Density Functional Theory from Effective Actions

Dick Furnstahl

Department of Physics Ohio State University

September, 2005

Collaborators: A. Bhattacharyya, S. Bogner, H.-W. Hammer, S. Puglia, S. Ramanan, A. Schwenk, B. Serot Outline Overview Action Topics Summary Outline

Overview: Microscopic DFT

Effective Actions and DFT

Issues and Ideas and Open Problems

Summary

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Outline

Overview: Microscopic DFT

Effective Actions and DFT

Issues and Ideas and Open Problems

Summary

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan DFT from Microscopic NN··· N Interactions

What? Constructive density functional theory (DFT) for nuclei Why now? Progress in chiral EFT Application of RG (e.g., low-momentum interactions) Advances in computational tools and methods How? Use framework of effective actions with EFT principles EFT interactions and operators evolved to low-momentum Few-body input not enough (?) =⇒ input from many-body Merge with other energy functional developments

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Density Functional Theory (DFT) with Coulomb

Dominant application: inhomogeneous Atomization Energies of Hydrocarbon Molecules electron gas 20 Interacting point electrons 0 in static potential of -20

atomic nuclei -40

“Ab initio” calculations of -60 Hartree-Fock DFT Local Spin Density Approximation atoms, molecules, crystals, DFT Generalized Gradient Approximation surfaces, . . . % deviation from experiment -80 -100 H C C H CH C H C H C H HF is good starting point, 2 2 2 2 4 2 4 2 6 6 6 molecule DFT/LSD is better, DFT/GGA is better still, . . .

Dick Furnstahl DFT from Effective Actions Consequences: In Coulomb DFT, Hartree-Fock gives dominate contribution =⇒ correlations are small corrections =⇒ DFT works! cf. NN interactions =⇒ correlations HF =⇒ DFT fails?? However . . . the ﬁrst two depend on the resolution =⇒ different cutoffs third one is affected by Pauli blocking

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Sources of Nonperturbative Physics for NN

1 Strong short-range repulsion (“hard core”)

2 Iterated tensor (S12) interaction 3 Near zero-energy bound states

Dick Furnstahl DFT from Effective Actions However . . . the ﬁrst two depend on the resolution =⇒ different cutoffs third one is affected by Pauli blocking

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Sources of Nonperturbative Physics for NN

1 Strong short-range repulsion (“hard core”)

2 Iterated tensor (S12) interaction 3 Near zero-energy bound states

Consequences: In Coulomb DFT, Hartree-Fock gives dominate contribution =⇒ correlations are small corrections =⇒ DFT works! cf. NN interactions =⇒ correlations HF =⇒ DFT fails??

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Sources of Nonperturbative Physics for NN

1 Strong short-range repulsion (“hard core”)

2 Iterated tensor (S12) interaction 3 Near zero-energy bound states

Consequences: In Coulomb DFT, Hartree-Fock gives dominate contribution =⇒ correlations are small corrections =⇒ DFT works! cf. NN interactions =⇒ correlations HF =⇒ DFT fails?? However . . . the ﬁrst two depend on the resolution =⇒ different cutoffs third one is affected by Pauli blocking

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Wavelength and Resolution

Dick Furnstahl DFT from Effective Actions Low-momentum potential =⇒ much simpler wave function!

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan The Deuteron at Different Resolutions 0.5 10 3 3 S deuteron w.f. S1 deuteron w.f. 0.4 1

Argonne v ] 1 18 ] 3/2 0.3 Argonne v18 -3/2 (k)| [fm 0 (r) [fm 0.2

ψ 0.1 ψ |

0.1 0.01

0 0 1 2 3 4 5 0 2 4 6 8 10 -1 r [fm] k [fm ] Repulsive core =⇒ short-distance suppression =⇒ high-momentum components

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan The Deuteron at Different Resolutions 0.5 10 3 3 S deuteron w.f. S1 deuteron w.f. 0.4 1

-1 ] 1 Λ = 2 fm ] -1 -1 3/2 0.3 = 2.0 fm Λ = 3 fm -3/2 Λ -1 -1 Λ = 3.0 fm Λ = 4 fm -1 Argonne v18

Λ = 4.0 fm (k)| [fm 0 (r) [fm 0.2 Argonne v 18 ψ 0.1 ψ |

0.1 0.01

0 0 1 2 3 4 5 0 2 4 6 8 10 -1 r [fm] k [fm ] Repulsive core =⇒ short-distance suppression =⇒ high-momentum components Low-momentum potential =⇒ much simpler wave function!

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan

The Deuteron at Different Resolutions more

Integrand of −〈ψ | V |ψ 〉 for Λ =6.0 fm−1 d Λ d 6

10 4

3 5

2 0 1 0 0

2 2 0 k' (fm−1) 4 4 k (fm−1) −1 6 6

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan

The Deuteron at Different Resolutions more

Integrand of −〈ψ | V |ψ 〉 for Λ =5.0 fm−1 d Λ d 6

10 4

3 5

2 0 1 0 0

2 2 0 k' (fm−1) 4 4 k (fm−1) −1 6 6

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan

The Deuteron at Different Resolutions more

Integrand of −〈ψ | V |ψ 〉 for Λ =4.0 fm−1 d Λ d 6

10 4

3 5

2 0 1 0 0

2 2 0 −1 k' (fm ) 4 4 k (fm−1) −1 6 6

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan

The Deuteron at Different Resolutions more

Integrand of −〈ψ | V |ψ 〉 for Λ =3.0 fm−1 d Λ d 6

10 4

3 5

2 0 1 0 0

2 2 0 k' (fm−1) 4 4 k (fm−1) −1 6 6

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan

The Deuteron at Different Resolutions more

Integrand of −〈ψ | V |ψ 〉 for Λ =2.0 fm−1 d Λ d 6

10 4

3 5

2 0 1 0 0

2 2 0 −1 k' (fm ) 4 4 k (fm−1) −1 6 6

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan In-Medium Wave Functions (NN Only)

1.6 1.6

1 -1 3 -1 1.4 S0 at kF = 1.35 fm 1.4 S1 at kF = 1.35 fm -1 -1 [P = 0, k = 0.1 fm , m*/m = 1] [P = 0, k = 0.1 fm , m*/m = 1] 1.2 1.2

1 1 (r) (r) k 0.8 k 0.8 Ψ Ψ 0.6 Fermi gas 0.6 Fermi gas v18 v18 3 3 0.4 N LO 0.4 N LO

0.2 0.2

0 0 0 1 2 3 4 5 0 1 2 3 4 5 r [fm] r [fm]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan In-Medium Wave Functions (NN Only)

1.6 1.6

1 -1 3 -1 1.4 S0 at kF = 1.35 fm 1.4 S1 at kF = 1.35 fm -1 -1 [P = 0, k = 0.1 fm , m*/m = 1] [P = 0, k = 0.1 fm , m*/m = 1] 1.2 1.2

1 1 (r) (r) k 0.8 k 0.8 Ψ Ψ 0.6 Fermi gas 0.6 Fermi gas v18 v18 3 3 0.4 N LO 0.4 N LO -1 -1 Λ = 2.0 fm (v18) Λ = 2.0 fm (v18) -1 3 -1 3 0.2 Λ = 2.0 fm (N LO) 0.2 Λ = 2.0 fm (N LO)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 r [fm] r [fm]

Dick Furnstahl DFT from Effective Actions “Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.”

