covariant density functional theory

Outline DFT Action Dilute Renormalization Summary

Covariant Density Functional Theory

Dick Furnstahl

Department of PhysicsOhio State University

September, 2004

Collaborators: A. Bhattacharyya, J. Engel, H.-W. Hammer,J. Piekarewicz, S. Puglia, A. Schwenk, B. Serot

Dick Furnstahl Covariant DFT

Covariant Density Functional Theory

Dick Furnstahl

Department of PhysicsOhio State University

September, 2004

Collaborators: A. Bhattacharyya, J. Engel, H.-W. Hammer,J. Piekarewicz, S. Puglia, A. Schwenk, B. Serot

A=12 A~60

Density Functio

nal Theory

Selfconsis

tent Mean Field

Ab initiofew-body

calculations No-Core Shell Model G-matrix

r-process

rp-process

0Ñω ShellModel

Limits of nuclearexistence

neutrons

Many-body approachesfor ordinary nuclei

Figure 2: Top: the nuclear landscape - the territory of RIA physics. The black squares represent the stable nuclei and the nuclei with half-lives comparable to or longer than the age of the Earth (4.5 billion years). These nuclei form the "valley of stability". The yellow region indicates shorter lived nuclei that have been produced and studied in laboratories. By adding either protons or neutrons one moves away from the valley of stability, finally reaching the drip lines where the nuclear binding ends because the forces between neutrons and protons are no longer strong enough to hold these particles together. Many thousands of radioactive nuclei with very small or very large N/Z ratios are yet to be explored. In the (N,Z) landscape, they form the terra incognita indicated in green. The proton drip line is already relatively well delineated experimentally up to Z=83. In contrast, the neutron drip line is considerably further from the valley of stability and harder to approach. Except for the lightest nuclei where it has been reached experimentally, the neutron drip line has to be estimated on the basis of nuclear models - hence it is very uncertain due to the dramatic extrapolations involved. The red vertical and horizontal lines show the magic numbers around the valley of stability. The anticipated paths of astrophysical processes (r-process, purple line; rp-process, turquoise line) are shown. Bottom: various theoretical approaches to the nuclear many-body problem. For the lightest nuclei, ab initio calculations (Green’s Function Monte Carlo, no-core shell model) based on the bare nucleon-nucleon interaction, are possible. Medium-mass nuclei can be treated by the large-scale shell model. For heavy nuclei, the density functional theory (based on selfconsistent mean field) is the tool of choice. By investigating the intersections between these theoretical strategies, one aims at nothing less than developing the unified description of the nucleus.

Issues and Questions about Covariant DFT

How is covariant Kohn-Sham DFT more than Hartree?Where are the approximations?How do we include long-range effects?

What can you calculate in a DFT approach? Excited states?What about single-particle properties?

How do we carry out a covariant nuclear EFT and DFT?Functionals depending on just jµ or ρs also?Does point coupling vs. meson fields matter?What about “vacuum physics”?

How does pairing work in DFT? Does covariance matter?

Should we connect to the free NN interaction?What about chiral EFT or low-momentum interactions and RG?

Outline

(Relativistic) Kohn-Sham DFT

Effective Action Approach to EFT-Based Kohn-Sham DFT

Insights from Dilute Fermi Systems

Renormalization of Covariant Kohn-Sham DFT

Ongoing and Future Challenges

Outline DFT Action Dilute Renormalization Summary Intro Covariant

Density Functional Theory (DFT)

Dominant application:inhomogeneouselectron gas

Interacting point electronsin static potential of atomicnuclei

“Ab initio” calculations ofatoms, molecules,crystals, surfaces

1970 1975 1980 1985 1990 199510

Density Functional Theory

Hartree−Fock

Dominant application:inhomogeneouselectron gas

Interacting point electronsin static potential of atomicnuclei

“Ab initio” calculations ofatoms, molecules,crystals, surfaces

H2 C2 C2H2 CH4 C2H4 C2H6 C6H6

molecule

Hartree-FockDFT Local Spin Density ApproximationDFT Generalized Gradient Approximation

Atomization Energies of Hydrocarbon Molecules

Hohenberg-Kohn: There existsan energy functional Ev [ρ] . . .

Ev [ρ] = FHK [ρ] +

∫d3x v(x)ρ(x)

FHK is universal (same for anyexternal v ) =⇒ H2 to DNA!

Useful if you can approximatethe energy functional

Kohn-Sham procedure similarto nuclear “mean field”calculations

Ev [ρ] = FHK [ρ] +

∫d3x v(x)ρ(x)

Ev [ρ] = FHK [ρ] +

∫d3x v(x)ρ(x)

Kohn-Sham DFT

=⇒VKS

Interacting density with vHO ≡ Non-interacting density with vKS

Orbitals φi(x) in local potential vKS([ρ], x) [but no M∗(x)]

[−∇2/2m + vKS(x)]φi = εiφi =⇒ ρ(x) =N∑

|φi(x)|2

Find Kohn-Sham potential vKS(x) from δEv [ρ]/δρ(x)Solve self-consistently

Kohn-Sham DFT

=⇒VKS

Interacting density with vHO ≡ Non-interacting density with vKS

Orbitals φi(x) in local potential vKS([ρ], x) [but no M∗(x)]

[−∇2/2m + vKS(x)]φi = εiφi =⇒ ρ(x) =N∑

|φi(x)|2

Find Kohn-Sham potential vKS(x) from δEv [ρ]/δρ(x)Solve self-consistently

Relativistic DFT

QED extension by Rajagopal/Callaway and Macdonald/VoskoSimilar HK theorems with jµ(x) instead of ρ(x)

and (four-vector) external potential

F [jµ] = E [jµ]−∫

d3x jµ(x)Vµext(x)

Kohn-Sham relativistic DFTSchrodinger equation −→ Dirac equation

Applications: materials containing heavy elements (Zα ∼ 1)Heavy-atom energies, magnetic moments of Fe, Co, Ni, . . .

QHD formulation by Speicher, Dreizler, and Engel (1992)

Questions and open problemsTreatment of UV divergences??? Vacuum subtractions???Construction of exchange-correlation functional (LDA?)

Relativistic DFT

QED extension by Rajagopal/Callaway and Macdonald/VoskoSimilar HK theorems with jµ(x) instead of ρ(x)

and (four-vector) external potential

F [jµ] = E [jµ]−∫

d3x jµ(x)Vµext(x)

Kohn-Sham relativistic DFTSchrodinger equation −→ Dirac equation

Applications: materials containing heavy elements (Zα ∼ 1)Heavy-atom energies, magnetic moments of Fe, Co, Ni, . . .

QHD formulation by Speicher, Dreizler, and Engel (1992)

Questions and open problemsTreatment of UV divergences??? Vacuum subtractions???Construction of exchange-correlation functional (LDA?)

Outline DFT Action Dilute Renormalization Summary Thermo Action KLW EFT

Thermodynamic Interpretation of DFT

Consider a system of spins Si

on a lattice with interaction g

The partition function has theinformation about the energy,magnetization of the system:

Z = Tr e−βg∑

i,j Si Sj

The magnetization M is

M =⟨∑

[(∑i

)e−βg

∑i,j Si Sj

Thermodynamic Interpretation of DFT

Consider a system of spins Si

on a lattice with interaction g

The partition function has theinformation about the energy,magnetization of the system:

Z = Tr e−βg∑

i,j Si Sj

The magnetization M is

M =⟨∑

[(∑i

)e−βg

∑i,j Si Sj

Add A Magnetic Probe Source H

The source probes configurationsnear the ground state

Z[H] = e−βF [H] = Tr e−β(g∑

i,j Si Sj−H∑

i Si )

Variations of the source yield themagnetization

M =⟨∑

= −∂F [H]

F [H] is the Helmholtz free energy.Set H = 0 (or equal to a realexternal source) at the end

i,j Si Sj−H∑

i Si )

M =⟨∑

= −∂F [H]

i,j Si Sj−H∑

i Si )

M =⟨∑

= −∂F [H]

Legendre Transformation to Effective Action

Find H[M] by inverting

M =⟨∑

= −∂F [H]

Legendre transform to the Gibbsfree energy

Γ[M] = F [H] + H M

The ground-state magnetizationMgs follows by minimizing Γ[M]:

H =∂Γ[M]

∂M−→ ∂Γ[M]

∣∣∣∣Mgs

M =⟨∑

= −∂F [H]

Γ[M] = F [H] + H M

H =∂Γ[M]

∂M−→ ∂Γ[M]

∣∣∣∣Mgs

M =⟨∑

= −∂F [H]

Γ[M] = F [H] + H M

H =∂Γ[M]

∂M−→ ∂Γ[M]

∣∣∣∣Mgs

Variational Energy and the Effective Action

Consider generalized Hamiltonian including time-independent H:

H(H) = g∑i,j

SiSj − H∑

In the large β limit, Z =⇒ ground state of H(H) with energy

E(H) = limβ→∞

−1β

Separating out the pieces:

E(H) = 〈H(H)〉H = 〈H〉H − H⟨∑

= 〈H〉H − HM

Thus as T → 0, the effective action

Γ(M) = E(H) + HM = 〈H〉H

is the expectation value of H in the ground state generated by H

The true ground state (with H = 0) is the variational minimum!

