Density Functional Theory: Methods and Problems · 2004-10-13 · Density Functional Theory (DFT)...

Outline DFT Challenges Connecting Summary

Density Functional Theory:Methods and Problems

Dick Furnstahl

Department of PhysicsOhio State University

October, 2004

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

Dick Furnstahl DFT Methods/Problems


5

82

50

28

28

50

82

2082

28

20

126

A=10

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

pro

tons

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.



Questions about DFT and Nuclear Structure

How is Kohn-Sham DFT more than mean field?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 does pairing work in DFT?Are higher-order contributions important?

What about broken symmetries? (translation, rotation, . . . )

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



Outline

Effective Action Approach to EFT-Based Kohn-Sham DFT

Problems and Challenges

Connecting to Microscopic Approaches

Summary


Outline DFT Challenges Connecting Summary Intro Action Inversion DFT/EFT Results

Density Functional Theory (DFT)

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

-100

-80

-60

-40

-20

0

20

% d

evia

tion

from

exp

erim

ent

Hartree-FockDFT Local Spin Density ApproximationDFT Generalized Gradient Approximation

Atomization Energies of Hydrocarbon Molecules



Density Functional Theory (DFT)

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



Kohn-Sham DFT

VHO

=⇒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∑

i=1

|φi(x)|2

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



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−βgP

{i,j} Si Sj

The magnetization M is

M =⟨∑

i

Si

⟩=

1Z

Tr

[(∑i

Si

)e−βg

P{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(gP

{i,j} Si Sj−HP

i Si )

Variations of the source yield themagnetization

M =⟨∑

i

Si

⟩H

= −∂F [H]

∂H

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



Legendre Transformation to Effective Action

Find H[M] by inverting

M =⟨∑

i

Si

⟩H

= −∂F [H]

∂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]

∂M

∣∣∣∣Mgs

= 0



Variational Energy and the Effective Action

Consider generalized Hamiltonian including time-independent H:

H(H) = g∑{i,j}

SiSj − H∑

i

Si

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

E(H) = limβ→∞

−1β

logZ[H]

Separating out the pieces:

E(H) = 〈H(H)〉H = 〈H〉H − H⟨∑

i

Si

⟩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!



DFT as Analogous Legendre Transformation

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

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

R[L+J ψ†ψ]

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

ρ(x) ≡ 〈ρ̂(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)

= 0

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



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-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 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) =

δW0[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) =∑

i

δΓi [ρ]

δρ(x)



Generalizing the Kohn-Sham Inversion Approach

Sources: Kohn-Sham DFT, kinetic energy, pairingµN + J(x)ρ(x) with J(x) = δΓint[ρ]/δρ(x) → 0 in gsAdd 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

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

†↓ + ψ↓ψ↑〉 =⇒ set j to zero in gs

For all sources, use [J]gs = J0 + J1 + J2 + · · · = 0=⇒ solve for J0 iteratively: [J0]old =⇒ [J0]new = −J1 − J2 + · · ·



Why Use EFT for Energy Functionals

Similar to conventional “phenomenological” approachesbut with a rigorous foundation (DFT from effective action)extendable and could be matched 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

Power counting: what to sum at each order in an expansionnaturalness =⇒ estimates of truncation errorsevidence from Skyrme and RMF functionals for hierarchy

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



EFT and RG Make Physics Easier

There’s an old vaudeville joke about a doctor and his 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!”



“Simple” Many-Body Problem: Hard Spheres

Infinite potential at radius R

0 R

sin(kr+δ)

r

Scattering length a0 = R

Dilute nR3 � 1 =⇒ kFa0 � 1

What is the energy / particle?

k F

R

1/~



In Search of a Perturbative Expansion skip

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

0)

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

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

0 R

=⇒ 〈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



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∂

∂t+

−→∇ 2

2M

]ψ − C0

2(ψ†ψ)2 +

C2

16

[(ψψ)†(ψ

↔∇2ψ) + h.c.

