QMPT 540

Quantum Theory of Many-Particle Systems,

Phys. 540• N±2 tp propagator with two times

• Scattering of two particles in free space

• Bound states of two particles

• Ladder diagrams and SRC in the medium

• Scattering of mean-field particles in the medium

• Cooper problem and pairing instabilities

• Bound pair states

• Other questions about last class and assignments?

• Comments?

QMPT 540

Consequences for sp propagator in infinite systems

• Electron gas --> plasmon --> sp propagator

• example of influence of collective state on sp properties

• Nuclear matter --> SRC --> sp propagator

• example of influence of strong mutual repulsion on Fermi sea

• Diagrams with momentum conservation in infinite systems

• Volume terms from interactions cancel those arising from

momentum integrations

• Propagator (sp) doesn’t depend on spin/isospin quantum numbers

for spin/isospin saturated system

• Label propagators with wave vectors (and energy) with beginning

and end point labeled by discrete quantum numbers

QMPT 540

Examples

• First-order term

• includes extra summation over external quantum numbers to

facilitate coupling to total spin etc.

• 2nd order (internal labeling as convenient)

G(1)

V(k;E) = G(0)(k;E)

×{− i

1ν

∑

mαmβ

∫d3k′

(2π)3〈kmαk′mβ | V |kmαk′mβ〉

×∫

C↑

dE′

2πG(0)(k′;E′)

}G(0)(k;E)

Σ(2)(k;E) = (−1)i212

∫dE1

2π

∫dE2

2π

∫d3q

(2π)3

∫d3k′

(2π)31ν

∑

mαmβmγmδ

× 〈kmαk′ − q/2mβ | V |k − qmγk′ + q/2mδ〉× G(0)(k′ + q/2; E1 + E2)G(0)(k′ − q/2;E2)G(0)(k − q;E − E1)

× 〈k − qmγk′ + q/2mδ| V |kmαk′ − q/2mβ〉

QMPT 540

Self-energy in infinite system

• Exact self-energy similarly from

• toΣ(k;E) = −U(k)

−i1ν

∑

mαmβ

∫d3k′

(2π)3〈kmαk′mβ | V |kmαk′mβ〉

∫

C↑

dE′

2πG(k′;E′)

−i212

∫dE1

2π

∫dE2

2π

∫d3q

(2π)3

∫d3k′

(2π)31ν

∑

mαmβmγmδ

× 〈kmαk′ − q/2mβ | V |k − qmγk′ + q/2mδ〉× G(k − q;E − E1)G(k′ + q/2;E1 + E2)G(k′ − q/2;E2)

× 〈k − qmγk′ + q/2mδ| Γ(E,E1, E2) |kmαk′ − q/2mβ〉

Σ∗(γ, δ;E) = −〈γ| U |δ〉 − i

∫

C↑

dE′

2π

∑

µ,ν

〈γµ| V |δν〉G(ν, µ;E′)

+12

∫dE1

2π

∫dE2

2π

∑

ε,µ,ν,ζ,ρ,σ

〈γµ| V |εν〉G(ε, ζ;E1)G(ν, ρ;E2)

×G(σ, µ;E1 + E2 − E) 〈ζρ| Γ(E1, E2;E,E1 + E2 − E) |δσ〉

QMPT 540

Propagator

• DE

• only magnitude of wave vector needed

• Noninteracting propagator

• with

• Particle spectral function

• Hole spectral function

• Dispersion relation (check)

G(k;E) = G(0)(k;E) + G(0)(k;E)Σ(k;E)G(k;E)

=E − ε(k)− Re Σ(k;E) + iIm Σ(k;E)

(E − ε(k)− Re Σ(k;E))2 + (Im Σ(k;E))2

G(0)(k;E) =θ(k − kF )

E − ε(k) + iη+

θ(kF − k)E − ε(k)− iη

ε(k) =!2k2

2m+ U(k)

