NFNN + full 3NF
16O
PreliminaryVSRG [λ = 1.9 fm−1]+ 3NF contact only
Coupled ClusterN = 2
External PotentialsDFT from response of energyto perturbation of densities=⇒ Apply external fields
1.5 2.0 2.5radius [fm]
−10
−8
−6
−4
−2
0
2
4
6
8
10
12
(Eto
t− U
ext )/
A [M
eV]
12C NCFC8He NCFC (no coul.)
40 MeV
NN-only
hΩ of harmonic trap
5 MeV
PreliminaryVSRG [λ = 1.5 fm−1]
Near future: Full 3NF, more external fields, other nuclei . . .
DFT Validation Against Ab Initio Calculations“Coester Lines”
Compare systematics, e.g.,by varying 3NF coupling inHamiltonian
−2.0 −1.5 −1.0cE [strength of 3NF contact]
−120
−115
−110
−105
−100
−95
−90
−85
−80
−75
Bind
ing
Ener
gy [M
eV]
NN + <3NF>NN + 0,1,2-body 3NFNN + full 3NFDME
16O
PreliminaryVSRG [λ = 1.9 fm−1]+ 3NF contact only
Coupled ClusterN = 2
External PotentialsDFT from response of energyto perturbation of densities=⇒ Apply external fields
1.5 2.0 2.5radius [fm]
−10
−8
−6
−4
−2
0
2
4
6
8
10
12
(Eto
t− U
ext )/
A [M
eV]
12C NCFC8He NCFC (no coul.)
40 MeV
NN-only
hΩ of harmonic trap
5 MeV
PreliminaryVSRG [λ = 1.5 fm−1]
Near future: Full 3NF, more external fields, other nuclei . . .
Possible Reasons for the Poor Agreement1) DME averages out too much information
- COM P-dependence (spatial non-locality) - energy-dependence
Errors of 1 MeV/nucleonin infinite NM
Possible Reasons for the Poor Agreement
2.84
-10.1
DME
3.24
-7.40
CoupledCluster
2.953.352.472.73rch
-9.66-7.72-7.89-6.72E/A
DMECoupledClusterDMECoupled
Cluster
2) Gradient expansion breaks down when saturation not good
e.g., N3LO NM looks reasonable at lower densities despite poor saturation
Ab-initio results for O16 and Ca40pretty decent, but DME is poor
O16 Ca40 Ca48
Gradients no longer “small” sinceDME = expansion about NM?
Possible Reasons for the Poor Agreement3) Errors in the Hartree contribution => feedback via self-consistency
r
R
r
R
Exact DME
Treat Hartree exactly a-la Coulomb? [Negele and Vautherin, Sprung et al.]
* See B. Gebremariam’s talk for the failings of the standard DME and possible solutions.
Possible Reasons for the Poor Agreement4) Inadequacy of many-body approximations (I.e., LO Brueckner)
3-hole-line correction could contribute at the few MeVlevel, even at low Λ
5) Approximate treatment of 3NF
Coupled-cluster NM calculation to assess H.O.T. would help!
6) Implementation errors in HFBRAD (rearrangement terms, etc.)
Long-Range Chiral EFT=⇒ Enhanced Skyrme
Add long-range (π-exchange)contributions in the densitymatrix expansion (DME)
NN/NNN through N2LOderived [SKB,BG]
Refit the Skyrme parameters
Test for sensitities andimproved observables(isotope chains)
Spin-orbit couplings from 2π3NF particularly interesting
Can we “see” the pion inmedium to heavy nuclei?
2N forces 3N forces 4N forces
Scalar Part of Negele-Vautherin DME (NVDME)Compare exact and DME for angle-averaged quantities:
Fs,exch(R, r) =
∫dΩr ρq(r1, r2)ρq(r2, r1)
Scalar NVDME for exchange is pretty good but there isroom for improvement, especially at the surfaceWhat about the vector part (in spin space) sq?
Improved DME for∫
dΩr sn(r1, r2) · sn(r2, r1) [B.G. et al.]
Improved DME for∫
dΩr sn(r1, r2) · sn(r2, r1) [B.G. et al.]