]+

C′2

8(ψ

↔∇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

0)

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′〉:

=⇒∫

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)

2k

+ + + + O(k3)

+i

(M

4π

)2

(C0)3k2 − iC2k

2 − iC ′2k2 cos θ

Matching: C0 = 4πM a0 = 4π

M R , C2 = 4πM

a20r02 = 4π

MR3

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:

=⇒ ∝∫ ∞

kF

d3q(2π)3

1k2 − q2

Same renormalization fixes it! Particles −→ holes∫ ∞

kF

1k2 − q2 =

∫ ∞

0

1k2 − q2−

∫ kF

0

1k2 − q2

D→3−→ −∫ kF

0

1k2 − q2 ∝ a2

0k7F



T = 0 Energy Density from Hugenholtz Diagrams

LO :

NLO : +

NNLO : +

+ +

+

E =

∫d3x

[K (x) +

12

(ν − 1)

ν

4πa0

M[ρ(x)]2

+ d1a2

0

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



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) =δΓint[ρ]

δρ(x)=

∫

δρ(x)

δJ0(y)

−1δΓint[ρ]

δJ0(y)= − − + · · ·

= − + · · ·

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

��



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

LO :

��

��

+ +

+

J0(x) =

[− (ν − 1)

ν

4πa0

Mρ(x)

− c1a2

0

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}

[−∇

2

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)]



Check Out An Example

0 1 2 3 4 5r/b

0

1

2

3

4ρ(

r/b)

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

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 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

0.1

1

ener

gy/p

artic

le

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

Reasonable estimates =⇒ truncation errors understood


Outline DFT Challenges Connecting Summary Tau M∗ Pairing To Do

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

{1

2Mτ +

38

t0ρ2 +1

16t3ρ2+α +

116

(3t1 + 5t2)ρτ

+1

64(9t1 − 5t2)(∇ρ)2 − 3

4W0ρ∇ · J +

132

(t1 − t2)J2}



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

Γ[ρ, τ ] = W [J, η]−∫

J(x)ρ(x)−∫η(x)τ(x)

Two Kohn-Sham potentials:

J0(x) =δΓint[ρ, τ ]

δρ(x)and η0(x) =

δΓint[ρ, τ ]

δτ(x)

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

[i∂t +

∇ 2

2M− v(x) + J0(x)−∇ · η0(x)∇

]ψ

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

16

[(ψψ)†(ψ

↔∇2ψ) + h.c.

]+

C′2

8(ψ

↔∇ψ)† · (ψ

↔∇ψ) + . . .

+

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

Eint[ρ] = . . .+C2

8

[35

(6π2

ν

)2/3

ρ8/3]

+3C′

2

8

[35

(6π2

ν

)2/3

ρ8/3]

+ . . .

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

Eint[ρ, τ ] = . . .+C2

8

[ρτ +

34

(∇ρ)2]+3C′

2

8

[ρτ − 1

4(∇ρ)2]+ . . .



Power Counting Estimates Work for Gradients!

LO NLO LDA ρτ 10*∇ρ

0.01

0.1

1

ener

gy/p

artic

le

ap = as

ap=0

ν =2, as = 0.16, A = 240

τ -NNLO

LO NLO LDA ρτ 10*(∇ρ)0.001

0.01

0.1

1

ener

gy/p

artic

le

ap = as

ap=0

ν =4, as = 0.10, A = 140

τ -NNLO



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

0 1 2 3 4 5r/b

0

0.5

1

1.5

2ρ(

r/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

-0.04

-0.02

0

0.02

0.04

0.06

∆ρ(r

/b)

ap = as

ap = 2asν=2, NF=7, A=240

as=0.16, rs=2as/3



Effective Mass and the Single-Particle Spectrum

3

4

5

6

7

8

9

10

11

12

ρ τ ρ

n = 1

n = 2

n = 4

n = 4

ap = as

M*(0)/M = 0.93

n = 3

ρ τ ρ

l = 7

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

ρ τ ρ ρ τ ρ

l = 0l = 0

n = 3

n = 2

n = 1

n = 1

n = 1

l = 7

Uniform system: ερk − ερτk = π

ν [(ν − 1)a2srs + 2(ν + 1)a3

p]k2

F−k2

2M ρ



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

Rd4x [L+ J(x)ψ†(x)ψ(x) +

Rd4x′ ξ(x′,x)ψ†(x′)ψ(x)]

With Γ[ρ, ξ] = Γ0[ρ, ξ] + Γint[ρ, ξ],

G(x , x ′) =δWδξ

∣∣∣∣J

=δΓ

δξ

∣∣∣∣ρ

= Gks(x , x ′) + Gks

[ δΓint

δGks− δΓint

δρ

]Gks

� � ��

��

� ��

��

� ��

� ��



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 quantities may differ . . .



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−

Rd4x [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)



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)

∣∣∣∣ρ=ρgs,φ=φgs

= 0

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!