Sp(k;E) =−1π

Im Σ(k;E)(E − ε(k)− Re Σ(k;E))2 + (Im Σ(k;E))2

Sh(k;E) =1π

Im Σ(k;E)(E − ε(k)− Re Σ(k;E))2 + (Im Σ(k;E))2

G(k;E) =∫ ∞

εF

dE′ Sp(k;E′)E − E′ + iη

+∫ εF

−∞dE′ Sh(k;E′)

E − E′ − iη

QMPT 540

Other quantities

• Energy per particle

• Density but also from

• integral over momentum distribution

• Kinetic energy

• Pressure at T=0 from density derivative of energy per particle

• using

• and so at saturation (H-vH theorem)

EN0

N=

ν

2ρ

∫d3k

(2π)3

∫ εF

−∞dE

(!2k2

2m+ E

)Sh(k;E)

ρ =νk3

F

6π2

ρ =ν

2π2

∫ ∞

0dkk2 n(k)

n(k) =∫ εF

−∞dE Sh(k;E)

T

N=

ν

2π2ρ

∫ ∞

0dkk2 !2k2

2mn(k)

P = ρ2 ∂(E/N)∂ρ

= ρ[εF − (E/N)] E/N = E/(ρV )

εF = ∂E/∂N E/N = εF

QMPT 540

Self-energy in the electron gas

• Second-order is not appropriate for the electron gas

• Consider direct matrix element in 2nd-order self-energy

• with

• Performing E2 and k’ integration and identifying noninteracting

polarization propagator (including spin summations)

• generates “infrared” divergence due to q-4 contribution so

infinite! continues in higher order!

〈kmαk′ − qmβ | V |k − qmγk′mδ〉D ⇒ δmαmγ δmβmδV (q)

V (q) =4πe2

q2

Σ(2)D (k;E) = 2i

∫dE1

2π

∫d3q

(2π)3V 2(q)G(0)(k − q;E − E1)Π(0)(q;E1)

Σ(2)(k;E) = (−1)i212

∫dE1

2π

∫dE2

2π

∫d3q

(2π)3

∫d3k′

(2π)31ν

∑

mαmβmγmδ

× 〈kmαk′ − q/2mβ | V |k − qmγk′ + q/2mδ〉× G(0)(k′ + q/2; E1 + E2)G(0)(k′ − q/2;E2)G(0)(k − q;E − E1)

× 〈k − qmγk′ + q/2mδ| V |kmαk′ − q/2mβ〉

QMPT 540

Higher-order terms

• Replace by to generate 3rd order diagram and

so on

• with worse divergences

• Remedy: sum all divergent terms replacing noninteracting

polarization propagator by the RPA one (already encountered for

discussion of plasmon)

• So evaluate

Π(0) Π(0)2V Π(0)

ΣRPA(k;E) = 2i∫

dE1

2π

∫d3q

(2π)3V 2(q)G(0)(k − q;E − E1)ΠRPA(q;E1)

QMPT 540

Ring approximation

• Add Fock term: represented by

• Approximation known as

• Self-consistent formulation

• Note

– medium modification important so should be included self-consistently

– drawback: f-sum rule no longer conserved and no plasmon (see later)

• Evaluate self-energy using Lehmann representation

• RPA generates discrete poles (plasmons)

• Plasmon term above Fermi energy (im part)

GW

Π(q, E) = − 1π

∫ ∞

0dE′ Im Π(q, E′)

E − E′ + iη+

1π

∫ 0

−∞dE′ Im Π(q, E′)

E − E′ − iη

Im ΣRPAp (k;E) =

π

2

∫d3q

(2π)3θ(|k − q|− kF )θ(qc − q)

× δ (E − Ep(q)− ε(k − q))

∂Π(0)(q, E)∂E

∣∣∣∣∣E=Ep(q)

−1

G(0)W (0)

QMPT 540

Development

• Algebra somewhat lengthy

• When q=0 point included, integrand requires expansion yielding a

logarithmic singularity (Hedin 1967)

• From delta-function: occurs at (+ for k>kF)

• Results (only plasmon)