H(H) = g∑i,j

SiSj − H∑

E(H) = limβ→∞

−1β

E(H) = 〈H(H)〉H = 〈H〉H − H⟨∑

= 〈H〉H − HM

Γ(M) = E(H) + HM = 〈H〉H

H(H) = g∑i,j

SiSj − H∑

E(H) = limβ→∞

−1β

E(H) = 〈H(H)〉H = 〈H〉H − H⟨∑

= 〈H〉H − HM

Γ(M) = E(H) + HM = 〈H〉H

H(H) = g∑i,j

SiSj − H∑

E(H) = limβ→∞

−1β

E(H) = 〈H(H)〉H = 〈H〉H − H⟨∑

= 〈H〉H − HM

Γ(M) = E(H) + HM = 〈H〉H

DFT as Analogous Legendre Transformation

In analogy to the spin system, add source J(x) coupled todensity operator ρ(x) ≡ ψ†(x)ψ(x) to the partition function:

Z[J] = e−W [J] ∼ Tr e−β(H+J ρ) −→∫D[ψ†]D[ψ] e−

∫[L+J ψ†ψ]

The (time-dependent) density ρ(x) in the presence of J(x) is

ρ(x) ≡ 〈ρ(x)〉J =δW [J]

δJ(x)

Invert to find J[ρ] and Legendre transform from J to ρ:

Γ[ρ] = −W [J] +

∫J ρ with J(x) =

δΓ[ρ]

δρ(x)−→ δΓ[ρ]

δρ(x)

∣∣∣∣ρgs(x)

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

∫[L+J ψ†ψ]

ρ(x) ≡ 〈ρ(x)〉J =δW [J]

δJ(x)

Γ[ρ] = −W [J] +

∫J ρ with J(x) =

δΓ[ρ]

δρ(x)

∣∣∣∣ρgs(x)

∫[L+J ψ†ψ]

ρ(x) ≡ 〈ρ(x)〉J =δW [J]

δJ(x)

Γ[ρ] = −W [J] +

∫J ρ with J(x) =

δΓ[ρ]

δρ(x)

∣∣∣∣ρgs(x)

Covariant DFT as Legendre Transformation

In analogy to the spin system, add source V µ(x) coupled tocurrent operator jµ(x) ≡ ψ(x)γµψ(x) to the partition function:

Z[V ] = e−W [V ] ∼ Tr e−β(H+V ·j) −→∫D[ψ†]D[ψ] e−

∫[L+Vµ ψγ

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

jµ(x) ≡ 〈jµ(x)〉V =δW [V ]

δvµ(x)

Invert to find V µ[j] and Legendre transform from V µ to jµ:

Γ[j] = −W [V ] +

∫v · j with Vµ(x) =

δΓ[j]δjµ(x)

−→ δΓ[j]δjµ(x)

∣∣∣∣jgs(x)

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

∫[L+Vµ ψγ

δvµ(x)

Γ[j] = −W [V ] +

δΓ[j]δjµ(x)

∣∣∣∣jgs(x)

∫[L+Vµ ψγ

δvµ(x)

Γ[j] = −W [V ] +

δΓ[j]δjµ(x)

∣∣∣∣jgs(x)

What About the Scalar Density?

Can add additional sources and Legendre transformations

In nonrelativistic DFT, add to Lagrangian + η(x) ∇ψ†∇ψ

Γ[ρ, τ ] = W [J, η]−∫

J(x)ρ(x)−∫η(x)τ(x)

=⇒ Skyrme HF energy functional E [ρ, τ, J] of densityand kinetic energy density

In covariant DFT, add to Lagrangian + S(x)ψψ

Γ[jµ, ρs] = W [Vµ,S]−∫

V (x) · j(x)−∫

S(x)ρs(x)

=⇒ RMF energy functional E [ρv , ρs] [with jµ = (ρv ,0)]

What About the Scalar Density?

Can add additional sources and Legendre transformations

In nonrelativistic DFT, add to Lagrangian + η(x) ∇ψ†∇ψ

Γ[ρ, τ ] = W [J, η]−∫

=⇒ Skyrme HF energy functional E [ρ, τ, J] of densityand kinetic energy density

In covariant DFT, add to Lagrangian + S(x)ψψ

Γ[jµ, ρs] = W [Vµ,S]−∫

V (x) · j(x)−∫

S(x)ρs(x)

=⇒ RMF energy functional E [ρv , ρs] [with jµ = (ρv ,0)]

Possible Effective Actions

Couple source to local Lagrangian field, e.g., J(x)ϕ(x)

Γ[φ] where φ(x) = 〈ϕ(x)〉 =⇒ 1PI effective actionArises from fermion L’s by introducing auxiliary fields (mesons!)Kohn-Sham via special saddlepoint evaluation

Couple source to non-local composite op, e.g., J(x , x ′)ϕ(x)ϕ(x ′)

Γ[G, φ] =⇒ 2PI effective action [CJT]

Source coupled to local composite operator, e.g., J(x)ϕ2(x)

1.5PI effective action? Almost:Kohn-Sham from inversion method (point coupling!)Problem from new divergences =⇒ polynomial J(x) counterterms

“Sentenced to death” by Banks and Rabyenergy interpretation? variational?reprieve?

Kohn-Luttinger-Ward Theorem (1960)

T → 0 diagrammatic expansion of Ω(µ,V ,T ) in external v(x)=⇒ same as F (N,V ,T ≡ 0) with µ0 and no anomalous diagrams

Ω(µ, V, T ) = Ω0(µ) + + + + · · ·

with G0(µ, T )

T→0−→ F (N, V, T = 0) = E0(N) + + + · · ·

with G0(µ0)

Uniform Fermi system with no external potential (degeneracy ν):

µ0(N) = (6π2N/νV )2/3 ≡ k2F/2M ≡ ε0

If symmetry of non-interacting and interacting systems agree

Ω(µ, V, T ) = Ω0(µ) + + + + · · ·

with G0(µ, T )

T→0−→ F (N, V, T = 0) = E0(N) + + + · · ·

with G0(µ0)

µ0(N) = (6π2N/νV )2/3 ≡ k2F/2M ≡ ε0

Ω(µ, V, T ) = Ω0(µ) + + + + · · ·

with G0(µ, T )

T→0−→ F (N, V, T = 0) = E0(N) + + + · · ·

with G0(µ0)

µ0(N) = (6π2N/νV )2/3 ≡ k2F/2M ≡ ε0

Kohn-Luttinger Inversion Method [F & W, sec. 30]

Find F (N) = Ω(µ) + µN with µ(N) from N(µ) = −(∂Ω/∂µ)TV

expand about non-interacting system (subscripts label expansion):

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

invert N = −(∂Ω(µ)/∂µ)TV order-by-order in expansionN appears in 0th order only: N = −[∂Ω0/∂µ)]µ=µ0 =⇒ µ0(N)first order has two terms, which lets us solve for µ1:

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Same pattern to all orders: µi is determined by functions of µ0

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

invert N = −(∂Ω(µ)/∂µ)TV order-by-order in expansion

N appears in 0th order only: N = −[∂Ω0/∂µ)]µ=µ0 =⇒ µ0(N)first order has two terms, which lets us solve for µ1:

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

invert N = −(∂Ω(µ)/∂µ)TV order-by-order in expansionN appears in 0th order only: N = −[∂Ω0/∂µ)]µ=µ0 =⇒ µ0(N)

first order has two terms, which lets us solve for µ1:

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Ω(µ) = Ω0(µ) + Ω1(µ) + Ω2(µ) + · · ·µ = µ0 + µ1 + µ2 + · · ·

F (N) = F0(N) + F1(N) + F2(N) + · · ·

0 = [∂Ω1/∂µ]µ=µ0 + µ1[∂2Ω0/∂µ

2]µ=µ0 =⇒ µ1 = − [∂Ω1/∂µ]µ=µ0

[∂2Ω0/∂µ2]µ=µ0

Apply this inversion to F = Ω + µN:

F (N) = Ω0(µ0) + µ0N︸︷︷︸F0

+ Ω1(µ0) + µ1N + µ1

[∂Ω0

]µ=µ0︸︷︷︸

+ Ω2(µ0) + µ2N + µ2

[∂Ω0

]µ=µ0

[∂Ω1

]µ=µ0

+12µ2

[∂2Ω0

∂µ2

]µ=µ0︸︷︷︸

+ · · ·

µi always cancels from Fi for i ≥ 1:

F (N) = F0(N) + Ω1(µ0)︸︷︷︸F1

+Ω2(µ0)−12

[∂Ω1/∂µ]2µ=µ0

[∂2Ω0/∂µ2]µ=µ0︸︷︷︸F2

+ · · ·

F (N) = Ω0(µ0) + µ0N︸︷︷︸F0

+ Ω1(µ0) + µ1N + µ1

[∂Ω0

]µ=µ0︸︷︷︸

+ Ω2(µ0) + µ2N + µ2

[∂Ω0

]µ=µ0

[∂Ω1

]µ=µ0

+12µ2

[∂2Ω0

∂µ2

]µ=µ0︸︷︷︸

+ · · ·

F (N) = F0(N) + Ω1(µ0)︸︷︷︸F1

+Ω2(µ0)−12

[∂Ω1/∂µ]2µ=µ0

[∂2Ω0/∂µ2]µ=µ0︸︷︷︸F2

+ · · ·

F (N) = Ω0(µ0) + µ0N︸︷︷︸F0

+ Ω1(µ0) + µ1N + µ1

[∂Ω0

]µ=µ0︸︷︷︸

+ Ω2(µ0) + µ2N + µ2

[∂Ω0

]µ=µ0

[∂Ω1

]µ=µ0

+12µ2

[∂2Ω0

∂µ2

]µ=µ0︸︷︷︸

+ · · ·

F (N) = F0(N) + Ω1(µ0)︸︷︷︸F1

+Ω2(µ0)−12

[∂Ω1/∂µ]2µ=µ0

[∂2Ω0/∂µ2]µ=µ0︸︷︷︸F2

+ · · ·

F (N) = Ω0(µ0) + µ0N︸︷︷︸F0

+ Ω1(µ0) + µ1N + µ1

[∂Ω0

]µ=µ0︸︷︷︸

+ Ω2(µ0) + µ2N + µ2

[∂Ω0

]µ=µ0

[∂Ω1

]µ=µ0

+12µ2

[∂2Ω0

∂µ2

]µ=µ0︸︷︷︸

+ · · ·

F (N) = F0(N) + Ω1(µ0)︸︷︷︸F1

+Ω2(µ0)−12

[∂Ω1/∂µ]2µ=µ0

[∂2Ω0/∂µ2]µ=µ0︸︷︷︸F2

+ · · ·

F (N) = Ω0(µ0) + µ0N︸︷︷︸F0

+ Ω1(µ0) + µ1N + µ1

[∂Ω0

]µ=µ0︸︷︷︸

+ Ω2(µ0) + µ2N + µ2

[∂Ω0

]µ=µ0

[∂Ω1

]µ=µ0

+12µ2

[∂2Ω0

∂µ2

]µ=µ0︸︷︷︸

+ · · ·

F (N) = F0(N) + Ω1(µ0)︸︷︷︸F1

+Ω2(µ0)−12

[∂Ω1/∂µ]2µ=µ0

[∂2Ω0/∂µ2]µ=µ0︸︷︷︸F2

+ · · ·

Generalizing the KLW Inversion Approach

Zeroth order is non-interacting system =⇒ easy to solveCommon feature of all generalizations =⇒ Kohn-Sham system!Here it has chemical potential µ0 and external potential v(x)

=⇒ fill levels up to µ0, which is known by counting up to N

But we still have a hard problem in finite systems

finding density ρ(x) in non-uniform system is complicated=⇒ it is not the density of the non-interacting system

for a self-bound system (nucleus!), there is no [net] v(x)

Introduce space-time dependent sourcesZeroth order system is always easy =⇒ single-particle orbitalsEnergy functional is very complicated before approximationNew feature: source −→ 0 in ground state (unlike µ)

But we still have a hard problem in finite systemsfinding density ρ(x) in non-uniform system is complicated

=⇒ it is not the density of the non-interacting systemfor a self-bound system (nucleus!), there is no [net] v(x)

But we still have a hard problem in finite systemsfinding density ρ(x) in non-uniform system is complicated

=⇒ it is not the density of the non-interacting systemfor a self-bound system (nucleus!), there is no [net] v(x)

Three generalizations: Kohn-Sham DFT, other sources, pairing1. µN + J(x)ρ(x) with J(x) = δF [ρ]/δρ(x) → 0 in gs

2. Add a source coupled to the kinetic energy density

+ η(x)τ(x) where τ(x) ≡ 〈∇ψ† ·∇ψ〉

=⇒ M∗(x) in the Kohn-Sham equation (cf. Skyrme)

[−∇2

2M+ vKS(x)

]ψα = εαψα =⇒

[−∇ 1

M∗(x)∇ + vKS(x)

]ψα = εαψα

3. Add a source coupled to the divergent pair density=⇒ e.g., j〈ψ†

↑ψ†↓ + ψ↓ψ↑〉 =⇒ set j to zero in gs

Same inversion method, but use [J]gs = J0 + J1 + J2 + · · · = 0=⇒ solve for J0 iteratively: [J0]old =⇒ [J0]new = −J1 − J2 + · · ·

Three generalizations: Kohn-Sham DFT, other sources, pairing1. µN + J(x)ρ(x) with J(x) = δF [ρ]/δρ(x) → 0 in gs2. Add a source coupled to the kinetic energy density

+ η(x)τ(x) where τ(x) ≡ 〈∇ψ† ·∇ψ〉

[−∇2

2M+ vKS(x)

[−∇ 1

M∗(x)∇ + vKS(x)

]ψα = εαψα

+ η(x)τ(x) where τ(x) ≡ 〈∇ψ† ·∇ψ〉

[−∇2

2M+ vKS(x)

[−∇ 1

M∗(x)∇ + vKS(x)

]ψα = εαψα

+ η(x)τ(x) where τ(x) ≡ 〈∇ψ† ·∇ψ〉

[−∇2

2M+ vKS(x)

[−∇ 1

M∗(x)∇ + vKS(x)

]ψα = εαψα

Three generalizations: Kohn-Sham DFT, other sources, pairing1. µN + V µ(x)jµ(x) with V µ(x) = δF [j]/δµ(x) → 0 in gs2. Add a source coupled to the scalar density

+ S(x)ρs(x) where S(x) ≡ 〈ψψ〉

=⇒ M∗(x) = M − S0(x) in the Kohn-Sham Dirac equation[−iα·∇+βM+W0(x)

[−iα·∇+βM∗(x)+W0(x)

]ψα = εαψα

3. Add a source coupled to the divergent pair density=⇒ e.g., j〈ψTηψ〉 =⇒ set j to zero in gs

Same inversion method, but use [S]gs = S0 + S1 + S2 + · · · = 0=⇒ solve for S0 iteratively: [S0]old =⇒ [S0]new = −S1−S2 + · · ·

Kohn-Sham Via Inversion Method

Inversion method for effective action DFT [Fukuda et al.]order-by-order matching in λ (e.g., EFT expansion parameters)

J[ρ, λ] = J0[ρ] + λJ1[ρ] + λ2J2[ρ] + · · ·W [J, λ] = W0[J] + λW1[J] + λ2W2[J] + · · ·Γ[ρ, λ] = Γ0[ρ] + λΓ1[ρ] + λ2Γ2[ρ] + · · ·

zeroth order is a noninteracting system with potential J0(x)

Γ0[ρ] = W0[J0]−∫

d4x J0(x)ρ(x) =⇒ ρ(x) =δW [J0]

δJ0(x)

=⇒ this is the Kohn-Sham system with the exact density!

Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’sFind J0 for the ground state by completing self-consistency loop:

J0 → W1 → Γ1 → J1 → W2 → Γ2 → · · · =⇒ J0(x) =∑

δΓi [ρ]

δρ(x)

Γ0[ρ] = W0[J0]−∫

d4x J0(x)ρ(x) =⇒ ρ(x) =δW [J0]

δJ0(x)

J0 → W1 → Γ1 → J1 → W2 → Γ2 → · · · =⇒ J0(x) =∑

δΓi [ρ]

δρ(x)

Γ0[ρ] = W0[J0]−∫

d4x J0(x)ρ(x) =⇒ ρ(x) =δW [J0]

δJ0(x)

Diagonalize W0[J0] by introducing KS orbitals =⇒ sum of εi ’s

Find J0 for the ground state by completing self-consistency loop:

J0 → W1 → Γ1 → J1 → W2 → Γ2 → · · · =⇒ J0(x) =∑

δΓi [ρ]

δρ(x)

Γ0[ρ] = W0[J0]−∫

d4x J0(x)ρ(x) =⇒ ρ(x) =δW [J0]

δJ0(x)

J0 → W1 → Γ1 → J1 → W2 → Γ2 → · · · =⇒ J0(x) =∑

δΓi [ρ]

δρ(x)

Why Use EFT for Energy Functionals

Similar to conventional “phenomenological” approachesbut with a rigorous foundation (DFT from effective action)extendable and can be connected to chiral EFT

or vlow k for NN and few-body

Eliminating model dependences (cf. “minimal” models!)framework for building a “complete” functionalmore efficient renormalization

New insight into analytic structure of functionale.g., logs in low-density expansion in kFas from RG

Power counting: what to sum at each order in an expansionnaturalness =⇒ estimates of truncation errorsevidence from Skyrme and RMF functionals for hierarchyfor covariant EFT, requires special renormalization

Outline DFT Action Dilute Renormalization Summary DFT/EFT Results Tau M∗ Pairing

“Simple” Many-Body Problem: Hard Spheres

Infinite potential at radius R

sin(kr+δ)

Scattering length a0 = R

Dilute nR3 1 =⇒ kFa0 1

What is the energy / particle?

In Search of a Perturbative Expansion

For free-space scattering at momentum k 1/R, we shouldrecover a perturbative expansion in kR for scattering amplitude:

f0(k) ∝ 1k cot δ(k)− ik

−→ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a3

with a0 = R and r0 = 2R/3 for hard-core spheres

Perturbation theory in the hard-core potential won’t work:

=⇒ 〈k|V |k′〉 ∝∫

dx eik·x V (x) e−ik′·x −→∞

Standard solution: Solve nonperturbatively, then expand

EFT approach: k 1/R means we probe at low resolution=⇒ replace potential with a simpler but general interaction

−→ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a3

=⇒ 〈k|V |k′〉 ∝∫

−→ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a3

=⇒ 〈k|V |k′〉 ∝∫

−→ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a3

=⇒ 〈k|V |k′〉 ∝∫

Nonrelativistic EFT for “Natural” Dilute Fermions

A simple, general interaction is a sum of delta functions andderivatives of delta functions. In momentum space (cf. Skyrme),

〈k|Veft|k′〉 = C0 +12

C2(k2 + k′2) + C′2k · k′ + · · ·

Or, Left has most general local (contact) interactions:

Left = ψ†[i∂

−→∇ 2

]ψ − C0

2(ψ†ψ)2 +

[(ψψ)†(ψ

↔∇2ψ) + h.c.