Consider Hartree-Fock with One-Pion ExchangeNV used a different approach from the scalar expansion:
sq(R + r/2,R− r/2)NVDME−→ i
2j0(rkF) r× Jq(R)
Look at∫
dr dR V1π(r) sn(r1, r2) · sn(r2, r1):
BG et al.: use phase-space averaging for finite nuclei
Consider Hartree-Fock with One-Pion ExchangeNV used a different approach from the scalar expansion:
sq(R + r/2,R− r/2)NVDME−→ i
2j0(rkF) r× Jq(R)
Look at∫
dr dR V1π(r) sn(r1, r2) · sn(r2, r1):
BG et al.: use phase-space averaging for finite nuclei
Outline
Overview: UNEDF project and Ab Initio DFT
Current Status and Possible Problems
Outlook and Additional Issues for Nuclear DFT
Extras
(Some) Important Issues for Nuclear DFTDFT for self-bound systems
Does DFT even exist? (HK theorem for intrinsic states?)Effective actions: symmetry breaking and zero modesGame plans proposed:
J. Engel: find intrinsic functional (one-d boson system)B. Giraud et al.: use harmonic oscillator trapJ. Braun et al.: deal with soliton zero modes usingFadeev-Popov method [e.g., Calzetta/Hu cond-mat/0508240]
Organization about mean field: need convergent expansionNeed freedom to choose background field to fix densityCan DFT deal with nuclear short-range correlations?Claim: need low-momentum interactions (χEFT→ Vlow k )
Effectiveness of approximations (e.g., DME)What about single-particle spectra?
Connect to Green’s function formulation?Bartlett claims good reproduction for Coulomb systems
How to best deal with long-range correlations? pairing?Alternative functionals (e.g., T. Papenbrock)
Outline
Overview: UNEDF project and Ab Initio DFT
Current Status and Possible Problems
Outlook and Additional Issues for Nuclear DFT
Extras
Content of Extras
Misconceptions
DFT from perturbative chiral interactions + DME (Kaiser et al.)
DFT from RG (Jens Braun)
Non-empirical pairing functional for nuclei
Power counting in functionals
BBG and G-matrix
Misconceptions vs. Correct Interpretations
DFT is a Hartree-(Fock) approximation to an effective interactionDFT can accomodate all correlations in principle,but in practice they may be included perturbatively(which can fail for some V ’s)
Nuclear matter is strongly nonperturbative in the potential“perturbativeness” is highly resolution dependent
(Fill in the blank) is responsible for nuclear saturationanother resolution-dependent inference
Evolving low-momentum interactions loses important informationlong-range physics is preservedrelevant short-range physics encoded in potential
Low-momentum NN potentials are just like G-matricesimportant distinction: conventional G-matrix stillhas high-momentum, off-diagonal matrix elements
DFT from Perturbative Chiral Interactions + DME
N. Kaiser et al. in ongoing series of papers (nucl-th/0212049,0312058, 0312059, 0406038, 0407116, 0509040, 0601100, . . . )
Fourier transform of expanded density matrix defines amomentum-space medium insertion, leading to EDF:
Three-body forces from explicit ∆, e.g.,
Perturbative expansion for energy tuned to nuclear matter
Many analytic results =⇒ qualitative insight, checks forquantitative calculations with low-momentum interactions
Density Functional RG for Nuclei
!!!!["] =12
Vlow k
is the density functional!![!]start from mean-field (background potential )
and include many-body correlationsU!
+ ( will be included later)!V3N
S![!†,!] =!!†
""t !
12m
! + (1! #)U!
#! +
12
!!†!#Vlow k!
†!
!2 !1 0 1 20.0
0.2
0.4
0.6
0.8
1.0
"#1
"#0.75
"#0.5
"#0.25
"#0
Example: RG flow of the ground-state density for 1d model (“smeared-out” -function interaction) !
•density basis expansion scales favorably to heavy nuclei•allows for a calculation of ground-state energy and density from nuclear forces•in production: results for 16O
Density Functional: !![!] = ln!D"†D" e!S!["†,"]+
R "!!"# ·("†")
Observables Sensitive to 3N Interactions?
Study systematics along isotopic chains
Example: kink in radius shift 〈r2〉(A)− 〈r2〉(208)
-0.2
0
0.2
0.4
0.6
202 204 206 208 210 212 214
isot
opic
shi
ft r2 (A
)-r2 (2
08) [
fm2 ]
total nucleon number A
Pb isotopes
exp.SkI3SLy6NL3
Associated phenomenologically with behavior of spin-orbitisoscalar to isovector ratio fixed in original Skyrme
Clues from chiral EFT contributions?Kaiser et al.: ratio of isoscalar to isovector spin-orbit
Non-Empirical Pairing Functional for NucleiT. Duguet (Saclay); T. Lesinski, K. Bennaceur, J. Meyer (Lyon)
New spherical code BSLHFBspherical Bessel function basisfinite-range / non-local pairinginteractions in EDFoperator representation: sum ofseparable terms (rank 2 for Vlow k )
Pairing at lowest order in NN(nuclear + Coulomb)
use Vlow k at Λ ≈ 2 fm−1 asNN pairing interactionUse SLY5 Skyrme for ph EDFwith fixed m∗
0/m = 0.7
Studied m∗(k , kF) for cutoffs Λ(K. Hebeler, T.D., A. Schwenk)
consistent ph/pp scales neededVlow k ok with m∗
Skyrme(kF)
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
0 0.5 1 1.5 2 2.5 3 3.5 4
V [
MeV
fm
3 ]
k [fm-1]
Vlow k k’=k
Vfit k’=k
Vlow k k’=0.05000
Vfit k’=0.05000
Vlow k k’=1
Vfit k’=1
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
∆1S 0
(kF,
kF)
[de
g.]