• Real part singular for k=kF

E = ±Ep(0) + ε(k)

rs = 2

QMPT 540

Continuum contribution

• Employ corresponding dispersion relation

• Insert in self-energy

• For energies above the Fermi energy

• similarly for energies below

• Real part from

• Continuum only

• Much smaller than plasmon terms

ΠRPAc (q, E) = − 1

π

∫ E+

E−

dE′ Im ΠRPAc (q, E′)

E − E′ + iη+

1π

∫ −E−

−E+

dE′ Im ΠRPAc (q, E′)

E − E′ − iη

Im ΣRPAc (k;E) = 2

∫d3q

(2π)3V 2(q)

∫ E+

E−

dE′ Im ΠRPAc (q, E′) θ(|k − q|− kF )δ(E − E′ − ε(k − q))

Re ΣRPA(k;E) = ΣHF (k) +Pπ

∫ +∞

−∞dE′ |Im ΣRPA(k;E′)|

E − E′

QMPT 540

Spectral functions

• From self-energy and Dyson equation -> spectral functions

• Wave vectors deep inside Fermi sea: two distinct features

• Because of the vanishing of the imaginary part in certain regions

there may be multiple solutions to

• outside these domains -> more than one delta-function solution

possible with

• Almost realized for k/kF=0.2 at rs=2

• Peak is referred to as the “plasmaron”

EQ(k) =!2k2

2m+ Re ΣRPA(k;EQ(k))

ZQ(k) =

(1− ∂Re ΣRPA(k;E)

∂E

∣∣∣∣E=EQ(k)

)−1

QMPT 540

Plot of spectral functions

• rs = 2

• Only plasmon

• Integrated strength:

• n(k) for rs = 1 and 2

QMPT 540

GW approximation

• Diagrammatically

• Ingredients of GW

• Note that we can write

• For dressed propagators

• Using Lehmann for sp propagators yields

Π(0)(k; q, E) =∫

dE′

2πiG(0)(k + q/2;E + E′)G(0)(k − q/2;E′)

Πf (k; q, E) =∫

dE′

2πiG(k + q/2;E + E′)G(k − q/2;E′)

Πf (k; q, E) =∫ ∞

εF

dE

∫ εF

−∞dE′ Sp(k + q/2; E)Sh(k − q/2; E′)

E − (E − E′) + iη

−∫ εF

−∞dE

∫ ∞

εF

dE′ Sh(k + q/2; E)Sp(k − q/2; E′)E + (E′ − E)− iη

QMPT 540

Development

• For positive energies

• By integrating over wave vector k the dressed version of the

Lindhard function is generated

• Obeys usual dispersion relation

• Plot for rs = 2 (dashed)

• rs = 4 (solid)

• noninteracting q/kF = 0.25

• Huge spreading for self-consistency

Im Πf (k; q, E) = −π

∫ ∞

εF

dE

∫ εF

−∞dE′Sp(k + q/2; E)Sh(k − q/2; E′)δ(E − (E − E′))

= −π

∫ E+εF

εF

dE Sp(k + q/2; E)Sh(k − q/2; E − E)

Πf (q, E) =∫

d3k

(2π)3Πf (k; q, E)

Πf (q, E) = − 1π

∫ ∞

0dE′ Im Πf (q, E′)

E − E′ + iη+

1π

∫ 0

−∞dE′ Im Πf (q, E′)

E − E′ − iη

QMPT 540

Polarization propagator

• As usual

• and the screened Coulomb interaction becomes

• Demonstrating that it has the same analytical properties as

Π(q, E) =Πf (q, E)

1− 2V (q)Πf (q, E)

W (q, E) ≡ V (q) + ∆W (q, E)

= V (q)− 1π

∫ ∞

0dE′ Im W (q, E′)

E − E′ + iη+

1π

∫ 0

−∞dE′ Im W (q, E′)

E − E′ − iη

W (q, E) = V (q) + V (q)2Πf (q, E)V (q)+V (q)2Πf (q, E)V (q)2Πf (q, E)V (q) + ...

= V (q) + 2V (q){Πf (q, E) + Πf (q, E)2V (q)Πf (q, E) + ...