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Conventional Wisdom on Nuclear Many-Body

Hans Bethe in review of nuclear matter (1971):

“The theory must be such that it can deal with any nucleon-nucleon (NN) force, including hard or ‘soft’ core, tensor forces, and other complications. It ought not to be necessary to tailor the NN force for the sake of making the computation of nuclear matter (or ﬁnite nuclei) easier, but the force should be chosen on the basis of NN experiments (and possibly subsidiary experimental evidence, like the binding energy of H3).”

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Conventional Wisdom on Nuclear Many-Body

Hans Bethe in review of nuclear matter (1971):

“Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required.”

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan EFT and RG Make Physics Easier There’s an old vaudeville joke about a doctor and patient . . .

Patient: Doctor, doctor, it hurts when I do this! Doctor: Then don’t do that. Weinberg’s Third Law of Progress in Theoretical Physics: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”

Dick Furnstahl DFT from Effective Actions 0 ¢¡¤£ Q —

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Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Chiral Effective Field Theory for Two Nucleons

1S0 3S1 80 150

40 100 Epelbaum, Meißner, et al. 50 0 Also Entem, Machleidt 0 -40 0 0.1 0.2 0.3 0 0.1 0.2 0.3 L + match at low energy πN 3P0 1D2 80 12 Qν 1π 2π 4N 40 8

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Dick Furnstahl DFT from Effective Actions

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Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Chiral Effective Field Theory for Two Nucleons

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Dick Furnstahl DFT from Effective Actions

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Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Chiral Effective Field Theory for Two Nucleons

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Dick Furnstahl DFT from Effective Actions

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Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Chiral Effective Field Theory for Two Nucleons

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Chiral Effective Field Theory for Two Nucleons

1S0 3S1 80 150

40 100 Epelbaum, Meißner, et al. 50 0 Also Entem, Machleidt 0 -40 0 0.1 0.2 0.3 0 0.1 0.2 0.3 L + match at low energy πN 3P0 1D2 80 12 Qν 1π 2π 4N 40 8

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan How do you go from Chiral EFT to a Potential?

E.g., see Evgeny Epelbaum review: nucl-th/0509032 Method of unitary transformations (e.g., Okubo) P space has nucleons only, Q space has the pions

Use chiral expansion in {p, mπ}/Λ Energy-independent potential Consistent operators constructed with power counting

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan State of the Art: N3LO

0 150 60 1 3 1 S S P 0 1 -10 1 40 100

20 -20 50 0 -30 Phase Shift [deg] 0 -20 0 10 3 3 3 P0 -10 P1 40 P2 0 -20 -10 20 -30 Phase Shift [deg] -20 0 15 0 ε 1 3 4 1 D D 10 2 -10 1 2

5 -20 0 Phase Shift [deg] -2 0 -30 40 0 3 3 ε2 30 D2 5 D3 -1

20 -2 0 10 -3

Phase Shift [deg] -5 0 -4 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Lab. Energy [MeV] Lab. Energy [MeV] Lab. Energy [MeV]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan State of the Art: N3LO

0 3 0 1 3 3 -1 2.5 F F -1 F 3 2 2 3 -2 1.5 -3 -2 1 -4 0.5 -3 Phase Shift [deg] -5 0 3 0 2.5 3 ε 3 F 6 3 -1 G 2 4 3 -2 1.5 4 1 -3 2 0.5 Phase Shift [deg] -4 0 0 0 0 0.5 3 1 3 0.4 -0.2 G5 -0.5 H5 H4 0.3 -0.4 -1 0.2 -0.6 0.1 -1.5 Phase Shift [deg] -0.8 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Lab. Energy [MeV] Lab. Energy [MeV] Lab. Energy [MeV]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan State of the Art: N3LO

Chiral NNLO 15 Chiral NNNLO Nijmegen PWA

[mb/sr] 10 Ω /d σ

d 5

0.6

0.4 y A 0.2

0 60 120 180 θ [deg]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan State of the Art: N3LO

110 0.06 500 100 A 400 y 90 300 0.04 100 110 200 0.02

100 dσ/dΩ 0 1000 0.2 20 A 18 y 0.15 16 100 115 120 125 0.1

dσ/dΩ 0.05

10 0 100 0.4

0 10

-0.4

dσ/dΩ Ay 1 -0.8 0 60 120 180 0 60 120 180 θ [deg] θ [deg]

Dick Furnstahl DFT from Effective Actions How do we organize (3, 4, ··· )–body forces? EFT!

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Many-Body Forces are Inevitable!

What if we have three Triton Binding Energy nucleons interacting? -7.5 Successive two-body scatterings with short-lived high-energy intermediate -8.0 states unresolved =⇒ must be absorbed into three-body AV18 force CD Bonn -8.5 Binding Energy (MeV)

Experiment

0 1 2 3 4 5 6 7 8 -1 Λ (fm ) [Bogner, Nogga, Schwenk]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan Many-Body Forces are Inevitable!

Experiment

0 1 2 3 4 5 6 7 8 -1 Λ (fm ) How do we organize [Bogner, Nogga, Schwenk] (3, 4, ··· )–body forces? EFT!

Dick Furnstahl DFT from Effective Actions “≈ 2nd Order”

Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan (Approximate) Nuclear Matter with NN and NNN

Hartree-Fock

-1 Λ = 1.6 fm 5 -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 0 Λ = 2.1 fm [no V3N]

-5 E/A [MeV]

-10

-15 0.8 1 1.2 1.4 1.6 -1 kF [fm ]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan (Approximate) Nuclear Matter with NN and NNN

Hartree-Fock “≈ 2nd Order”

-1 -1 Λ = 1.6 fm Λ = 1.6 fm 5 -1 5 -1 Λ = 1.9 fm Λ = 1.9 fm -1 -1 Λ = 2.1 fm Λ = 2.1 fm -1 -1 Λ = 2.3 fm Λ = 2.3 fm -1 -1 0 Λ = 2.1 fm [no V3N] 0 Λ = 2.1 fm [no V3N]

-5 -5 E/A [MeV] E/A [MeV]

-10 -10

-15 -15 0.8 1 1.2 1.4 1.6 0.8 1 1.2 1.4 1.6 -1 -1 kF [fm ] kF [fm ]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan (Nuclear) Many-Body Physics: “Old” vs. “New”

Inﬁnite # of low-energy One Hamiltonian for all potentials; different problems and energy/length resolutions =⇒ different dof’s scales (not QCD!) and Hamiltonians There is no best potential Find the “best” potential =⇒ use a convenient one! Two-body data may be Many-body data needed and sufﬁcient; many-body forces many-body forces inevitable as last resort Exploit divergences (cutoff Avoid (hide) divergences dependence as tool) Choose approximations (e.g., Power counting determines diagrams) by “art” diagrams and truncation error

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Intro Vlowk Philosophy ChPT NM Plan My Favored Scenario for DFT (Today!)