↔∇ψ)† · (ψ

↔∇ψ)− D0

6(ψ†ψ)3 + . . .

Dimensional analysis =⇒ C2i ∼ 4πM R2i+1 , D2i ∼ 4π

M R2i+4

Nonrelativistic EFT for “Natural” Dilute Fermions

A simple, general interaction is a sum of delta functions andderivatives of delta functions. In momentum space (cf. Skyrme),

〈k|Veft|k′〉 = C0 +12

C2(k2 + k′2) + C′2k · k′ + · · ·

Or, Left has most general local (contact) interactions:

Left = ψ†[i∂

−→∇ 2

]ψ − C0

2(ψ†ψ)2 +

[(ψψ)†(ψ

↔∇2ψ) + h.c.

↔∇ψ)† · (ψ

↔∇ψ)− D0

6(ψ†ψ)3 + . . .

Dimensional analysis =⇒ C2i ∼ 4πM R2i+1 , D2i ∼ 4π

M R2i+4

Renormalization

Reproduce: f0(k) ∝ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a3

Consider the leading potential V (0)EFT(x) = C0δ(x) or

〈k|V (0)eft |k

′〉 =⇒ =⇒ C0

Choosing C0 ∝ a0 gets the first term. Now 〈k|VG0V |k′〉:

Renormalization

0 − a20r0/2)k2 +O(k3a3

〈k|V (0)eft |k

′〉 =⇒ =⇒ C0

=⇒∫

d3q(2π)3

1k2 − q2 + iε

−→∞!

=⇒ Linear divergence!

Renormalization

0 − a20r0/2)k2 +O(k3a3

〈k|V (0)eft |k

′〉 =⇒ =⇒ C0

=⇒∫ Λc d3q

(2π)3

1k2 − q2 + iε

−→ Λc

2π2 −ik4π

+O(k2/Λc)

=⇒ If cutoff at Λc , then can absorb into V (0), but all powers of k2

Renormalization

0 − a20r0/2)k2 +O(k3a3

〈k|V (0)eft |k

′〉 =⇒ =⇒ C0

=⇒∫

dDq(2π)3

1k2 − q2 + iε

D→3−→ − ik4π

Dimensional regularization with minimal subtraction=⇒ only one power of k !

Dim. reg. + minimal subtraction =⇒ simple power counting:

P/2− k

P/2 + k

P/2− k′

P/2 + k′

iT (k, cos θ) − iC0 − M

4π(C0)

+ + + + O(k3)

(C0)3k2 − iC2k

2 − iC ′2k2 cos θ

Matching: C0 = 4πM a0 = 4π

M R , C2 = 4πM

a20r02 = 4π

3 , · · ·

Recovers effective range expansion order-by-order withperturbative diagrammatic expansion

one power of k per diagramestimate truncation error from dimensional analysis

Dim. reg. + minimal subtraction =⇒ simple power counting:

P/2− k

P/2 + k

P/2− k′

P/2 + k′

iT (k, cos θ) − iC0 − M

4π(C0)

+ + + + O(k3)

(C0)3k2 − iC2k

2 − iC ′2k2 cos θ

Matching: C0 = 4πM a0 = 4π

M R , C2 = 4πM

a20r02 = 4π

3 , · · ·

Recovers effective range expansion order-by-order withperturbative diagrammatic expansion

one power of k per diagramestimate truncation error from dimensional analysis

Now Sum Over Fermions in the Fermi Sea

Leading order V (0)EFT(x) = C0δ(x)

=⇒ ∝ a0k6F

At the next order, we get a linear divergence again:

=⇒ ∝∫ ∞

d3q(2π)3

1k2 − q2

Same renormalization fixes it! Particles −→ holes∫ ∞

1k2 − q2 =

∫ ∞

1k2 − q2−

∫ kF

1k2 − q2

D→3−→ −∫ kF

1k2 − q2 ∝ a2

=⇒ ∝ a0k6F

=⇒ ∝∫ ∞

d3q(2π)3

1k2 − q2

1k2 − q2 =

∫ ∞

1k2 − q2−

∫ kF

1k2 − q2

D→3−→ −∫ kF

1k2 − q2 ∝ a2

=⇒ ∝ a0k6F

=⇒ ∝∫ ∞

d3q(2π)3

1k2 − q2

1k2 − q2 =

∫ ∞

1k2 − q2−

∫ kF

1k2 − q2

D→3−→ −∫ kF

1k2 − q2 ∝ a2

T = 0 Energy Density from Hugenholtz Diagrams

= ρk2

+ (ν − 1)2

3π(kFa0)

+ (ν − 1)4

35π2 (11− 2 ln 2)(kFa0)2

+ (ν − 1)(0.076 + 0.057(ν − 3)

)(kFa0)

+ (ν − 1)1

10π(kFr0)(kFa0)

+ (ν + 1)1

5π(kFap)

3 + · · ·

= ρk2

+ (ν − 1)2

3π(kFa0)

+ (ν − 1)4

35π2 (11− 2 ln 2)(kFa0)2

+ (ν − 1)(0.076 + 0.057(ν − 3)

)(kFa0)

+ (ν − 1)1

10π(kFr0)(kFa0)

+ (ν + 1)1

5π(kFap)

3 + · · ·

= ρk2

+ (ν − 1)2

3π(kFa0)

+ (ν − 1)4

35π2 (11− 2 ln 2)(kFa0)2

+ (ν − 1)(0.076 + 0.057(ν − 3)

)(kFa0)

+ (ν − 1)1

10π(kFr0)(kFa0)

+ (ν + 1)1

5π(kFap)

3 + · · ·

= ρk2

+ (ν − 1)2

3π(kFa0)

+ (ν − 1)4

35π2 (11− 2 ln 2)(kFa0)2

+ (ν − 1)(0.076 + 0.057(ν − 3)

)(kFa0)

+ (ν − 1)1

10π(kFr0)(kFa0)

+ (ν + 1)1

5π(kFap)

3 + · · ·

= ρk2

+ (ν − 1)2

3π(kFa0)

+ (ν − 1)4

35π2 (11− 2 ln 2)(kFa0)2

+ (ν − 1)(0.076 + 0.057(ν − 3)

)(kFa0)

+ (ν − 1)1

10π(kFr0)(kFa0)

+ (ν + 1)1

5π(kFap)

3 + · · ·]

NLO : +

NNLO : +

∫d3x

[K (x) +

(ν − 1)

M[ρ(x)]2

+ d1a2

2M[ρ(x)]7/3

+ d2 a30[ρ(x)]8/3

+ d3 a20 r0[ρ(x)]8/3

+ d4 a3p[ρ(x)]8/3 + · · ·

What can EFT do for DFT?

Effective action as a path integral =⇒ construct W [J] = − ln Z [J],order-by-order in EFT expansion

For dilute system, same diagrams as in DR/MS expansion

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

E.g., J(x) = J0(x) + JLO(x) + JNLO(x) + . . .Two relations involving J0:

ρ(x) =δW0[J0]

δJ0(x)and J0(x)|ρ=ρgs

=δΓinteracting[ρ]

δρ(x)

∣∣∣∣ρ=ρgs

Interpretation: J0 is the external potential that yields for anoninteracting system the exact density

This is the Kohn-Sham potential!Two conditions on J0 =⇒ Self-consistency

ρ(x) =δW0[J0]

δρ(x)

∣∣∣∣ρ=ρgs

ρ(x) =δW0[J0]

δρ(x)

∣∣∣∣ρ=ρgs

New Feynman Rules

Conventional diagrammatic expansion of propagator:

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

Non-local, state-dependent Σ∗(x, x′;ω) vs. local J0(x):

J0(x) = − − + (perms.) + · · ·

= − + (perms.) + · · ·

New Feynman rules =⇒ “inverse density-density correlator”

New Feynman Rules

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

J0(x) = − − + (perms.) + · · ·

= − + (perms.) + · · ·

New Feynman Rules

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

J0(x) = − − + (perms.) + · · ·

= − + (perms.) + · · ·

New Feynman Rules

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

J0(x) = − − + (perms.) + · · ·

= − + (perms.) + · · ·

Kohn-Sham J0 According to the EFT ExpansionSimplifying with the local density approximation (LDA)

J0(x) =

− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

J0(x) =

[− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

J0(x) =

[− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

J0(x) =

[− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

J0(x) =

[− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

J0(x) =

[− (ν − 1)

Mρ(x)

− c1a2

2M[ρ(x)]4/3

− c2 a30[ρ(x)]5/3

− c3 a20 r0[ρ(x)]5/3

− c4 a3p[ρ(x)]5/3 + · · ·

Dilute Fermi Gas in a Harmonic Trap

(Generic)Iteration procedure:

1. Guess an initial density profile ρ(r) (e.g., Thomas-Fermi)

2. Evaluate local single-particle potential vKS(r) ≡ v(r)− J0(r)

3. Solve for lowest N states (including degeneracies): ψα, εα

[−∇

2M+ vKS(r)

]ψα(x) = εαψα(x)