}V (q)

= V (q) + 2V 2(q)Π(q, E)

Π

QMPT 540

More development

• Imaginary part of screened Coulomb can be written as

• Since the self-energy can be written as

• Insert spectral representation and perform energy integration

• First term (particle domain)

• Hole domain

Im W (q, E) = 2V 2(q)Im Π(q, E)

= 2V 2(q)Im Πf (q, E)

(1− 2V (q) Re Πf (q, E))2 + (2V (q) Im Πf (q, E))2

∆W = 2V 2Π

Σ∆W (k;E) = i

∫dE1

2π

∫d3q

(2π)3G(k − q;E − E1)∆W (q;E1)

Σ∆W (k;E) − 1π

∫d3q

(2π)3

∫ ∞

0dE′

∫ ∞

εF

dESp(k − q; E)Im W (q, E′)

E − E′ − E + iη

− 1π

∫d3q

(2π)3

∫ 0

−∞dE′

∫ εF

−∞dE

Sh(k − q; E)Im W (q, E′)E − E′ − E − iη

Im Σ∆W (k;E) =∫

d3q

(2π)3

∫ E−εF

0dE′Sp(k − q;E − E′)Im W (q, E′)

Im Σ∆W (k;E) = −∫

d3q

(2π)3

∫ 0

E−εF

dE′Sh(k − q;E − E′)Im W (q, E′)

QMPT 540

Final steps

• Still need generalization of HF term

• So GW self-energy becomes

• with DE self-consistency loop complete

• Holm and von Barth (1998) calculation employed sum of Gaussians

• with GW(0) to initiate parameters

• GW(0) keeps screened Coulomb fixed but determines G self-

consistently

• Both schemes require several iteration steps for the parameters

ΣV (k) = −∫

d3k′

(2π)3V (k − k′)n(k′)

ΣGW (k;E) = ΣV (k)− 1π

∫ ∞

εF

dE′ Im Σ∆W (k;E′)E − E′ + iη

+1π

∫ εF

−∞dE′ Im Σ∆W (k;E′)

E − E′ − iη

S(k;E) =∑

n

Wn(k)√2πΓn(k)

exp{− (E − En(k))2

2Γ2n(k)

}

QMPT 540

Results for GW

• No more plasmon! (no more f-sum rule too)

• Im part of noninteracting dressed polarization propagator has

substantial imaginary part in large domain where plasmon used to

reside -> spreads plasmon strength

• Meaning?

• Self-energy rs=4 for k=0 and k=kF

• GW solid

• GW(0) dashed

• similar to G(0)W(0)

QMPT 540

Spectral functions

• Still satellites in GW(0) dashed

• rs = 4

• No satellites in GW (observed in Na)

• GW less correlated

• Illustrated by quasiparticle strength

• and momentum distribution

QMPT 540

Difficulties

• Bandwidth: difference between quasiparticle energies kF and 0

• Increases with self-consistency

• Problems?!?!

• Self-consistency physical

• Self-energy and plasmon worse

• Vertex corrections (exchange)?

• but correlation energy…

• Possible solution: Faddeev approach

QMPT 540

Energy per particle electron gas

• Fundamental issue related to DFT applications

• Quantum Monte Carlo available

• -XC in table (XC = exchange correlation energy per particle)

• famous paper: cited 6065 times (07/30/2009) and counting

• Green’s function results: GW very close to QMC

• Particle number violations possible unless fully self-consistent

rs 1 2 4 5 10 20QMC 0.5180 0.2742 0.1466 0.1198 0.0644 0.0344

0.5144 0.2729 0.1474 0.1199 0.0641 0.0344GW 0.5160(2) 0.2727(5) 0.1450(5) 0.1185(5) 0.0620(9) 0.032(1)

0.2741 0.1465GW (0) 0.5218(1) 0.2736(1) 0.1428(1) 0.1158(1) 0.0605(4) 0.030(1)

G(0)W (0) 0.5272(1) 0.2821(1) 0.1523(1) 0.1247(1) 0.0665(2) 0.0363(5)RPA 0.5370 0.2909 0.1613 0.1340 0.0764 0.0543−Ex/N 0.4582 0.2291 0.1145 0.0916 0.0458 0.0229

QMPT 540

Assignment

• Choose a topic for your presentation!

