ab initio density functional theory for nucleintg/talks/furnstahl_bh_2008.pdf · ab initio density...

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

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




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]

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


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 + · · ·



energyfunctional

HFBsolver

DME

ρρA[ ], B[ ], ...

J0(r) =δEint[ρ]

δρ(r)⇐⇒ [−∇2

2m−J0(x)]ψα = εαψα =⇒ ρ(x) =

∑α

nα|ψα(x)|2


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 + · · ·



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]


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


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.]

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




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




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: 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!

ab initio density functional theory for nucleintg/talks/furnstahl_bh_2008.pdf · ab initio density...

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