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
Outline DFT Challenges Connecting Summary
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.
Dick Furnstahl DFT Methods/Problems
Outline DFT Challenges Connecting Summary
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?
Dick Furnstahl DFT Methods/Problems
Outline DFT Challenges Connecting Summary
Outline
Effective Action Approach to EFT-Based Kohn-Sham DFT
Problems and Challenges
Connecting to Microscopic Approaches
Summary
Dick Furnstahl DFT Methods/Problems
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
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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
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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
Dick Furnstahl DFT Methods/Problems
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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
]
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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!
Dick Furnstahl DFT Methods/Problems
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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 !
Dick Furnstahl DFT Methods/Problems
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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?
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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)
Dick Furnstahl DFT Methods/Problems
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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 + · · ·
Dick Furnstahl DFT Methods/Problems
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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
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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!”
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“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/~
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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
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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 !
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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 + · · ·
]
Dick Furnstahl DFT Methods/Problems
Outline DFT Challenges Connecting Summary Intro Action Inversion DFT/EFT Results
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
Dick Furnstahl DFT Methods/Problems
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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”
��������� ��
Dick Furnstahl DFT Methods/Problems
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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 + · · ·
]
Dick Furnstahl DFT Methods/Problems
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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)]
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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}
Dick Furnstahl DFT Methods/Problems
Outline DFT Challenges Connecting Summary Tau M∗ Pairing To Do
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)
Dick Furnstahl DFT Methods/Problems
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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]+ . . .
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
Outline DFT Challenges Connecting Summary Tau M∗ Pairing To Do
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
Dick Furnstahl DFT Methods/Problems
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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
Dick Furnstahl DFT Methods/Problems
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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
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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
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How Do G and Gks Yield the Same Density?
Claim: ρks(x) = −iνG0KS(x , x+) equals ρ(x) = −iνG(x , x+)
Start with� � �����
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Simple diagrammatic demonstration:� � � � � � � � �
Densities agree by construction!
But other quantities may differ . . .
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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
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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)
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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!
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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
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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
]
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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
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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)
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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?
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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
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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
]
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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
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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)
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=⇒ independent of Λ
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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)
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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
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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]
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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 ]
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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
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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)
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Non-localities from near-on-shellparticle-hole excitations
+ + + + · · ·
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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
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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?
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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)
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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
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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
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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
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