• Suggestions

– Dispersive optical model

– Spectral function of dilute Fermi gases

– Quasiparticle DFT

– Faddeev summation for self-energy

– Neutron matter equation of state

QMPT 540

Quantum Theory of Many-Particle Systems,

Phys. 540• Self-energy in infinite systems

• Diagram rules

• Electron gas various forms of GW approximation

• Other questions about last class and assignments?

• Comments?

QMPT 540

Nucleons in nuclear matter

• NN interaction requires summation of ladder diagrams

• Study influence of SRC on sp propagator

• Formulate in self-consistent form

• employs noninteracting but dressed convolution of sp propagators

〈kmαmα′ | Γpphh(K, E) |k′mβmβ′〉= 〈kmαmα′ | V |k′mβmβ′〉 + 〈kmαmα′ | ∆Γpphh(K, E) |k′mβmβ′〉

= 〈kmαmα′ | V |k′mβmβ′〉 +12

∑

mγmγ′

∫d3q

(2π)3〈kmαmα′ | V |qmγmγ′〉

× Gfpphh(K, q;E) 〈qmγmγ′ | Γpphh(K, E) |k′mβmβ′〉

Gfpphh(K, q;E) =

∫ ∞

εF

dE′∫ ∞

εF

dE′′Sp(q + K/2;E′)Sp(K/2− q;E′′)E − E′ − E′′ + iη

−∫ εF

−∞dE′

∫ εF

−∞dE′′Sh(q + K/2;E′)Sh(K/2− q;E′′)

E − E′ − E′′ − iη

QMPT 540

Lehmann representation

• Dispersion relation (arrows for location of poles in energy plane)

• Corresponding self-energy

• Energy integral can be performed (note arrows)

〈kmαmα′ | ∆Γpphh(K, E) |k′mβmβ′〉

=−1π

∫ ∞

2εF

dE′ Im 〈kmαmα′ | ∆Γpphh(K, E′) |k′mβmβ′〉E − E′ + iη

+1π

∫ 2εF

−∞dE′ Im 〈kmαmα′ | ∆Γpphh(K, E′) |k′mβmβ′〉

E − E′ − iη

≡ 〈kmαmα′ | ∆Γ↓(K, E) |k′mβmβ′〉+ 〈kmαmα′ | ∆Γ↑(K, E) |k′mβmβ′〉

Σ∆Γ(k;E) = −i1ν

∑

mαmα′

∫d3k′

(2π)3

∫dE′

2πG(k′;E′)

× 〈 12 (k − k′)mαmα′ | ∆Γpphh(K, E + E′) | 1

2 (k − k′)mαmα′〉

Σ∆Γ(k;E) =1ν

∑

mαmα′

∫d3k′

(2π)3

∫ εF

−∞dE′ 〈 1

2 (k − k′)mαmα′ | ∆Γ↓(E + E′) | 12 (k − k′)mαmα′〉Sh(k′, E′)

− 1ν

∑

mαmα′

∫d3k′

(2π)3

∫ ∞

εF

dE′ 〈 12 (k − k′)mαmα′ | ∆Γ↑(E + E′) | 1

2 (k − k′)mαmα′〉Sp(k′, E′)

≡ ∆Σ↓(k;E) + ∆Σ↑(k;E)

QMPT 540

Final steps and implementation

• Total self-energy

• includes HF-like term

• Practical calculations first done with mean-field sp propagators

in scattering equation

Σ(k;E) = ΣV (k)− 1π

∫ ∞

εF

dE′ Im Σ(k;E′)E − E′ + iη

+1π

∫ εF

−∞dE′ Im Σ(k;E′)

E − E′ − iη

= ΣV (k) + ∆Σ↓(k;E) + ∆Σ↑(k;E)

ΣV (k) =∫

d3k′

(2π)31ν

∑

mαmα′

〈 12 (k − k′)mαmα′ | V | 1

2 (k − k′)mαmα′〉n(k′)