Construct a chiral EFT to a given order (N3LO at present) including many-body forces (N3LO has leading 4-body) choose cutoff regulator Λ as large as possible up to breakdown scale to minimize truncation error Evolve Λ down with RG (to Λ ≈ 2 fm−1 for ordinary nuclei) all interactions and other operators Generate density functional in effective action form direct construction (e.g., DME++) or match to ﬁnite-density EFT expansion

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Outline

Overview: Microscopic DFT

Effective Actions and DFT

Issues and Ideas and Open Problems

Summary

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Density Functional Theory (DFT)

Hohenberg-Kohn: There exists an energy functional Ev [ρ] ... Z 3 Ev [ρ] = FHK[ρ] + d x v(x)ρ(x)

FHK is universal (same for any external v) =⇒ H2 to DNA! Introduce orbitals and minimize energy functional =⇒ Egs, ρgs Useful if you can approximate the energy functional

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing DFT as Effective Action Effective action is generically the Legendre transform of a generating functional with external source Partition function in presence of J(x) coupled to density:

Z R † Z[J] = e−W [J] ∼ Tr e−β(Hb+J ρb) −→ D[ψ†]D[ψ] e− [L+J ψ ψ]

The density ρ(x) in the presence of J(x) is [we want J = 0] δW [J] ρ(x) ≡ hρ(x)i = b J δJ(x)

Invert to ﬁnd J[ρ] and Legendre transform from J to ρ: Z δΓ[ρ] Γ[ρ] = W [J] − J ρ and J(x) = − δρ(x)

Dick Furnstahl DFT from Effective Actions Substitute and separate out the pieces: Z Z † E0(J) = hHb(J)iJ = hHbiJ + Jhψ ψiJ = hHbiJ + J ρ(J)

Expectation value of Hb in ground state generated by J[ρ] Z 1 hHbiJ = E0(J) − J ρ = Γ[ρ] β

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Partition Function in Zero Temperature Limit Consider Hamiltonian with time-independent source J(x): Z Hb(J) = Hb + J ψ†ψ If ground state is isolated (and bounded from below), h i e−βHb = e−βE0 |0ih0| + Oe−β(E1−E0)

As β → ∞, Z[J] =⇒ ground state of Hb(J) with energy E0(J)

−W [J] −β(Hb+J ρ) 1 1 Z[J] = e ∼ Tr e b =⇒ E0(J) = lim − log Z[J] = W [J] β→∞ β β

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Partition Function in Zero Temperature Limit Consider Hamiltonian with time-independent source J(x): Z Hb(J) = Hb + J ψ†ψ If ground state is isolated (and bounded from below), h i e−βHb = e−βE0 |0ih0| + Oe−β(E1−E0)

As β → ∞, Z[J] =⇒ ground state of Hb(J) with energy E0(J)

−W [J] −β(Hb+J ρ) 1 1 Z[J] = e ∼ Tr e b =⇒ E0(J) = lim − log Z[J] = W [J] β→∞ β β Substitute and separate out the pieces: Z Z † E0(J) = hHb(J)iJ = hHbiJ + Jhψ ψiJ = hHbiJ + J ρ(J)

Expectation value of Hb in ground state generated by J[ρ] Z 1 hHbiJ = E0(J) − J ρ = Γ[ρ] β Dick Furnstahl DFT from Effective Actions The true ground state (with J = 0) is a variational minimum So more sources should be better! (e.g., Γ[ρ, τ, J, ··· ]) Universal dependence on external potential is trivial: Z Z Z Γ[ρ] = W [J]− J ρ = Wv=0[J +v]− [(J +v)−v] ρ = Γv=0[ρ] + v ρ

But functionals change with resolution or field redefinitions =⇒ only stationary points are observables If uniform, find spontaneously broken ground state; if finite . . . NOTE: Beware of new UV divergences! [For Minkowski-space version of this, see Weinberg Vol. II ]

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Putting it all together . . .

1 J→0 δΓ[ρ] J→0 δΓ[ρ] Γ[ρ] = hHbiJ −→ E0 and J(x) = − −→ = 0 β δρ(x) δρ(x) ρgs(x)

=⇒ For static ρ(x), Γ[ρ] ∝ the DFT energy functional FHK!

Dick Furnstahl DFT from Effective Actions Universal dependence on external potential is trivial: Z Z Z Γ[ρ] = W [J]− J ρ = Wv=0[J +v]− [(J +v)−v] ρ = Γv=0[ρ] + v ρ

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Putting it all together . . .

1 J→0 δΓ[ρ] J→0 δΓ[ρ] Γ[ρ] = hHbiJ −→ E0 and J(x) = − −→ = 0 β δρ(x) δρ(x) ρgs(x)

=⇒ For static ρ(x), Γ[ρ] ∝ the DFT energy functional FHK! The true ground state (with J = 0) is a variational minimum So more sources should be better! (e.g., Γ[ρ, τ, J, ··· ])

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Putting it all together . . .

1 J→0 δΓ[ρ] J→0 δΓ[ρ] Γ[ρ] = hHbiJ −→ E0 and J(x) = − −→ = 0 β δρ(x) δρ(x) ρgs(x)

=⇒ For static ρ(x), Γ[ρ] ∝ the DFT energy functional FHK! The true ground state (with J = 0) is a variational minimum So more sources should be better! (e.g., Γ[ρ, τ, J, ··· ]) Universal dependence on external potential is trivial: Z Z Z Γ[ρ] = W [J]− J ρ = Wv=0[J +v]− [(J +v)−v] ρ = Γv=0[ρ] + v ρ

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Paths to the Effective Action Density Functional

1 Follow Coulomb Kohn-Sham DFT Calculate asymmetric nuclear matter as function of density =⇒ LDA functional + standard Kohn-Sham procedure Add semi-empirical gradient expansion 2 RG approach [Polonyi/Schwenk] 3 Auxiliary ﬁeld method [Faussurier, Valiev/Fernando] Eliminate ψ†ψ in favor of auxiliary ﬁeld ϕ Loop expansion about expectation value φ Kohn-Sham: Use freedom to require density unchanged 4 Inversion method [Fukuda et al., Valiev/Fernando] =⇒ systematic Kohn-Sham DFT

Dick Furnstahl DFT from Effective Actions Plan: Make this work by construction inversion method (“point-coupling”) auxiliary ﬁelds (e.g., “mesons” in covariant DFT)

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham DFT

=⇒ VKS VHO

Interacting density in VHO ≡ Non-interacting density in VKS Orbitals {φi (x)} in local potential VKS([ρ], x)

A 2 X 2 [−∇ /2m + VKS(x)]φi = εi φi =⇒ ρ(x) = |φi (x)| i=1

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham DFT

=⇒ VKS VHO

Interacting density in VHO ≡ Non-interacting density in VKS Orbitals {φi (x)} in local potential VKS([ρ], x)

A 2 X 2 [−∇ /2m + VKS(x)]φi = εi φi =⇒ ρ(x) = |φi (x)| i=1 Plan: Make this work by construction inversion method (“point-coupling”) auxiliary ﬁelds (e.g., “mesons” in covariant DFT)

Dick Furnstahl DFT from Effective Actions Interpretation: J0 is the external potential that yields for a noninteracting system the exact density This is the Kohn-Sham potential!

Two conditions involving J0 =⇒ Self-consistency

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing What can Power Counting do for DFT?

Given W [J] as an EFT expansion, how do we ﬁnd Γ[ρ]? Z Γ[ρ] = W [J] − Jρ

Inversion method: order-by-order inversion from W [J] to Γ[ρ]

Decompose J(x) = J0(x) + JLO(x) + JNLO(x) + ...

Two conditions on J0:

δW0[J0] δΓinteracting[ρ] ρ(x) = and J0(x)| = δJ (x) ρ=ρgs δρ(x) 0 ρ=ρgs

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing What can Power Counting do for DFT?

Given W [J] as an EFT expansion, how do we ﬁnd Γ[ρ]? Z Γ[ρ] = W [J] − Jρ

Inversion method: order-by-order inversion from W [J] to Γ[ρ]

Decompose J(x) = J0(x) + JLO(x) + JNLO(x) + ...

Two conditions on J0:

δW0[J0] δΓinteracting[ρ] ρ(x) = and J0(x)| = δJ (x) ρ=ρgs δρ(x) 0 ρ=ρgs

Interpretation: J0 is the external potential that yields for a noninteracting system the exact density This is the Kohn-Sham potential!