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] + · · ·

W̃ [J, j , λ] = W̃0[J, j] + W̃1[J, j] + W̃2[J, j] + · · ·Γ̃[ρ, φ, λ] = Γ̃0[ρ, φ] + Γ̃1[ρ, φ] + Γ̃2[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)

)= Ei

(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 = + + + + · · ·

iG =

(〈Tψ↑(x)ψ†↑(x

′)〉0 〈Tψ↑(x)ψ↓(x ′)〉0〈Tψ†↓(x)ψ†↑(x

′)〉0 〈Tψ†↓(x)ψ↓(x ′)〉0

)≡

(iG0

ks iF 0ks

iF 0ks† −iG0

ks

)In frequency space, the Green’s functions are

iG0ks(x, x

′;ω) =∑

i

[ui(x) u∗i (x′)ω − Ei + iη

+vi(x′) v∗i (x)

ω + Ei − iη

]

iF 0ks(x, x

′;ω) = −∑

i

[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∑

i

|vi(x)|2

and the pair density is (warning: divergent!)

φ(x) =∑

i

[u∗i (x)vi(x) + ui(x)v∗i (x)]

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

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

J0(x)∣∣∣ρ=ρgs

=δΓ̃int[ρ, φ]

δρ(x)

∣∣∣∣∣ρ=ρgs

and j0(x)∣∣∣φ=φgs

=δΓ̃int[ρ, φ]

δφ(x)

∣∣∣∣∣φ=φgs



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

e−W =

∫D(ψ†ψ) e−

Rd4x [ψ†

α( ∂∂τ −

∇ 22M −µ)ψα +

C02 ψ

†↑ψ

†↓ψ↓ψ↑+ j(ψ↑ψ↓+ψ†

↓ψ†↑)] + 1

2 ζ j2

cf. adding integration over auxiliary field∫

D(∆∗,∆) e−1

|C0|R|∆|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?



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

µ0

0

1

εk

vk2 uk

2

j0

ρ = 2∑

k

v2k =

∫d3k

(2π)3

(1− ξk

Ek

)φ = 2

∑k

uk vk = −∫

d3k(2π)3

j0Ek

+ 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|

=12

∫d3k

(2π)3

1ε0

k

=⇒ M4πas

= −12

∫d3k

(2π)3

[1

Ek− 1ε0

k

]



Observables With Kohn-Sham Pairing

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

EV

= (Γ0 + Γ1)|j0=− 12 |C0|φ =

∫d3k

(2π)3

[ξk −Ek +

12

j20Ek

]+[µ0 −

14|C0|ρ

]ρ

with

ρ =

∫d3k

(2π)3

(1− ξk

Ek

)and φ = −

∫d3k

(2π)3

j0Ek

+ 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 →

k2F

2M



Dimensional Regularization (DR) skip

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

C0(Λ) =4πas

M1

1− asΛ=

4πas

M+

4πa2s

MΛ +O(Λ2) = C(1)

0 + C(2)0 + · · ·

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

2

)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

2

)3−D∫

dDk(2π)D

(ε0k )β

Ek=

MΛ

2πµβ0

(1− δβ,2

x2

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:

ρ =

∫d3k

(2π)3

(1−

ε0k − µ0

Ek

)= 0− I(1) + µ0 I(0)

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

1|C0(Λ)|

=12

I(0) or1|C0|

=12

(I(0)− ζ(0)

)Dick Furnstahl DFT Methods/Problems


Anomalous Density in Finite Systems

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

φ(x) =∑

i

[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

φ =

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− 1ε0

k

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

c (x)

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

φ =

∫ kc d3k(2π)3 j0

1√(ε0

k − µ0)2 + j20

− Pε0

k − µ0

kc→∞−→ finite

Compare convergence in uniform system and in nuclei with LDA

0.01 0.1 1 10Energy Cutoff

10-6

10-5

10-4

10-3

10-2

10-1

100

Frac

tion

Mis

sing

subtraction 1subtraction 2

µ0 = 1j0 = 0.1

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

r [fm]

∆(r

) [M

eV]



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

−40

−30

−20

−10

0

r [fm]

U(r

) [M

eV]

µ + Ec

µ

µ − Ec

r1

r2

I II III

5 10 15 20 25 30 35 40 45 500

1

2

3

4

∆ [M

eV]

5 10 15 20 25 30 35 40 45 50−600

−500

−400

−300

−200

−100

Ec [MeV]

g eff [M

eV fm

3 ]