Σ(0)∆Γ(k;E) =

1ν

∑

mαmα′

∫d3k′

(2π)3〈 1

2 (k − k′)mαmα′ | ∆Γ(0)↓ (E + ε(k′)) | 1

2 (k − k′)mαmα′〉 θ(kF − k′)

− 1ν

∑

mαmα′

∫d3k′

(2π)3〈 1

2 (k − k′)mαmα′ | ∆Γ(0)↑ (E + ε(k′)) | 1

2 (k − k′)mαmα′〉 θ(k′ − kF )

QMPT 540

Results for mean-field input

• Realistic interactions generate pairing instabilities

• Avoid with “gap” is sp spectrum according to

• Hole strength spread so “sp energy” below QP energy --> gap

• Requires full knowledge of self-energy

• Self-consistent determination of sp spectrum

• Subsequent determination of self-energy constrained by sp

spectrum -> imaginary part exhibits momentum dependent

domains

ε(k) =∫

dE E Sh(k, E)∫dE Sh(k, E)

k < kF

ε(k) =∫

dE E SQ(k, E)∫dE SQ(k,E)

= EQ(k) k > kF

QMPT 540

Self-energy below Fermi energy

• Wave vectors 0.1 (solid), 0.51 (dotted), and 2.1 fm-1 (dashed)

• kF = 1.36 fm-1

• Im part

• Spectral functions with peaks at

• QP spectral function

• with and

EQ(k) =!2k2

2m+ ReΣ(k;EQ(k))

SQ(k, E) =1π

Z2Q(k) | W (k) |

(E − EQ(k))2 + (ZQ(k)W (k))2

W (k) = Im Σ(k;EQ(k))ZQ(k) =

{1−

(∂Re Σ(k;E)

∂E

)

E=EQ(k)

}−1

QMPT 540

Spectral functions

• Near kF about 70% of sp strength is contained in QP peak

• additional ~13% distributed below the Fermi energy

• remaining strength (~17%) above the Fermi energy

• Note “gap”

• Distribution narrows

• towards kF

QMPT 540

Spectral functions above the Fermi energy

• Wave vectors 0.79 (dotted), 1.74 (solid), and 3.51 fm-1 (dashed)

• Common distribution -> SRC

• For 0.79 -> 17%

• !F + 100 MeV -> 13%

• above 500 MeV -> 7%

• Without tensor force:

• strength only 10.5%

• Momentum distribution

• n(0) same for different methods

• and interactions (not at kF)

• Solid: self-consistent

QMPT 540

Self-consistent results

• Wave vectors 0.0 (solid), 1.36 (dotted), and 2.1 fm-1 (dashed)

• Limited set of Gaussians --> self-consistent

• Spectral functions

• Main difference:

– common strength

• Slightly less correlated

• ZF from 0.72 to 0.75

QMPT 540

SRC and saturation

• Energy sum rule

• Contribution from high momenta after energy integral for

different densities for Reid potential

EN0

N=

ν

2ρ

∫d3k

(2π)3

∫ εF

−∞dE

(!2k2

2m+ E

)Sh(k;E)

QMPT 540

Saturation properties of nuclear matter

• Colorful and continuing story

• Initiated by Brueckner: proper treatment of SRC in medium ->

ladder diagrams but only include pp propagation

• Brueckner G-matrix but Bethe-Goldstone equation…

• Dispersion relation

• Include HF term in “BHF” self-energy

• Below Fermi energy: no imaginary part

〈kmαmα′ | G(K, E) |k′mβmβ′〉 = 〈kmαmα′ | V |k′mβmβ′〉

+12

∑

mγmγ′

∫d3q

(2π)3〈kmαmα′ | V |qmγmγ′〉 θ(|q + K/2| − kF ) θ(|K/2 − q| − kF )