4. Compute a new density ρ(r) =∑N

α=1 |ψα(x)|2other observables are functionals of ψα, εα

5. Repeat 2.–4. until changes are small (“self-consistent”)

Looks like a Skyrme Hartree-Fock calculation! [Except for M∗(r)]

[−∇

2M+ vKS(r)

[−∇

2M+ vKS(r)

[−∇

2M+ vKS(r)

[−∇

2M+ vKS(r)

[−∇

2M+ vKS(r)

[−∇

2M+ vKS(r)

Check Out An Example

0 1 2 3 4 5r/b

C0 = 0 (exact)

Dilute Fermi Gas in Harmonic TrapNF=7, A=240, g=2, as=-0.160

E/A <kFas> <r2>1/2

6.750 -0.524 2.598

0 1 2 3 4 5r/b

C0 = 0 (exact)Kohn-Sham LO

E/A <kFas> <r2>1/2

6.750 -0.524 2.598 5.982 -0.578 2.351

0 1 2 3 4 5r/b

C0 = 0 (exact)Kohn-Sham LOKohn-Sham NLO (LDA)

E/A <kFas> <r2>1/2

6.750 -0.524 2.598 5.982 -0.578 2.351 6.254 -0.550 2.472

0 1 2 3 4 5r/b

C0 = 0 (exact)Kohn-Sham LOKohn-Sham NLO (LDA)Kohn-Sham NNLO (LDA)

E/A <kFas> <r2>1/2

6.750 -0.524 2.598 5.982 -0.578 2.351 6.254 -0.550 2.472 6.227 -0.553 2.459

Power Counting Terms in Energy Functionals

Scale contributions according to average density or 〈kF〉

LO NLO NNLO0.01

ν=4, as=-0.1, A=140ν=4, as=+0.1, A=140ν=2, as=+.16, A=330

Reasonable estimates =⇒ truncation errors understood

LO NLO NNLO0.01

ν=4, as=-0.1, A=140ν=4, as=+0.1, A=140ν=2, as=+.16, A=330

LO NLO NNLO0.01

ν=4, as=-0.1, A=140ν=4, as=+0.1, A=140ν=2, as=+.16, A=330

LO NLO NNLO0.01

ν=4, as=-0.1, A=140ν=4, as=+0.1, A=140ν=2, as=+.16, A=330

LO NLO NNLO0.01

ν=4, as=-0.1, A=140ν=4, as=+0.1, A=140ν=2, as=+.16, A=330

Beyond Kohn-Sham LDA: Kinetic Energy Density

Coulomb meta-GGA DFT uses E [ρ, τ(ρ)], with τ ≡ 〈∇ψ† ·∇ψ〉But τ is expanded in terms of ρ

τ(x) =35

(3π2)2/3 ρ5/3 +1

36(∇ρ)2

ρ+ · · ·

=⇒ same Kohn-Sham equation

J0(x) =δEint[ρ]

δρ(x)=⇒

[−∇2

2M+ J0(x)

]ψα = εαψα

In Skyrme HF, ρ and τ are treated independently in E [ρ, τ, J]

E [ρ, τ, J] =

∫d3x

2Mτ +

t0ρ2 +1

16t3ρ2+α +

(3t1 + 5t2)ρτ

64(9t1 − 5t2)(∇ρ)2 − 3

4W0ρ∇ · J +

(t1 − t2)J2

Beyond Kohn-Sham LDA: Kinetic Energy Density

Coulomb meta-GGA DFT uses E [ρ, τ(ρ)], with τ ≡ 〈∇ψ† ·∇ψ〉But τ is expanded in terms of ρ

τ(x) =35

(3π2)2/3 ρ5/3 +1

36(∇ρ)2

ρ+ · · ·

=⇒ same Kohn-Sham equation

J0(x) =δEint[ρ]

δρ(x)=⇒

[−∇2

2M+ J0(x)

]ψα = εαψα

In Skyrme HF, ρ and τ are treated independently in E [ρ, τ, J]

E [ρ, τ, J] =

∫d3x

2Mτ +

t0ρ2 +1

16t3ρ2+α +

(3t1 + 5t2)ρτ

64(9t1 − 5t2)(∇ρ)2 − 3

4W0ρ∇ · J +

(t1 − t2)J2

To do this in DFT/EFT, add to Lagrangian + η(x) ∇ψ†∇ψ

Γ[ρ, τ ] = W [J, η]−∫

Two Kohn-Sham potentials:

J0(x) =δEint[ρ, τ ]

δρ(x)and η0(x) =

δEint[ρ, τ ]

δτ(x)

Quadratic part of Lagrangian in W0 diagonalized:∫d4x ψ†

[i∂t +

2M− v(x) + J0(x)−∇ · η0(x)∇

Kohn-Sham equation =⇒ defines 1/2M∗(x) ≡ 1/2M − η0(x)

To do this in DFT/EFT, add to Lagrangian + η(x) ∇ψ†∇ψ

Γ[ρ, τ ] = W [J, η]−∫

Two Kohn-Sham potentials:

J0(x) =δEint[ρ, τ ]

δρ(x)and η0(x) =

δEint[ρ, τ ]

δτ(x)

Quadratic part of Lagrangian in W0 diagonalized:∫d4x ψ†

[i∂t +

2M− v(x) + J0(x)−∇ · η0(x)∇

Kohn-Sham equation =⇒ defines 1/2M∗(x) ≡ 1/2M − η0(x)

First Step: Hartree-Fock Diagrams Only

Consider bowtie diagrams from vertices with derivatives:

Left = . . .+C2

[(ψψ)†(ψ

↔∇2ψ) + h.c.

↔∇ψ)† · (ψ

↔∇ψ) + . . .

Energy density in Kohn-Sham LDA (ν = 2):

Eint[ρ] = . . .+C2

ρ8/3]

+3C′

ρ8/3]

+ . . .

Energy density in Kohn-Sham with τ (ν = 2):

Eint[ρ, τ ] = . . .+C2

[ρτ +

(∇ρ)2]+3C′

[ρτ − 1

4(∇ρ)2]+ . . .

Left = . . .+C2

[(ψψ)†(ψ

↔∇2ψ) + h.c.

↔∇ψ)† · (ψ

↔∇ψ) + . . .

Eint[ρ] = . . .+C2

ρ8/3]

+3C′

ρ8/3]

+ . . .

Eint[ρ, τ ] = . . .+C2

[ρτ +

(∇ρ)2]+3C′

[ρτ − 1

4(∇ρ)2]+ . . .

Left = . . .+C2

[(ψψ)†(ψ

↔∇2ψ) + h.c.

↔∇ψ)† · (ψ

↔∇ψ) + . . .

Eint[ρ] = . . .+C2

ρ8/3]

+3C′

ρ8/3]

+ . . .

Eint[ρ, τ ] = . . .+C2

[ρτ +

(∇ρ)2]+3C′

[ρτ − 1

4(∇ρ)2]+ . . .

Power Counting Estimates Work for Gradients!

LO NLO LDA ρτ 10*∇ρ

ap = as

ν =2, as = 0.16, A = 240

τ -NNLO

LO NLO LDA ρτ 10*(∇ρ)0.001

ap = as

ν =4, as = 0.10, A = 140

τ -NNLO

Kohn-Sham LDA ρ vs. ρτ [Anirban Bhattacharyya]

0 1 2 3 4 5r/b

ρ-DFT, ap = as

ρτ-DFT, ap = as

ρ-DFT, ap = 2as

ρτ-DFT, ap = 2 as

ν=2, NF=7, A=240as=0.16, rs=2as/3

Kohn-Sham LDA ρ vs. ρτ : Differences

ap = as E/A√〈r2〉

ρ 7.66 2.87ρτ 7.65 2.87

ap = 2as E/A√〈r2〉

ρ 8.33 3.10ρτ 8.30 3.09

0 1 2 3 4 5r/b

∆ρ(r

ap = as

ap = 2asν=2, NF=7, A=240

as=0.16, rs=2as/3

Effective Mass and the Single-Particle Spectrum

0 1 2 3 4 5r/b

M* / M

ap = as

ap = 2as

ν=2, NF=7, A=240

as=0.16, rs=2as/3

Effective mass M∗ related to single-particle levels

Effective Mass and the Single-Particle Spectrum

ρ τ ρ

ap = as

M*(0)/M = 0.93

ρ τ ρ

M*(0)/M = 0.73ap = 2as

ρ τ ρ ρ τ ρ

l = 0l = 0

Uniform system: ερk − ερτk = π

ν [(ν − 1)a2srs + 2(ν + 1)a3

F−k2

Analogous Construction for Covariant Case

Study covariant vs. nonrel. DFT/EFT in controlled expansionConfine system in a Woods-Saxon “trap” with Vext and Sext

Choose Vext,Sext so no spin-orbit from external potentials

First: Consider LO covariant effective Lagrangian

Left = ψ[i∂µγµ −M]ψ − Cs

2(ψψ)(ψψ)− Cv

2(ψγµψ)(ψγµψ)

Compare observables from two DFT functionalsSources coupled to vector Vµψγµψ and scalar Sψψ densities

=⇒ two Kohn-Sham potentials, scalar and vector

Vµ0 (x) =

δEint[jµ, ρs]

δjµ(x)and S0(x) =

δEint[jµ, ρs]

δρs(x)

Source coupled to vector Vµψγµψ densities only=⇒ one (vector) Kohn-Sham potential (plus external fields)