Two conditions involving J0 =⇒ Self-consistency

Dick Furnstahl DFT from Effective Actions LO :

E.g., apply to short-range LO contribution: Hartree-Fock

Z Z ∞ Z ∞ 0 1 3 dω dω 0 0 0 W1[J] = ν(ν − 1)C0 d x GJ (x, x; ω)GJ (x, x; ω ) 2 −∞ 2π −∞ 2π 1 (ν − 1) Z XF = − C d 3x [ρ (x)]2 where ρ (x) ≡ ν |ψ (x)|2 2 ν 0 J J α α

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Treat Source J(x) as a Background Field Effective action as a path integral =⇒ construct W [J], order-by-order in an expansion (e.g., EFT power counting) Propagators (lines) are in the background ﬁeld J(x) X θ(α − F) θ(F − α) G0(x, x0; ω) = ψ (x)ψ∗ (x0) + J α α ω − + iη ω − − iη α α α ∇2 where ψ (x) satisﬁes: − + v(x) − J(x) ψ (x) = ψ (x) α 2M α α α

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Treat Source J(x) as a Background Field Effective action as a path integral =⇒ construct W [J], order-by-order in an expansion (e.g., EFT power counting) Propagators (lines) are in the background ﬁeld J(x) X θ(α − F) θ(F − α) G0(x, x0; ω) = ψ (x)ψ∗ (x0) + J α α ω − + iη ω − − iη α α α ∇2 where ψ (x) satisﬁes: − + v(x) − J(x) ψ (x) = ψ (x) α 2M α α α E.g., apply to short-range LO contribution: Hartree-Fock

LO :

Dick Furnstahl DFT from Effective Actions Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’s Find J0 for the ground state via self-consistency loop:

X δΓi [ρ] J → W → Γ → J → W → Γ → · · · =⇒ J (x) = 0 1 1 1 2 2 0 δρ(x) i>0

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Kohn-Sham Via Inversion Method (cf. KLW [1960]) Inversion method for effective action DFT [Fukuda et al.] order-by-order matching in λ (e.g., EFT expansion) 2 diagrams =⇒ W [J, λ] = W0[J] + λW1[J] + λ W2[J] + ··· 2 assume =⇒ J[ρ, λ] = J0[ρ] + λJ1[ρ] + λ J2[ρ] + ··· 2 derive =⇒ Γ[ρ, λ] = Γ0[ρ] + λΓ1[ρ] + λ Γ2[ρ] + ··· Start with exact expressions for Γ and ρ

Z δW [J] Γ[ρ] = W [J] − d 4x J(x)ρ(x) =⇒ ρ(x) = δJ(x) =⇒ plug in expansions with ρ treated as order unity

Dick Furnstahl DFT from Effective Actions Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’s Find J0 for the ground state via self-consistency loop:

X δΓi [ρ] J → W → Γ → J → W → Γ → · · · =⇒ J (x) = 0 1 1 1 2 2 0 δρ(x) i>0

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Zeroth order is noninteracting system with potential J0(x) Z 4 δW0[J0] Γ0[ρ] = W0[J0] − d x J0(x)ρ(x) =⇒ ρ(x) = δJ0(x)

=⇒ Kohn-Sham system with the exact density! J0 ≡ VKS

Dick Furnstahl DFT from Effective Actions Find J0 for the ground state via self-consistency loop:

X δΓi [ρ] J → W → Γ → J → W → Γ → · · · =⇒ J (x) = 0 1 1 1 2 2 0 δρ(x) i>0

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Zeroth order is noninteracting system with potential J0(x) Z 4 δW0[J0] Γ0[ρ] = W0[J0] − d x J0(x)ρ(x) =⇒ ρ(x) = δJ0(x)

=⇒ Kohn-Sham system with the exact density! J0 ≡ VKS Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’s

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Zeroth order is noninteracting system with potential J0(x) Z 4 δW0[J0] Γ0[ρ] = W0[J0] − d x J0(x)ρ(x) =⇒ ρ(x) = δJ0(x)

=⇒ Kohn-Sham system with the exact density! J0 ≡ VKS Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’s Find J0 for the ground state via self-consistency loop:

X δΓi [ρ] J → W → Γ → J → W → Γ → · · · =⇒ J (x) = 0 1 1 1 2 2 0 δρ(x) i>0

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Potential

∗ 0 Local J0(x) [cf. non-local, state-dependent Σ (x, x ; ω)] δΓ [ρ, τ] δΓ [ρ, τ] e.g., J (x) = int and η (x) = int 0 δρ(x) 0 δτ(x) Direct derivatives (e.g., DME++) are easiest, or use “inverse density-density correlator”

1 δΓint[ρ] δρ(x) − δΓint[ρ]   − − · · · J0(x) = = Z = + δρ(x) δJ0(y) δJ0(y)  

= − + · · ·

New Feynman rules for Γint =⇒ anomalous diagrams

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Potential

∗ 0 Local J0(x) [cf. non-local, state-dependent Σ (x, x ; ω)] δΓ [ρ, τ] δΓ [ρ, τ] e.g., J (x) = int and η (x) = int 0 δρ(x) 0 δτ(x) Direct derivatives (e.g., DME++) are easiest,

or use “inverse density-density correlator”

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New Feynman rules for Γint =⇒ anomalous diagrams

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Potential

∗ 0 Local J0(x) [cf. non-local, state-dependent Σ (x, x ; ω)] δΓ [ρ, τ] δΓ [ρ, τ] e.g., J (x) = int and η (x) = int 0 δρ(x) 0 δτ(x) Direct derivatives (e.g., DME++) are easiest,

or use “inverse density-density correlator”

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New Feynman rules for Γint =⇒ anomalous diagrams

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Potential

∗ 0 Local J0(x) [cf. non-local, state-dependent Σ (x, x ; ω)] δΓ [ρ, τ] δΓ [ρ, τ] e.g., J (x) = int and η (x) = int 0 δρ(x) 0 δτ(x) Direct derivatives (e.g., DME++) are easiest,

or use “inverse density-density correlator”

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New Feynman rules for Γint =⇒ anomalous diagrams

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Potential

∗ 0 Local J0(x) [cf. non-local, state-dependent Σ (x, x ; ω)] δΓ [ρ, τ] δΓ [ρ, τ] e.g., J (x) = int and η (x) = int 0 δρ(x) 0 δτ(x) Direct derivatives (e.g., DME++) are easiest,

or use “inverse density-density correlator”

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New Feynman rules for Γint =⇒ anomalous diagrams

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Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Example: Dilute EFT Ingredients See “Crossing the Border” [nucl-th/0008064] 1 Use the most general L with low-energy dof’s consistent with global and local symmetries of underlying theory 2 † ∂ ∇ C0 † 2 D0 † 3 Left = ψ i ∂t + 2M ψ − 2 (ψ ψ) − 6 (ψ ψ) + ... 2 Declaration of regularization and renormalization scheme

natural a0 =⇒ dimensional regularization and min. subtraction 3 Well-deﬁned power counting =⇒ small expansion parameters

=⇒ kF Λ ∼ / =⇒ use the separation of scales Λ with 1 R kFa0, etc.

6 7 kF : kF : + O O

k 2 3 2 4 E = ρ F + (k a ) + (11 − 2 ln 2)(k a )2 + ··· 2M 5 3π F 0 35π2 F 0

Dick Furnstahl DFT from Effective Actions Same functional as dilute Fermi gas with ti ↔ Ci equivalent a0 ≈ −2–3 fm but |kFap|, |kFr0| < 1 (with ap < 0) missing non-analytic terms, NNN, . . .