kF [fm-1]
Vfit
Vlow k
AV18
Pairing Gaps from Vlow k + Coulomb Near Data!
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2820
∆n LC
S [M
eV]
Ca
5028
Ni
8250N
Sn
184126
Pb
Vlow k, Λ=1.8Vlow k, Λ=1.8 + Coulomb
Exp.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2820
∆p LC
S [M
eV]
N=28
5028
N=50
50Z
N=82
82
N=126
Current limitationsThree-body force missingNo density/spin/isospinfluctuations (Milan: +40%!?)Phenom. ph functional andmomentum-independent m∗
0
Upgrade plansOther observables (Lesinski),deformed nuclei (Rotival)Incorporate NNN (Lesinski)Check fluctuationsConstruct ph part (BG, VR)
Power Counting in Skyrme and RMF Functionals?
Old NDA analysis:[Friar et al., rjf et al.]
c[ψ†ψ
f 2πΛ
]l [∇Λ
]n
f 2πΛ2
=⇒ρ←→ ψ†ψτ ←→ ∇ψ† · ∇ψJ←→ ψ†∇ψ
Density expansion?
17≤ ρ0
f 2πΛ≤ 1
4
for 1000 ≥ Λ ≥ 5002 3 4 5
power of density1
5
10
50
100
500
1000
ener
gy/p
artic
le (M
eV)
ε0
natural (Λ=600 MeV)Skyrme ρn
RMFT-II ρn netRMFT-I ρn net
kF = 1.35 fm−1
Bethe-Brueckner-Goldstone Power Counting
Strong short-range repulsion=⇒ Sum V ladders =⇒ G
vs.
Vlow k momentumdependence + phase space=⇒ perturbative
Λ: |P/2 ± k| > kF and |k| < Λ
F: |P/2 ± k| < kF
P/2
k
Λ
kF
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6kf [fm
−1]
0.01
0.1
1
10
3rd o
rder
/ 2
nd o
rder
AV18Λ = 2.1 fm−1
0 0.5 1 1.5 2 2.5k [fm−1]
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
V(k
,k) [
fm]
1S0 V(k,k)3S1 V(k,k)
Λ = 2 fm−1
Bethe-Brueckner-Goldstone Power Counting
Strong short-range repulsion=⇒ Sum V ladders =⇒ G
vs.
Vlow k momentumdependence + phase space=⇒ perturbative
Λ: |P/2 ± k| > kF and |k| < Λ
F: |P/2 ± k| < kF
P/2
k
Λ
kF
0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6kf [fm
−1]
0.001
0.01
0.1
1
10
3rd o
rder
/ 2
nd o
rder
N3LO [600 MeV]λ = 2.0 fm-1
Λ = 2.0 fm-1
0 0.5 1 1.5 2 2.5k [fm−1]
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
V(k
,k) [
fm]
1S0 V(k,k)3S1 V(k,k)
Λ = 2 fm−1
Compare Potential and G Matrix: AV18
Compare Potential and G Matrix: AV18
Compare Potential and G Matrix: AV18
Compare Potential and G Matrix: AV18
Compare Potential and G Matrix: N3LO (500 MeV)
Compare Potential and G Matrix: N3LO (500 MeV)
Compare Potential and G Matrix: N3LO (500 MeV)
Compare Potential and G Matrix: N3LO (500 MeV)
Hole-Line Expansion Revisited (Bethe, Day, . . . )
Consider ratio of fourth-order diagrams to third-order:
k
m
a
b
l
c k
m
abl
c
n
b
km
a lc
pq
b
“Conventional” G matrix still couples low-k and high-kadd a hole line =⇒ ratio ≈
∑n≤kF〈bn|(1/e)G|bn〉 ≈ κ ≈ 0.15
no new hole line =⇒ ratio ≈ −χ(r = 0) ≈ −1 =⇒ sum allorders
Low-momentum potentials decouple low-k and high-kadd a hole line =⇒ still suppressedno new hole line =⇒ also suppressed (limited phase space)freedom to choose single-particle U =⇒ use for Kohn-Sham
=⇒ Density functional theory (DFT) should work!