Vµ0 (x) =

δEint[jµ]δjµ(x)

Analogous Construction for Covariant Case

Study covariant vs. nonrel. DFT/EFT in controlled expansionConfine system in a Woods-Saxon “trap” with Vext and Sext

Choose Vext,Sext so no spin-orbit from external potentials

First: Consider LO covariant effective Lagrangian

Left = ψ[i∂µγµ −M]ψ − Cs

2(ψψ)(ψψ)− Cv

2(ψγµψ)(ψγµψ)

Compare observables from two DFT functionalsSources coupled to vector Vµψγµψ and scalar Sψψ densities

=⇒ two Kohn-Sham potentials, scalar and vector

Vµ0 (x) =

δEint[jµ, ρs]

δjµ(x)and S0(x) =

δEint[jµ, ρs]

δρs(x)

Source coupled to vector Vµψγµψ densities only=⇒ one (vector) Kohn-Sham potential (plus external fields)

Vµ0 (x) =

δEint[jµ]δjµ(x)

Consider bowtie diagrams with scalar and four-vector vertices:

Case ρv , ρs: Quadratic part of Lagrangian in W0 diagonalized∫d4x ψ

[i∂µγµ −M + S0(x)− γ0V 0

0 (x)]ψ

Kohn-Sham equation =⇒ defines M∗(x) ≡ M − S0(x)

Case ρv : Quadratic part of Lagrangian in W0 diagonalized∫d4x ψ

[i∂µγµ −M − γ0V 0

0 (x)]ψ

Scalar diagram evaluated in LDA

How do observables compare?Binding energies, densitiesWhat about the single-particle spectra?

[i∂µγµ −M + S0(x)− γ0V 0

0 (x)]ψ

[i∂µγµ −M + S0(x)− γ0V 0

0 (x)]ψ

Kohn-Sham LDA ρv vs. ρv , ρs

0 1 2 3 4 5r (fm)

0.25ρ P (f

)ρv/ρs, A = 16ρv, A = 16ρv/ρs, A = 40ρv, A=40

Kohn-Sham LDA ρv vs. ρv , ρs

0 1 2 3 4

q (fm-1)

ρv/ρs, A = 16ρv, A = 16ρv/ρs, A = 40ρv, A = 40

Kohn-Sham LDA ρv vs. ρv , ρs: Differences

A = 16 BE/A√〈r2〉

ρv 27.5 1.97ρv , ρs 27.1 1.99

A = 40 BE/A√〈r2〉

ρv 29.4 2.57ρv , ρs 27.8 2.56

0 1 2 3 4 5r (fm)

∆ρP (f

Energy Spectra ρv , ρs vs. ρv

A = 16

ρv,ρsρv -70

A = 40

ρv,ρsρv

How is the Full G Related to Gks?

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

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

Z [J, ξ] = eiW [J,ξ] =

∫DψDψ† ei

∫d4x [L+ J(x)ψ†(x)ψ(x) +

∫d4x′ ξ(x′,x)ψ†(x′)ψ(x)]

With Γ[ρ, ξ] = Γ0[ρ, ξ] + Γint[ρ, ξ],

G(x , x ′) =δWδξ

∣∣∣∣J

∣∣∣∣ρ

= Gks(x , x ′) + Gks

[ δΓint

δGks− δΓint

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

Z [J, ξ] = eiW [J,ξ] =

∫DψDψ† ei

∫d4x [L+ J(x)ψ†(x)ψ(x) +

∫d4x′ ξ(x′,x)ψ†(x′)ψ(x)]

∣∣∣∣J

∣∣∣∣ρ

[ δΓint

δGks− δΓint

+ + + + · · · =⇒ = +x′

x=⇒ Σ∗(x,x′;ω)

Z [J, ξ] = eiW [J,ξ] =

∫DψDψ† ei

∫d4x [L+ J(x)ψ†(x)ψ(x) +

∫d4x′ ξ(x′,x)ψ†(x′)ψ(x)]

∣∣∣∣J

∣∣∣∣ρ

[ δΓint

δGks− δΓint

How Do G and Gks Yield the Same Density?

Claim: ρks(x) = −iνG0KS(x , x+) equals ρ(x) = −iνG(x , x+)

Start with

Simple diagrammatic demonstration:

Densities agree by construction!

But other observables may differ; spectral functions?

Start with

Pairing in DFT/EFT from Effective Action

Natural framework for spontaneous symmetry breakinge.g., test for zero-field magnetization M in a spin systemintroduce an external field H to break rotational symmetryLegendre transform Helmholtz free energy F (H):

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

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

Pairing in DFT/EFT from Effective Action

With pairing, the broken symmetry is a U(1) [phase] symmetrystandard effective action treatment in condensed matter uses

contact interaction, auxiliary pairing field ∆(x),and 1PI Γ[∆]

to leading order in the loop expansion (mean field)=⇒ BCS weak-coupling gap equation

Here: Combine the EFT expansion and the inversion methodexternal current j coupled to pair density breaks symmetrynatural generalization of Kohn-Sham DFT

Generalizing Effective Action to Include Pairing

Generating functional with sources J, j coupled to densities:

Z [J, j] = e−W [J,j] =

∫D(ψ†ψ) e−

∫d4x [L+ J(x)ψ†

αψα + j(x)(ψ†↑ψ

†↓+ψ↓ψ↑)]

Densities found by functional derivatives wrt J, j :

ρ(x) ≡ 〈ψ†(x)ψ(x)〉J,j =δW [J, j]δJ(x)

∣∣∣∣j

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

∣∣∣∣J

Effective action Γ[ρ, φ] by functional Legendre transformation:

Γ[ρ, φ] = W [J, j]−∫

d4x J(x)ρ(x)−∫

d4x j(x)φ(x)

Z [J, j] = e−W [J,j] =

∫D(ψ†ψ) e−

†↓+ψ↓ψ↑)]

∣∣∣∣j

∣∣∣∣J

Γ[ρ, φ] = W [J, j]−∫

d4x J(x)ρ(x)−∫

d4x j(x)φ(x)

Z [J, j] = e−W [J,j] =

∫D(ψ†ψ) e−

†↓+ψ↓ψ↑)]

∣∣∣∣j

∣∣∣∣J

Γ[ρ, φ] = W [J, j]−∫

d4x J(x)ρ(x)−∫

d4x j(x)φ(x)

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]

δρ(x)= J(x) and

δE [ρ, φ]

δφ(x)= j(x)

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

δE [ρ, φ]

δρ(x)

∣∣∣∣ρ=ρgs,φ=φgs

=δE [ρ, φ]

δφ(x)

This is Hohenberg-Kohn DFT extended to pairing!

We need a method to carry out the inversionFor Kohn-Sham DFT, apply inversion methodsWe need to renormalize!

δE [ρ, φ]

δρ(x)= J(x) and

δE [ρ, φ]

δφ(x)= j(x)

δE [ρ, φ]

δρ(x)

=δE [ρ, φ]

δφ(x)

δE [ρ, φ]

δρ(x)= J(x) and

δE [ρ, φ]

δφ(x)= j(x)

δE [ρ, φ]

δρ(x)

=δE [ρ, φ]

δφ(x)

δE [ρ, φ]

δρ(x)= J(x) and

δE [ρ, φ]

δφ(x)= j(x)

δE [ρ, φ]

δρ(x)

=δE [ρ, φ]

δφ(x)

Kohn-Sham Inversion Method Revisited

Order-by-order matching in EFT expansion parameter λ

J[ρ, φ, λ] = J0[ρ, φ] + J1[ρ, φ] + J2[ρ, φ] + · · ·j[ρ, φ, λ] = j0[ρ, φ] + j1[ρ, φ] + j2[ρ, φ] + · · ·

W [J, j , λ] = W0[J, j] + W1[J, j] + W2[J, j] + · · ·Γ[ρ, φ, λ] = Γ0[ρ, φ] + Γ1[ρ, φ] + Γ2[ρ, φ] + · · ·

0th 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(h0(x)− µ0 j0(x)

j0(x) −h0(x) + µ0

)(ui(x)vi(x)

(ui(x)vi(x)

where h0(x) ≡ −∇2

2M+ v(x)− J0(x)

with conventional orthonormality relations for ui , vi

Diagrammatic Expansion of Wi

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

Γint = + + + + · · ·

(〈Tψ↑(x)ψ†↑(x

′)〉0 〈Tψ↑(x)ψ↓(x ′)〉0〈Tψ†↓(x)ψ†↑(x

′)〉0 〈Tψ†↓(x)ψ↓(x ′)〉0

ks iF 0ks

iF 0ks† −iG0

)In frequency space, the Green’s functions are

iG0ks(x, x

′;ω) =∑

[ui(x) u∗i (x′)ω − Ei + iη

+vi(x′) v∗i (x)

ω + Ei − iη

iF 0ks(x, x

′;ω) = −∑

[ui(x) v∗i (x′)ω − Ei + iη

−ui(x′) v∗i (x)

ω + Ei − iη

Kohn-Sham Self-Consistency Procedure

Same iteration procedure as in Skyrme or RMF with pairing

In terms of the orbitals, the fermion density is

ρ(x) = 2∑

|vi(x)|2

and the pair density is (warning: divergent!)