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Comparing Skyrme and Dilute Functionals Skyrme energy density functional (for N = Z)

Z τ 3 1 1 E[ρ, τ, J] = d 3x + t ρ2 + (3t + 5t )ρτ+ (9t − 5t )(∇ρ)2 2M 8 0 16 1 2 64 1 2 3 1 − W ρ∇ · J + t ρ2+α + ··· 4 0 16 3

Dilute ρτ energy density functional for ν = 4 (Vexternal = 0) Z τ 3 1 1 E[ρ, τ, J] = d 3x + C ρ2 + (3C + 5C0 )ρτ+ (9C − 5C0 )(∇ρ)2 2M 8 0 16 2 2 64 2 2 3 c c 1 − C00ρ∇ · J + 1 C2ρ7/3 + 2 C3ρ8/3 + D ρ3 + ··· 4 2 2M 0 2M 0 16 0

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Comparing Skyrme and Dilute Functionals Skyrme energy density functional (for N = Z)

Z τ 3 1 1 E[ρ, τ, J] = d 3x + t ρ2 + (3t + 5t )ρτ+ (9t − 5t )(∇ρ)2 2M 8 0 16 1 2 64 1 2 3 1 − W ρ∇ · J + t ρ2+α + ··· 4 0 16 3

Same functional as dilute Fermi gas with ti ↔ Ci equivalent a0 ≈ −2–3 fm but |kFap|, |kFr0| < 1 (with ap < 0) missing non-analytic terms, NNN, . . .

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Power Counting Terms in Energy Functionals

Scale contributions according to average density or hkFi

ν =4, as = 0.10, A = 140

0.1

0.01 energy/particle

ap = as

ap=0 0.001 LO NLO LDA ρτ 10*(∇ρ) τ -NNLO Reasonable estimates =⇒ truncation errors understood Where to truncate for nuclei?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Covariant DFT as Legendre Transformation To probe the system, add a source V µ(x) coupled to current operator bjµ(x) ≡ ψ(x)γµψ(x) to the partition function:

Z R µ Z[V ] = e−W [V ] ∼ Tr e−β(Hb+V ·bj) −→ D[ψ†]D[ψ] e− [L+Vµ ψγ ψ]

The (time-dependent) current jµ(x) in presence of V µ(x) is

µ µ δW [V ] j (x) = ρv (x), jv (x) ≡ hψ(x)γ ψ(x)iV = δVµ(x)

Invert to ﬁnd V µ[j] and Legendre transform from V µ to jµ: Z µ δΓ[j] δΓ[j] Γ[j] = −W [V ] + V · j with V (x) = −→ = 0 δjµ(x) δjµ(x) jgs(x)

µ =⇒ For static j (x), Γ[j] ∝ the DFT energy functional E[ρv ]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing What About the Scalar Density?

Can add additional sources and Legendre transformations In nonrelativistic DFT, add to Lagrangian + η(x) ∇ψ†∇ψ Z Z Γ[ρ, τ] = W [J, η] − J(x)ρ(x) − η(x)τ(x)

=⇒ Skyrme HF energy functional E[ρ, τ, J] of density and kinetic energy density (see A. Bhattacharyya talk) In covariant DFT, add to Lagrangian + S(x) ψψ Z Z µ µ Γ[j , ρs] = W [V , S] − V (x) · j(x)− S(x)ρs(x)

µ =⇒ RMF energy functional E[ρv , ρs] [with j = (ρv , 0)] Generates point-coupling functional

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Pairing in DFT/EFT from Effective Action

Natural framework for spontaneous symmetry breaking e.g., test for zero-ﬁeld magnetization M in a spin system introduce an external ﬁeld H to break rotational symmetry Legendre transform Helmholtz free energy F(H):

invert M = −∂F(H)/∂H =⇒ Γ[M] = F[H(M)] + MH(M)

since H = ∂Γ/∂M, minimize Γ to ﬁnd ground state

Dick Furnstahl DFT from Effective Actions Effective action Γ[ρ, φ] by functional Legendre transformation: Z Z Γ[ρ, φ] = W [J, j] − d 4x J(x)ρ(x) − d 4x j(x)φ(x)

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Generalizing Effective Action to Include Pairing

Generating functional with sources J, j coupled to densities: Z −W [J,j] † − R d 4x [L + J(x)ψ† ψ + j(x)(ψ†ψ†+ψ ψ )] Z[J, j] = e = D(ψ ψ) e α α ↑ ↓ ↓ ↑

Densities found by functional derivatives wrt J, j:

† δW [J, j] ρ(x) ≡ hψ (x)ψ(x)iJ,j = δJ(x) j

† † δW [J, j] φ(x) ≡ hψ↑(x)ψ↓(x) + ψ↓(x)ψ↑(x)iJ,j = δj(x) J

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Generalizing Effective Action to Include Pairing

Generating functional with sources J, j coupled to densities: Z −W [J,j] † − R d 4x [L + J(x)ψ† ψ + j(x)(ψ†ψ†+ψ ψ )] Z[J, j] = e = D(ψ ψ) e α α ↑ ↓ ↓ ↑

Densities found by functional derivatives wrt J, j:

† δW [J, j] ρ(x) ≡ hψ (x)ψ(x)iJ,j = δJ(x) j

† † δW [J, j] φ(x) ≡ hψ↑(x)ψ↓(x) + ψ↓(x)ψ↑(x)iJ,j = δj(x) J Effective action Γ[ρ, φ] by functional Legendre transformation: Z Z Γ[ρ, φ] = W [J, j] − d 4x J(x)ρ(x) − d 4x j(x)φ(x)

Dick Furnstahl DFT from Effective Actions We need a method to carry out the inversion For Kohn-Sham DFT, apply inversion methods We need to renormalize!

Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Γ[ρ, φ] ∝ ground-state (free) energy functional E[ρ, φ] at ﬁnite temperature, the proportionality constant is β The sources are given by functional derivatives wrt ρ and φ

δE[ρ, φ] δE[ρ, φ] = J(x) and = j(x) δρ(x) δφ(x)

but the sources are zero in the ground state =⇒ determine ground-state ρ(x) and φ(x) by stationarity:

δE[ρ, φ] δE[ρ, φ] = = 0 δρ(x) δφ(x) ρ=ρgs,φ=φgs ρ=ρgs,φ=φgs This is Hohenberg-Kohn DFT extended to pairing!

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Γ[ρ, φ] ∝ ground-state (free) energy functional E[ρ, φ] at ﬁnite temperature, the proportionality constant is β The sources are given by functional derivatives wrt ρ and φ

δE[ρ, φ] δE[ρ, φ] = J(x) and = j(x) δρ(x) δφ(x)

but the sources are zero in the ground state =⇒ determine ground-state ρ(x) and φ(x) by stationarity:

δE[ρ, φ] δE[ρ, φ] = = 0 δρ(x) δφ(x) ρ=ρgs,φ=φgs ρ=ρgs,φ=φgs This is Hohenberg-Kohn DFT extended to pairing! We need a method to carry out the inversion For Kohn-Sham DFT, apply inversion methods We need to renormalize!