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 µ0

0

1

εk

vk2 uk

2

j0

ukvk

��

��

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



On-Going and Future Challenges

Long-range effects

Gradient expansions

Restoring brokensymmetries

Auxiliary fieldKohn-Sham Theory

Covariant DFT:Dilute system

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

��

� � � � �

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

+ + + + · · ·


Outline DFT Challenges Connecting Summary Perturbative DFT/RG

Comparing Skyrme and Dilute Functionals

Skyrme energy density functional (for N = Z )

E [ρ, τ, J] =

∫d3x

{τ

2M+

38

t0ρ2 +1

16(3t1 + 5t2)ρτ

+1

64(9t1 − 5t2)(∇ρ)2 +

116

t3ρ2+α − 34

W0ρ∇ · J + · · ·}

Dilute ρτ energy density functional for ν = 4 (Vexternal= 0)

E [ρ, τ, J] =

∫d3x

{τ

2M+

38

C0ρ2 +

116

(3C2 + 5C′2)ρτ

+1

64(9C2 − 5C′

2)(∇ρ)2 +c1

2MC2

0ρ7/3 +

c2

2MC3

0ρ8/3 + · · ·

Same functional as dilute Fermi gas with ti ↔ Ci

equivalent as ≈ −2–3 fm but |kFap|, |kFrs| < 1 (with ap < 0)missing non-analytic terms



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

=⇒ correlations are small corrections =⇒ works!

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

P/2

k

Λ

kF

|k| < Λand

Match at finite density =⇒ perturbative?



Polonyi-Schwenk RG Approach to DFT

Non-interacting fermions inbackground mean-fieldpotential V at λ = 0

Gradually switch off backgroundpotential and turn on themicroscopic interaction Uas λ→ 1

Sλ,1[ψ†, ψ] =

∫dx ψ†α(x)

(∂

∂t− ∇2

x

2M+ (1− λ)Vλ;α(x)

)ψα(x)

Sλ,2[ψ†, ψ] =λ

2

∫ ∫(ψ†ψ) · U · (ψ†ψ)



Density Functional = Effective Action for Density

Effective action Γ[ρ] = −W [J] + J · ρ is minimal at thephysical (zero source) ground state density:

δΓ[ρ]

δρ

∣∣∣∣ρgs

= 0 =⇒ Egs = E [ρgs] = limβ→∞

1β

Γ[ρgs]

The effective action

is m

inimal at

the physical (=zero source) ground state density, i.e.,

, with g.s. energy

Density functional = effective action for the density

curvature will include

exchange-correlations-1=+ interactions

Curvature will include correlations(δ2Γ[ρ]

δρ δρ

∣∣∣∣ρgs

)−1

=δ2W [J]

δJ δJ

∣∣∣∣J=0

=

The effective action

is m

inimal at

the physical (=zero source) ground state density, i.e.,

, with g.s. energy

Density functional = effective action for the density

curvature will include

exchange-correlations-1=+ interactions

+ interactions



Evolution of Effective Action with Parameter λ

∆ background Hartree exchange-correlations

∂λΓλ[ρ] = ∂λ[(1− λ)Vλ] · ρ+12ρ · U · ρ+

12

Tr

[U ·(δ2Γλ[ρ]

δρ δρ

)−1]

Expand density functional about evolving ground-state density

Γλ[ρ] = Γ[ρgs,λ](0) +

∑n≥2

∫· · ·∫

1n!

Γ[ρgs,λ](n) · (ρ− ρgs,λ)1 · · · (ρ− ρgs,λ)n

change in background potential Hartree contribution exchange-correlations

All exchange-corr. via dressed ph propagator

Evolution of the effective action w

ith control parameter

Expand density functional around evolving g.s. density

Evolution equations

for expansion coeff.(build up m

any corr. asFerm

i liquid RG

)

Evolution equations for expansioncoefficients build up correlationsthrough dressed ph propagator



Summary

Effective action =⇒ framework for Kohn-Sham DFTEFT provides systematic expansionsConnects to exact Green’s functionExploit renormalization (e.g., pair density)

Problems and ChallengesPairingLong-range effectsRestoring broken symmetriesMatching to Vlow k and/or chiral EFT

Alternatives to pursueRG for density functional


Density Functional Theory: Methods and Problems · 2004-10-13 · Density Functional Theory (DFT)...

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Transcript of Density Functional Theory: Methods and Problems · 2004-10-13 · Density Functional Theory (DFT)...