φ(x) =∑

[u∗i (x)vi(x) + ui(x)v∗i (x)]

The chemical potential µ0 is fixed by∫ρ(x) = A

Diagrams for Γ[ρ, φ] = −E [ρ, φ] (with LDA+) yields KS potentials

J0(x)∣∣∣ρ=ρgs

=δΓint[ρ, φ]

δρ(x)

∣∣∣∣∣ρ=ρgs

and j0(x)∣∣∣φ=φgs

=δΓint[ρ, φ]

δφ(x)

∣∣∣∣∣φ=φgs

Divergences: Uniform SystemGenerating functional with constant sources µ and j :

e−W =

∫D(ψ†ψ) e−

∫d4x [ψ†

α( ∂∂τ −

∇ 22M −µ)ψα +

C02 ψ

†↑ψ

†↓ψ↓ψ↑+ j(ψ↑ψ↓+ψ†

↓ψ†↑)]

+ 12 ζ j2

cf. adding integration over auxiliary field∫

D(∆∗,∆) e−1

|C0|∫|∆|2

=⇒ shift variables to eliminate ψ†↑ψ†↓ψ↓ψ↑ for ∆∗ψ↑ψ↓

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

Same linear divergence as in 2-to-2 scattering

Strategy: Add counterterm 12ζ j2 to L

additive to W (cf. |∆|2) =⇒ no effect on scatteringEnergy interpretation? Finite part?

e−W =

∫D(ψ†ψ) e−

∫d4x [ψ†

α( ∂∂τ −

∇ 22M −µ)ψα +

C02 ψ

†↑ψ

†↓ψ↓ψ↑+ j(ψ↑ψ↓+ψ†

↓ψ†↑)]

+ 12 ζ j2

D(∆∗,∆) e−1

|C0|∫|∆|2

e−W =

∫D(ψ†ψ) e−

∫d4x [ψ†

α( ∂∂τ −

∇ 22M −µ)ψα +

C02 ψ

†↑ψ

†↓ψ↓ψ↑+ j(ψ↑ψ↓+ψ†

↓ψ†↑)] + 1

2 ζ j2

D(∆∗,∆) e−1

|C0|∫|∆|2

Kohn-Sham Non-interacting System

Canonical Bogoliubov transformation solves exactly

W0[µ0, j0] =

∫d3k

(2π)3 (ξk − Ek ) +12ζ(0)j20

where ξk ≡ ε0k − µ0 and Ek ≡√ξ2

k + j20

Kohn-Sham potential j0 plays the role of a constant gap

vk2 uk

ρ = 2∑

∫d3k

(2π)3

(1− ξk

)φ = 2

uk vk = −∫

d3k(2π)3

+ ζ(0)j0

Renormalizing the “Gap” Equation

Leading-order (LO) calculation requires Γ1[ρ, φ] =⇒ j1 = δΓ1/δφ ∼ C0Tr F + CTC

Choose LO counterterms (“CTC”) so that Γ1 is a function of ρand the renormalized φ only

“Gap” equation from j = j0 + j1 = 0 =⇒ linear divergence

j0 = −j1 = −12|C0|φ

uniform=

12|C0| j0

∫ d3k(2π)3

1√(ε0

k − µ0)2 + j20

− ζ(0)

Conventional approach: Subtract equation for as to eliminate C0

M4πas

+1|C0|

∫d3k

(2π)3

=⇒ M4πas

= −12

∫d3k

(2π)3

Ek− 1ε0

j0 = −j1 = −12|C0|φ

uniform=

12|C0| j0

∫ d3k(2π)3

1√(ε0

k − µ0)2 + j20

− ζ(0)

M4πas

+1|C0|

∫d3k

(2π)3

=⇒ M4πas

= −12

∫d3k

(2π)3

Ek− 1ε0

j0 = −j1 = −12|C0|φ

uniform=

12|C0| j0

∫ d3k(2π)3

1√(ε0

k − µ0)2 + j20

− ζ(0)

M4πas

+1|C0|

∫d3k

(2π)3

=⇒ M4πas

= −12

∫d3k

(2π)3

Ek− 1ε0

Observables With Kohn-Sham Pairing

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

= (Γ0 + Γ1)|j0=− 12 |C0|φ =

∫d3k

(2π)3

[ξk −Ek +

]+[µ0 −

14|C0|ρ

∫d3k

(2π)3

(1− ξk

)and φ = −

∫d3k

(2π)3

+ ζ(0)j0

Explicitly finite and dependence on ζ(0) cancels out

Recover normal state from j0 → 0:

EV→ 3

5µ0ρ−

14|C0|ρ2 and ρ→ 1

3π2 k3F and µ0 →

Dimensional Regularization (DR)

DR/PDS =⇒ explicit Λ to “check” for cutoff dependencecf. Papenbrock & Bertsch DR/MS calculation =⇒ Λ = 0

C0(Λ) =4πas

1− asΛ=

4πa2s

MΛ +O(Λ2) = C(1)

0 + C(2)0 + · · ·

Basic free-space integral =⇒ beachball renormalization in Γ2:(Λ

)3−D ∫ dDu(2π)D

1t2 − u2 + iε

PDS−→ − 14π

(Λ + it)

=⇒ independent of Λ

Dimensional Regularization and Pairing

The basic DR/PDS integral in D dimensions, with x ≡ j0/µ0, is

I(β) ≡(Λ

)3−D∫

dDk(2π)D

(ε0k )β

2πµβ0

(1− δβ,2

)+ (−)β+1 M3/2

√2π

[µ20(1 + x2)](β+1/2)/2 P0

β+1/2

( −1√1 + x2

Check the density equation =⇒ Λ dependence cancels:

∫d3k

(2π)3

ε0k − µ0

)= 0− I(1) + µ0 I(0)

The gap equation implies ζ(0) is naturally taken from 1/C0(Λ):

1|C0(Λ)|

I(0) or1|C0|

(I(0)− ζ(0)

I(β) ≡(Λ

)3−D∫

dDk(2π)D

(ε0k )β

2πµβ0

(1− δβ,2

)+ (−)β+1 M3/2

√2π

[µ20(1 + x2)](β+1/2)/2 P0

β+1/2

( −1√1 + x2

)Check the density equation =⇒ Λ dependence cancels:

∫d3k

(2π)3

ε0k − µ0

)= 0− I(1) + µ0 I(0)

1|C0(Λ)|

I(0) or1|C0|

(I(0)− ζ(0)

I(β) ≡(Λ

)3−D∫

dDk(2π)D

(ε0k )β

2πµβ0

(1− δβ,2

)+ (−)β+1 M3/2

√2π

[µ20(1 + x2)](β+1/2)/2 P0

β+1/2

( −1√1 + x2

)Check the density equation =⇒ Λ dependence cancels:

∫d3k

(2π)3

ε0k − µ0

)= 0− I(1) + µ0 I(0)

1|C0(Λ)|

I(0) or1|C0|

(I(0)− ζ(0)

)Dick Furnstahl Covariant DFT

Anomalous Density in Finite Systems

How do we renormalize the pair density in a finite system?

φ(x) =∑

[u∗i (x)vi(x) + ui(x)v∗i (x)] −→∞

cf. scalar density ρs =∑

i ψ(x)ψ(x) for solitons or relativistic nuclei

In the uniform limit, φ can be defined with a subtraction

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− 1ε0

kc→∞−→ finite

Apply this in a local density approximation (Thomas-Fermi)

φ(x) = 2Ec∑i

ui(x)vi(x)− j0(x)M kc(x)

2π2 with Ec =k2

2M+ vKS(x)− µ

φ(x) =∑

[u∗i (x)vi(x) + ui(x)v∗i (x)] −→∞

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− 1ε0

φ(x) = 2Ec∑i

2π2 with Ec =k2

2M+ vKS(x)− µ

φ(x) =∑

[u∗i (x)vi(x) + ui(x)v∗i (x)] −→∞

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− 1ε0

φ(x) = 2Ec∑i

2π2 with Ec =k2

2M+ vKS(x)− µ

Bulgac Renormalization [Bulgac/Yu PRL 88 (2002) 042504]

Convergence is very slow as the energy cutoff is increased=⇒ Bulgac/Yu: make a different subtraction

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− Pε0

k − µ0

Compare convergence in uniform system and in nuclei with LDA

0.01 0.1 1 10Energy Cutoff

subtraction 1subtraction 2

µ0 = 1j0 = 0.1

0 2 4 6 8 10 120

r [fm]

Even Better! [Bulgac, PRC 65 (2002) 051305]

Convergence is rapid above Fermi surface but not below=⇒ scale set by Fermi energy rather than gap

Solution: Energy cutoff around µ

0 5 10 15 20 25−50

r [fm]

µ + Ec

µ − Ec

I II III

5 10 15 20 25 30 35 40 45 500

∆ [M

5 10 15 20 25 30 35 40 45 50−600

−500

−400

−300

−200

−100

Ec [MeV]

g eff [M

Higher Order: Induced Interaction

As j0 → 0, ukvk peaks at µ0

Leading order T = 0:∆LO/µ0 = 8

e2 e−1/N(0)|C0|

= 8e2 e−π/2kF|as|

NLO modifies exponent=⇒ changes prefactor

∆NLO ≈ ∆LO/(4e)1/3

vk2 uk

j0ukvk

e2 e−1/N(0)|C0|

∆NLO ≈ ∆LO/(4e)1/3

vk2 uk

e2 e−1/N(0)|C0|

∆NLO ≈ ∆LO/(4e)1/3

vk2 uk

e2 e−1/N(0)|C0|

∆NLO ≈ ∆LO/(4e)1/3

vk2 uk

e2 e−1/N(0)|C0|

∆NLO ≈ ∆LO/(4e)1/3 µ0

vk2 uk

! " $#% '&)(+*(-, (.*/0(-,2143 56 7

Covariant Pairing

Point-coupling version of Capelle and Gross [PRB 59 (1999) 7140]Dirac-Bogoliubov-de Gennes equations

Couple to time-reversed pairs with definite Lorentz nature

Same basic DFT treatment with scalar source coupled to

ψTη0ψ with η0 ≡ γ1γ3

and zero-component of four-vector source coupled to

ψTη0vψ with η0

v ≡ γ0γ1γ3

Outline DFT Action Dilute Renormalization Summary Fermi-to-Dyson Dilute No-Sea

Fermi to Dyson in 1953 [recalled in Nature 427 (2004) 297]

Concerning a proposed pseudoscalar meson theory:

“There are two ways of doing calculations in theoreticalphysics”, he said. “One way, and this is the way I prefer, is tohave a clear physical picture of the process that you arecalculating. The other way is to have a precise andself-consistent mathematical formalism. You have neither.”