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Inversion Method Revisited

Order-by-order matching in EFT expansion parameter λ

2 W [J, j, λ] = W0[J, j] + λW1[J, j] + λ W2[J, j] + ··· 2 J[ρ, φ, λ] = J0[ρ, φ] + λJ1[ρ, φ] + λ J2[ρ, φ] + ··· 2 j[ρ, φ, λ] = j0[ρ, φ] + λj1[ρ, φ] + λ j2[ρ, φ] + ··· 2 Γ[ρ, φ, λ] = Γ0[ρ, φ] + λΓ1[ρ, φ] + λ Γ2[ρ, φ] + ···

th 0 order is Kohn-Sham system with potentials J0(x) and j0(x) =⇒ yields the exact densities ρ(x) and φ(x) introduce single-particle orbitals and solve (cf. HFB) h0(x) − µ0 j0(x) ui (x) ui (x) = Ei j0(x) −h0(x) + µ0 vi (x) vi (x) ∇2 where h (x) ≡ − + V (x) − J (x) 0 2M 0 with conventional orthonormality relations for ui , vi

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing

Diagrammatic Expansion of Wi

Same diagrams, but with Nambu-Gor’kov Green’s functions

Γ = + + + + int · · ·

† 0 0 ! 0 0 ! hT ψ↑(x)ψ↑(x )i0 hT ψ↑(x)ψ↓(x )i0 iGks iFks iG = † † 0 † 0 ≡ 0 † 0 hT ψ↓(x)ψ↑(x )i0 hT ψ↓(x)ψ↓(x )i0 iFks −iGks

In frequency space, the Green’s functions are

∗ 0 0 ∗ 0 0 X ui (x) ui (x ) vi (x ) vi (x) iGks(x, x ; ω) = + ω − Ei + iη ω + Ei − iη i

∗ 0 0 ∗ 0 0 X ui (x) vi (x ) ui (x ) vi (x) iFks(x, x ; ω) = − − ω − Ei + iη ω + Ei − iη i

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Basics Kohn-Sham Covariant Pairing Kohn-Sham Self-Consistency Procedure

Same iteration procedure as in Skyrme or RMF with pairing In terms of the orbitals, the fermion density is

X 2 ρ(x) = 2 |vi (x)| i and the pair density is (warning: divergent!)

X ∗ ∗ φ(x) = [ui (x)vi (x) + ui (x)vi (x)] i R The chemical potential µ0 is ﬁxed by ρ(x) = A Diagrams for eΓ[ρ, φ] = −E[ρ, φ] (with LDA+) yields KS potentials

δΓ [ρ, φ] δΓ [ρ, φ] eint eint J0(x) = and j0(x) = ρ=ρgs δρ(x) φ=φgs δφ(x) ρ=ρgs φ=φgs

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Outline

Overview: Microscopic DFT

Effective Actions and DFT

Issues and Ideas and Open Problems

Summary

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

How do we connect to the free NN··· N interaction? Chiral EFT −→RG low-momentum interactions: Power counting? What can you calculate in a DFT approach? What about single-particle properties? Excited states? How is Kohn-Sham DFT more than “mean ﬁeld”? Where are the approximations? How do we truncate? How do we include long-range effects (correlations)? What about broken symmetries? (translation, rotation, . . . ) What about the “Dirac sea” in covariant DFT? What about UV divergences in DFT pairing? Can we (should we) decouple pp and ph? Are higher-order contributions important?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

Dick Furnstahl DFT from Effective Actions Add a non-local source ξ(x0, x) coupled to ψ(x)ψ†(x0):

Z R 4 † R 4 0 0 † 0 Z [J, ξ] = eiW [J,ξ] = DψDψ† ei d x [L + J(x)ψ (x)ψ(x) + d x ψ(x)ξ(x,x )ψ (x )]

With Γ[ρ, ξ] = Γ0[ρ, ξ] + Γint[ρ, ξ],

0 δW δΓ 0 h1 δΓint δΓint i G(x, x ) = = = Gks(x, x ) + Gks + Gks

δξ J δξ ρ i δGks δρ

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Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

How is the Full G Related to Gks? [nucl-th/0410105]

x0 + + + + =⇒ = + = Σ∗(x, x0; ω) · · · x ⇒

Dick Furnstahl DFT from Effective Actions With Γ[ρ, ξ] = Γ0[ρ, ξ] + Γint[ρ, ξ],

0 δW δΓ 0 h1 δΓint δΓint i G(x, x ) = = = Gks(x, x ) + Gks + Gks

δξ J δξ ρ i δGks δρ

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Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

How is the Full G Related to Gks? [nucl-th/0410105]

x0 + + + + =⇒ = + = Σ∗(x, x0; ω) · · · x ⇒

Add a non-local source ξ(x0, x) coupled to ψ(x)ψ†(x0):

Z R 4 † R 4 0 0 † 0 Z [J, ξ] = eiW [J,ξ] = DψDψ† ei d x [L + J(x)ψ (x)ψ(x) + d x ψ(x)ξ(x,x )ψ (x )]

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

How is the Full G Related to Gks? [nucl-th/0410105]

x0 + + + + =⇒ = + = Σ∗(x, x0; ω) · · · x ⇒

Add a non-local source ξ(x0, x) coupled to ψ(x)ψ†(x0):

Z R 4 † R 4 0 0 † 0 Z [J, ξ] = eiW [J,ξ] = DψDψ† ei d x [L + J(x)ψ (x)ψ(x) + d x ψ(x)ξ(x,x )ψ (x )]

With Γ[ρ, ξ] = Γ0[ρ, ξ] + Γint[ρ, ξ],

0 δW δΓ 0 h1 δΓint δΓint i G(x, x ) = = = Gks(x, x ) + Gks + Gks

δξ J δξ ρ i δGks δρ

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Dick Furnstahl DFT from Effective Actions

Simple diagrammatic demonstration:

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Densities agree by construction! Is the Kohn-Sham basis a useful one for G?

Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

G and Gks Yield the Same Density by Construction

0 + +

Claim: ρks(x) = −iνGKS(x, x ) equals ρ(x) = −iνG(x, x )

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Dick Furnstahl DFT from Effective Actions Is the Kohn-Sham basis a useful one for G?

Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

G and Gks Yield the Same Density by Construction

0 + +

Claim: ρks(x) = −iνGKS(x, x ) equals ρ(x) = −iνG(x, x )

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Simple diagrammatic demonstration:

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Densities agree by construction!

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

G and Gks Yield the Same Density by Construction

0 + +

Claim: ρks(x) = −iνGKS(x, x ) equals ρ(x) = −iνG(x, x )

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Simple diagrammatic demonstration:

¡ ¢ ¢ ¡

Densities agree by construction! Is the Kohn-Sham basis a useful one for G?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing

How Close is GKS to G?

It depends on what sources are used!

0 δW δΓ 0 h1 δΓint δΓint i G(x, x ) = = = Gks(x, x ) + Gks + Gks δξ J δξ ρ i δGks δρ -20 A = 16 1p1/2

-30 1p3/2 Nonrel. M∗ in Γ[ρ] vs. Γ[ρ, τ] vs. ··· (see Anirban’s talk) Covariant case at LO: -40 Γ[ρv ] vs. Γ[ρv , ρs]

Higher orders? binding energy (MeV) -50

1s1/2 ρ ρv,ρs v -60

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Kohn-Sham DFT and “Mean-Field” Models

n(k)

1 ¡

kF k 1 Kohn-Sham propagator always has “mean-ﬁeld” structure =⇒ doesn’t mean that correlations aren’t included in Γ[ρ]!

2 † n(k) = hakaki is resolution dependent (not observable!) =⇒ operator related to experiment is more complicated 3 Is the Kohn-Sham basis a useful one for other observables?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Approximating and Fitting the Functional

Need a truncated expansion to carry out inversion method Chiral EFT expansion is well-defined Power counting for low-momentum interactions? Gradient expansions? Density matrix expansion Semiclassical expansions used in Coulomb DFT Derivative expansion techniques developed for (one-loop) effective actions? How should we “fine tune” a DFT functional? What does EFT say about what knobs to adjust? EFT tells about theoretical errors =⇒ use in fits (e.g., Bayesian)

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Long-range Effects

Long-range forces (e.g., pion exchange) =⇒ limits of DME++

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Non-localities from near-on-shell particle-hole excitations

+ + + + · · ·

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Symmetry Breaking and Zero Modes

What about breaking of translational, rotational invariance, particle number? No guidance from Coulomb DFT (?) Effective action =⇒ zero modes cf. soliton zero modes and projection methods Fadeev-Popov games? Energy functional for the intrinsic density? =⇒ J. Engel: one-dimensional laboratory

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing UV Divergences in Nonrelativistic and Relativistic Effective Actions

All low-energy effective theories have incorrect UV behavior Sensitivity to short-distance physics signalled by divergences but ﬁniteness (e.g., with cutoff) doesn’t mean not sensitive! =⇒ must absorb (and correct) sensitivity by renormalization Instances of UV divergences

nonrelativistic covariant scattering scattering pairing pairing anti-nucleons

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Power Counting Lost / Power Counting Regained Gasser, Sainio, Svarc =⇒ ChPT for πN with relativistic N’s loop and momentum expansions don’t agree =⇒ systematic power counting lost heavy-baryon EFT restores power counting by 1/M expansion Hua-Bin Tang (1996) [and with Paul Ellis]: “. . . EFT’s permit useful low-energy expansions only if we absorb all of the hard-momentum effects into the parameters of the Lagrangian.”

VB negative−energy states positive−energy states

q > Λ = C0 and x x x x x x x} x x x x } ⇒ } x x x x x ω VB −M holes µ +M

Becher/Leutwyler IR =⇒ Schindler-Gegelia-Scherer version

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Moving Dirac Sea Physics into Coefﬁcients

Absorb the “hard” part of a diagram into parameters, =⇒ the remaining “soft” part satisﬁes chiral power counting original πN prescription by H.B. Tang (expand, integrate term-by-term, and resum propagators) systematized for πN by Becher and Leutwyler: “infrared regularization” or IR not unique; e.g., Fuchs et al. additional ﬁnite subtractions in DR Extension of IR to multiple heavy particles [Lehmann/Prezeau]´ convenient reformulation by Schindler, Gegelia, Scherer tadpoles, NN loops in free space vanish! particle-particle loop reduces to nonrelativistic DR/MS result

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Consequences for Free-Space Natural Fermions

Tadpoles, NN loops in free space vanish! Leading order (LO) has scalar, vector, etc. vertices

Cs Cv µ Left = · · · − 2 (ψψ)(ψψ) − 2 (ψγ ψ)(ψγµψ) + ··· =⇒

At NLO, only particle-particle loop survives IR

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Only forward-going nucleons contribute =⇒ same scattering amplitude as nonrel. DR/MS for small k

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Comments on Vacuum Physics

Unlike QED DFT, “no sea” for nuclear structure is a misnomer include “vacuum physics” in coefﬁcients via renormalization Renormalization versus Renormalizability Renormalization is required to account for short-distance behavior but can be implicit Renormalizability at the hadronic level corresponds to making a model for the short-distance behavior not a good model phenomenologically Please don’t send me any more RHA papers to referee! Fixing short-distance behavior is not the same thing as throwing away negative-energy states For a long time, we looked for unique “relativistic effects”; these were largely misguided efforts

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Questions about DFT and Nuclear Structure

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing UV Divergences in Nonrelativistic and Relativistic Effective Actions

nonrelativistic covariant scattering scattering pairing pairing anti-nucleons

Dick Furnstahl DFT from Effective Actions 1 2 + 2 ζ j ]

1 2 Strategy: Add counterterm 2 ζ j to L additive to W (cf. |∆|2) =⇒ no effect on scattering Energy interpretation? Finite part?

Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Divergences: Dilute Fermi System Generating functional with constant sources µ and j:

Z 2 C −W † − R d 4x [ψ† ( ∂ − ∇ −µ)ψ + 0 ψ†ψ†ψ ψ + j(ψ ψ +ψ†ψ†)] e = D(ψ ψ) e α ∂τ 2M α 2 ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑

1 R 2 R ∗ − |∆| cf. adding integration over auxiliary ﬁeld D(∆ , ∆) e |C0| † † ∗ =⇒ shift variables to eliminate ψ↑ψ↓ψ↓ψ↑ for ∆ ψ↑ψ↓

New divergences because of j =⇒ e.g., expand to O(j2)

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Same linear divergence as in 2-to-2 scattering

Dick Furnstahl DFT from Effective Actions ]

Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Divergences: Dilute Fermi System Generating functional with constant sources µ and j:

Z 2 C −W † − R d 4x [ψ† ( ∂ − ∇ −µ)ψ + 0 ψ†ψ†ψ ψ + j(ψ ψ +ψ†ψ†) + 1 ζ j2 ] e = D(ψ ψ) e α ∂τ 2M α 2 ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑ 2

1 R 2 R ∗ − |∆| cf. adding integration over auxiliary ﬁeld D(∆ , ∆) e |C0| † † ∗ =⇒ shift variables to eliminate ψ↑ψ↓ψ↓ψ↑ for ∆ ψ↑ψ↓

New divergences because of j =⇒ e.g., expand to O(j2)

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Same linear divergence as in 2-to-2 scattering 1 2 Strategy: Add counterterm 2 ζ j to L additive to W (cf. |∆|2) =⇒ no effect on scattering Energy interpretation? Finite part?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Renormalized Uniform System Observables

To ﬁnd the energy density, evaluate Γ at the stationary point:

Z 3 2 E d k 1 j0 h 1 i = (Γ + Γ )| 1 = ξ −E + + µ − |C |ρ ρ 0 1 j0=− |C0|φ 3 k k 0 0 V 2 (2π) 2 Ek 4 with Z 3 Z 3 d k ξk d k j0 (0) ρ = 3 1 − and φ = − 3 + ζ j0 (2π) Ek (2π) Ek

Explicitly ﬁnite and dependence on ζ(0) cancels out Finite system =⇒ optimize renormalization (see Bulgac et al.)

Dick Furnstahl DFT from Effective Actions NLO modiﬁes exponent =⇒ changes prefactor 1/3 ∆NLO ≈ ∆LO/(4e)

Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Higher Order: Induced Interaction

1 As j0 → 0, uk vk peaks at µ0 2 u2 vk k Leading order T = 0: 8 −1/N(0)|C0| ∆LO/µ0 = e2 e = 8 e−π/2kF|as| e2 j0

ukvk 0 ε

µ0 k

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Same renormalization works (Furnstahl/Hammer) =⇒ energy interpretation? ﬁnite system?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary GreenF Mean-Field Symmetry Covariant Pairing Higher Order: Induced Interaction

1 As j0 → 0, uk vk peaks at µ0 2 u2 vk k Leading order T = 0: 8 −1/N(0)|C0| ∆LO/µ0 = e2 e = 8 e−π/2kF|as| e2 j0 NLO modiﬁes exponent

=⇒ changes prefactor ukvk 1/3 0 ∆NLO ≈ ∆LO/(4e) ε

µ0 k

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Same renormalization works (Furnstahl/Hammer) =⇒ energy interpretation? ﬁnite system?

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Outline

Overview: Microscopic DFT

Effective Actions and DFT

Issues and Ideas and Open Problems

Summary

Dick Furnstahl DFT from Effective Actions Outline Overview Action Topics Summary Summary

Plan: Chiral EFT −→ low momentum VNN , VNNN ,... −→ DFT for nuclei Effective action formalism provides framework Many issues to resolve (my list for today) gradient expansions (DME++, . . . ), long-range effects isospin dependence, many-body contributions, low-density limit symmetry breaking and restoration higher-order pairing how to ﬁne-tune? systematic covariant DFT . .