Fermi to Dyson in 1953 [recalled in Nature 427 (2004) 297]

I was slightly stunned, but ventured to ask him why he didnot consider the pseudoscalar meson theory to be aself-consistent mathematical formalism. He replied,“Quantum electrodynamics is a good theory because theforces are weak, and when the formalism is ambiguous wehave a clear physical picture to guide us. With thepseudoscalar meson theory there is no physical picture, andthe forces are so strong that nothing converges. To reachyour calculated results, you had to introduce arbitrary cut-offprocedures that are not based either on solid physics or onsolid mathematics.”

UV Divergences in Nonrelativisticand Relativistic Effective Actions

Sensitivity to short-distance physics signalled by divergencesbut finiteness (e.g., with cutoff) doesn’t mean not sensitive!=⇒ must absorb sensitivity via renormalization

Sources of UV divergences

nonrelativistic covariantscattering scattering

pairing pairinganti-nucleons

Power Counting Lost / Power Counting Regained

Gasser, Sainio, and Svarc =⇒ ChPT for πN with relativistic N’sloop and momentum expansions don’t agree

=⇒ systematic power counting lostheavy-baryon EFT restores power counting by 1/M expansion

Hua-Bin Tang (1996) [and with Paul Ellis]:

q > Λ

=⇒ C0 and

µ +M−Mωx x

negative−energy states

x x x xx x x

positive−energy states

x x x x

“. . . EFT’s permit useful low-energy expansions only if we ab-sorb all of the hard-momentum effects into the parameters ofthe Lagrangian.”

q > Λ

=⇒ C0 and

µ +M−Mωx x

x x x xx x x

x x x x

“When we include the nucleons relativistically, the anti-nucleon contributions are also hard-momentum effects.”

q > Λ

=⇒ C0 and

µ +M−Mωx x

x x x xx x x

x x x x

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 and Prezeau]convenient reformulation by Schindler, Gegelia, Schererparticle-particle loop reduces to nonrelativistic DR/MS resultparticle-hole loops in free space vanish!

Consequences for Free-Space Natural Fermions

Leading order (LO) has scalar, vector, etc. vertices

At NLO, only particle-particle loop survives IR

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

At NNLO, only particle-particle loop diagram survives IR

All other diagrams are zero in IR

Effective Action and Subtraction Prescription

Use effective action formalism to carry out EFT at finite densityKohn-Sham DFT using inversion methodW0[S0,V

µ0 ] with Kohn-Sham potentials S0,V

Zeroth order =⇒ non-interacting system=⇒ Tr ln(i 6∂ + µγ0 −M∗ − gv6V0) ≡ Tr ln G−1

KS (µ)Divergences as with pairing =⇒ counterterms in IR

Subtraction prescription: Define CT’s to cancel Tr ln at µ = 0Derivative expansion: −i Tr ln(i 6∂ −M∗ − gv6V0) =∫

d4x [Ueff(S0) + 12 Z1s(S0)∂µS0 ∂

µS0 + · · · ] ≡ − CT’s

=⇒ local polynomial in the fields: CT’s ⇐⇒ iTr ln G−1KS (0)

Shifts “hard” Dirac sea physics into coefficientsuse same coefficients for any Kohn-Sham fields S0(x),Vµ

0 (x)use the ground state problem to subtract at µ = 0

=⇒ fixed consequences for treating linear response (RPA)

d4x [Ueff(S0) + 12 Z1s(S0)∂µS0 ∂

µS0 + · · · ] ≡ − CT’s

d4x [Ueff(S0) + 12 Z1s(S0)∂µS0 ∂

µS0 + · · · ] ≡ − CT’s

“No-Sea Approximation” for the Ground State

Self-consistent equations for S0,Vµ0 from extremizing Γ[jµ, ρs]

these determine static S0(x) and V0(x) for ground state

Γ with gs densities is proportional to the (free) energyG−1

KS (µ) is diagonal in the single-particle basis ψα(x)eiωx0

replace CT’s in Γ with +i Tr ln G−1KS (0) (using S0(x),V0(x))

so W0 = −i Tr ln G−1KS (µ) + i Tr ln G−1

KS (0)

=⇒εα<µ∑α

(µ− εα)−εα<0∑α

(−εα)− [vac. sub.] =

0<εα<µ∑α

(µ− εα)

Similarly, for ρs(x) ∝ δW0/δS0(x) =⇒ Tr GKS(µ)− Tr GKS(0)

εα<µ∑

h−x −

εα<0∑

−x =

0<εα<µ∑

=⇒ we recover the “no-sea approximation” for the ground state

Consider Γ[jµ, ρs] with time-dependent fluctuations=⇒ S0(x) = S0(x) + S(x) V µ

0 (x) = V0(x) δµ0 + V µ(x)Γ is the generator of 1PI Green’s functions

=⇒ linear response from 2nd order terms in S, Vµ

CT’s =⇒ +i Tr ln G−1(0) still holds with S(x),Vµ(x)!

Expand −i Tr ln G−1(µ) + i Tr ln G−1(0) + . . . in S, V µ

ln(G−1KS + S) = ln(G−1

KS )[. . .− 1

2 GKS · S ·GKS · S + . . .]

so the contributions to RPA rings from the Tr ln’s are:

−i Tr ln G−1(µ) =⇒ p h + p − −i Tr ln G−1(0) =⇒ + −

Combining (only well defined together!) . . .

p h + p − − + − = p h − h −

=⇒ Subtraction prescription for RPA: include ph and h− pairs

Covariant EFT and Negative-Energy States

EFT: Absorb hard-momentum Dirac-sea physics into parameters

Good RMF/RPA “no-sea” phenomenology explained by EFTQCD vacuum effects automatically encoded in fit parameters!in principle we need all counterterms;

in practice under-determinedrelies on naturalness for usual truncation

Outline DFT Action Dilute Renormalization Summary Future Answers

On-Going and Future Challenges

Covariant diluteFermi system

Long-range effects

Gradient expansions

Auxiliary fieldKohn-Sham Theory

Restoring brokensymmetries

Long-range effects

Gradient expansions

Controlled laboratory for finite systemCompare to heavy baryon EFTFocus on spin-orbitThree-body forces?

Covariant pairingInduced interaction

Time-dependent Kohn-Sham theoryHigher order in effective action

formalismKohn-Sham part =⇒ RPAPoint-coupling version?

Long-range effects

Gradient expansions

Long-range forces (e.g., pion exchange)

Non-localities from near-on-shellparticle-hole excitations

+ + + + · · ·

Long-range effects

Gradient expansions

Semiclassical expansions usedin Coulomb DFT

Gradient expansion techniques for(one-loop) effective actions

Long-range effects

Gradient expansions

Auxiliary fields correspond tonon-dynamical meson fields

Apply saddlepoint evaluationrequiring density to be unchanged

Faussier and Valiev/Fernando formalism,but no higher-order calculations yet

How to separate ph and pp for pairing?

Revisit large N expansion?

Long-range effects

Gradient expansions

Translational and rotational invariance,particle number

Not addressed in Coulomb DFT

Energy functional for the intrinsic density?[Engel, Furnstahl, Schwenk]

Should we connect to the free NN interaction?In Coulomb DFT, Hartree-Fock gives dominate contribution

=⇒ correlations are small corrections =⇒ gradient expansion

cf. conventional NN interactions =⇒ correlations Hartree-Fock

But if we run a cutoff toward the Fermi surface . . .

F: |P/2 ± k| < kF

Λ: |P/2 ± k| > kF

|k| < Λand

Match at finite density =⇒ perturbative?

F: |P/2 ± k| < kF

Λ: |P/2 ± k| > kF

|k| < Λand

F: |P/2 ± k| < kF

Λ: |P/2 ± k| > kF

|k| < Λand

Summary Comments on Vacuum Physics

Unlike QED DFT, “no sea” for nuclear structure is a misnomerinclude “vacuum physics” in coefficients through renormalization

Renormalization versus RenormalizabilityRenormalization is required to account for short-distance

behavior but can be implicitRenormalizability at the hadronic level corresponds to making

a model for the short-distance behaviornot a good model phenomenologically

Fixing short-distance behavior is not the same thing asthrowing away negative-energy states

For a long time, we looked for unique “relativistic effects”;these were largely misguided efforts

covariant density functional theory

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