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =2.0 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 1.5 1.5 k (fm−1) 0 k' (fm−1) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =1.8 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 −1 0 k' (fm−1) 1.5 1.5 k (fm ) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =1.6 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 −1 1.5 1.5 0 k' (fm ) k (fm−1) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =1.4 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 1.5 1.5 −1 0 k' (fm−1) k (fm ) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =1.2 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 1.5 1.5 −1 0 k' (fm−1) k (fm ) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG

The Deuteron at Different Resolutions back

Integrand of −〈ψ | V |ψ 〉 for Λ =1.0 fm−1 d Λ d 20

15 14

12 10 10 5 8

0 6

0 0 4 0.5 0.5 2 1 1 −1 0 k' (fm−1) 1.5 1.5 k (fm ) 2 2

Dick Furnstahl DFT from Effective Actions Outline RG Effective Action as Energy Functional: Minkowski

back

See, e.g., Weinberg, Vol. II

Dick Furnstahl DFT from Effective Actions Outline RG

Polonyi-Schwenk RG Approach to DFT back

Gradually switch off background Non-interacting fermions in potential and turn on the background mean-ﬁeld microscopic interaction U potential V at λ = 0 as λ → 1

Z ∂ ∇2 S [ψ†, ψ] = dx ψ† (x) − x + (1 − λ)V (x) ψ (x) λ,1 α ∂t 2M λ;α α λ ZZ S [ψ†, ψ] = (ψ†ψ) · U · (ψ†ψ) λ,2 2

Dick Furnstahl DFT from Effective Actions Outline RG Density Functional = Effective Action for Density

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The effective action is minimal at Effective action Γ[ρ] = −W [J] + J · ρ is minimal at the the physphysicalical (=zer (zeroo sourc source)e) ground ground state de statensity, density: i.e., , with g.s. energy δΓ[ρ] 1 The effec=tiv0e actio=n⇒ Egs = E [ ρ gsis ]m =inimlimal at Γ[ρgs] δρ β→∞ β the pρhgsysical (=zero source) ground state density, i.e., Density functional = effective action for the density , with g.s. energy Curvature will include correlations Density functional c=u erffvecattivuer aec tiwoni flolr itnhec dleundsiety −1 exchang e-c2orrel atio!ns 2 curvaδturΓ[e wρi]ll include δ W [J] exch-a1nge-cor relations = δρ δρ-1 δJ δJ ρgs J=0

== = + interactions + interactions + interactions Dick Furnstahl DFT from Effective Actions Outline RG Evolution of Effective Action with Parameter λ

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∆ background Hartree exchange-correlations Evolution of the effective action with control para"meter −1# 1 1 δ2Γ [ρ] ∂ Γ [ρ] = ∂ [(1 − λ)V ] · ρ + ρ · U · ρ + Tr U · λ λ λ λ λ 2 2 δρ δρ

chaExpandnge in backg densityround pote functionalntial Hartree c aboutontributio evolvingn exchangeground-state-correlations density Expand density functional around evolving g.s. density X Z Z 1 Γ [ρ] = Γ[ρ ](0) + ··· Γ[ρ ](n) · (ρ − ρ ) ··· (ρ − ρ ) λ gs,λ n! gs,λ gs,λ 1 gs,λ n n≥2 Evolution equations forEvolution expansion equations coeff. for (buexpansionild up many c coefﬁcientsorr. as build up Fercorrelationsmi liquid RG) through dressed ph propagator

All exchange-corr. via drDickess Furnstahled ph propDFTag fromato Effectiver Actions Outline RG

Auxiliary Fields [Faussurier] back Introduce scalar ﬁeld ϕ coupled to ψ†ψ Construct Se[ψ†, ψ, ϕ] such that ψ, ψ† is only in ψ†[G−1(ϕ)]ψ and Z † † Dϕ eiSe[ψ ,ψ,ϕ] =⇒ eiS[ψ ,ψ]

Integrate out ψ†ψ =⇒ determinant =⇒ Tr ln[G−1(ϕ)] + ···

Keep only leading saddle point φ0(x) =⇒ Hartree ﬂuctuation corrections generate loop expansion freedom to choose mean ﬁeld [Kerman et al. (1983)] cf., H = (T + U) + (V − U) for arbitrary U Kohn-Sham: choose special saddle-point evaluation + reference local potential φxc such that −Tr Gxc(x, x ) = n(x) −1 expand Tr ln[Gxc + δφ] in δφ = φ − φxc =⇒ Γxc[n] with φxc(x) = δΓxc[n]/δn(x) −1 introduce orbitals {ψα, α} to diagonalize Tr ln[Gxc ]

Dick Furnstahl DFT from Effective Actions Dominant application: inhomogeneous Atomization Energies of Hydrocarbon Molecules electron gas 20 Interacting point electrons 0 in static potential of -20 atomic nuclei -40

0.5 10 3 3 S deuteron w.f. S1 deuteron w.f. 0.4 1

-1 ] 1 Λ = 2 fm ] -1 -1 3/2 0.3 = 2.0 fm Λ = 3 fm -3/2 Λ -1 -1 Λ = 3.0 fm Λ = 4 fm -1 Argonne v18

Λ = 4.0 fm (k)| [fm 0 (r) [fm 0.2 Argonne v 18 ψ 0.1 ψ |

0.1 0.01

0 0 1 2 3 4 5 0 2 4 6 8 10 -1 r [fm] k [fm ] Repulsive core =⇒ short-distance suppression =⇒ high-momentum components Low-momentum potential =⇒ much simpler wave function!

0.5 10 3 3 S deuteron w.f. S1 deuteron w.f. 0.4 1

-1 ] 1 Λ = 2 fm ] -1 -1 3/2 0.3 = 2.0 fm Λ = 3 fm -3/2 Λ -1 -1 Λ = 3.0 fm Λ = 4 fm -1 Argonne v18

Λ = 4.0 fm (k)| [fm 0 (r) [fm 0.2 Argonne v 18 ψ 0.1 ψ |

0.1 0.01

0 0 1 2 3 4 5 0 2 4 6 8 10 -1 r [fm] k [fm ] 1.6 1.6

1 -1 3 -1 1.4 S0 at kF = 1.35 fm 1.4 S1 at kF = 1.35 fm -1 -1 [P = 0, k = 0.1 fm , m*/m = 1] [P = 0, k = 0.1 fm , m*/m = 1] 1.2 1.2

1 1 (r) (r) k 0.8 k 0.8 Ψ Ψ 0.6 Fermi gas 0.6 Fermi gas v18 v18 3 3 0.4 N LO 0.4 N LO -1 -1 Λ = 2.0 fm (v18) Λ = 2.0 fm (v18) -1 3 -1 3 0.2 Λ = 2.0 fm (N LO) 0.2 Λ = 2.0 fm (N LO)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 r [fm] r [fm] 1.6 1.6

1 -1 3 -1 1.4 S0 at kF = 1.35 fm 1.4 S1 at kF = 1.35 fm -1 -1 [P = 0, k = 0.1 fm , m*/m = 1] [P = 0, k = 0.1 fm , m*/m = 1] 1.2 1.2

1 1 (r) (r) k 0.8 k 0.8 Ψ Ψ 0.6 Fermi gas 0.6 Fermi gas v18 v18 3 3 0.4 N LO 0.4 N LO -1 -1 Λ = 2.0 fm (v18) Λ = 2.0 fm (v18) -1 3 -1 3 0.2 Λ = 2.0 fm (N LO) 0.2 Λ = 2.0 fm (N LO)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 r [fm] r [fm] Scale contributions according to average density or hkFi

Reasonable estimates =⇒ truncation errors understood Where to truncate for nuclei?