E − ε(q + K/2) − ε(K/2 − q) + iη〈qmγmγ′ | G(K, E) |k′mβmβ′〉

〈kmαmα′ | G(K, E) |k′mβmβ′〉 = 〈kmαmα′ | V |k′mβmβ′〉 − 1π

∫ ∞

2εF

dE′ Im 〈kmαmα′ | ∆G(K, E′) |k′mβmβ′〉E − E′ + iη

≡ 〈kmαmα′ | V |k′mβmβ′〉+ 〈kmαmα′ | ∆G↓(K, E) |k′mβmβ′〉

ΣBHF (k;E)) =∫

d3k′

(2π)31ν

∑

mαmα′

θ(kF − k′) 〈 12 (k − k′)mαmα′ | G(k + k′;E + ε(k′) | 1

2 (k − k′) mαmα′〉

QMPT 540

BHF

• DE for k < kF yields solutions at

• with strength < 1

• Since there is no imaginary part below the Fermi energy, no

momenta above kF can admix -> problem with particle number

• Only sp energy is determined self-consistently

• Choice of auxiliary potential

– Standard only for k < kF (0 above)

– Continuous all k

• Only one calculation of G-matrix for standard choice

• Iterations for continuous choice

εBHF (k) =!2k2

2m+ ΣBHF (k; εBHF (k))

Us(k) = ΣBHF (k; εBHF (k))

Uc(k) = ΣBHF (k; εBHF (k))

QMPT 540

BHF

• Propagator

• Energy

• Rewrite using on-shell self-energy

• First term: kinetic energy free Fermi gas

• Compare

• so BHF obtained by replacing V by G

GBHF (k;E) =θ(k − kF )

E − εBHF (k) + iη+

θ(kF − k)E − εBHF (k)− iη

EA0

A=

ν

2ρ

∫d3k

(2π)3

(!2k2

2m+ εBHF

)θ(k − kF )

EA0

A=

4ρ

∫d3k

(2π)3!2k2

2m+

12ρ

∫d3k

(2π)3

∫d3k′

(2π)3∑

mαmα′

θ(kF − k)θ(kF − k′)

〈 12 (k − k′) mαmα′ | G(k + k′; εBHF (k) + εBHF (k′)) | 1

2 (k − k′) mαmα′〉

EHF =12

∑

p

θ(pF − p)[

p2

2m+ εHF (p)

]

= TFG +12

∑

pp′

θ(pF − p)θ(pF − p′) 〈pp′| V |pp′〉

QMPT 540

BHF

• Binding energy usually

within 10 MeV from

empirical volume term in

the mass formula even

for very strong repulsive

cores

• Repulsion always

completely cancelled by

higher-order terms

• Minimum in density never

coincides with empirical

value when binding OK ->

Coester bandLocation of minimum determined

by deuteron D-state probability

QMPT 540

Historical perspective• First attempt using scattering in the medium Brueckner 1954

• Formal development (linked cluster expansion) Golds!ne 1956

• Reorganized perturbation expansion (60s) Be"e & student# involving ordering in the number of hole lines BBG-expansion

• Variational Theory vs. Lowest Order BBG (70s) Clark (also crisis paper)

$ $ $ $ $ $ Pandharipand%

• Variational results & next hole-line terms (80s) Day, Wiring&

• New insights from experiment & theory NIKHEF Amsterdam

about what nucleons are up to in the nucleus (90s)

• 3-hole line terms with continuous choice (90s) Baldo et al.

• Ongoing...

QMPT 540

Some remarks

• Variational results gave more binding than G-matrix calculations

• Interest in convergence of Brueckner approach

• Bethe et al.: hole-line expansion

• G-matrix: sums all energy terms with 2 independent hole lines

(noninteracting ...)

• Dominant for low-density

• Phase space arguments suggests to group all terms with 3

independent hole lines as the next contribution

• Requires technique from 3-body problem first solved by Faddeev

-> Bethe-Faddeev summation

• Including these terms generates minima indicated by!in figure

• Better but not yet good enough

QMPT 540

More

• Variational results and 3-hole-line results more or less in

agreement

• Baldo et al. also calculated 3-hole-line terms with continuous

choice for auxiliary potential and found that results do not

depend on choice of auxiliary potential, furthermore 2-hole-line

with continuous choice is already sufficient!

• Conclusion: convergence OK for a given realistic two-body

interaction for the energy per particle

• Very recently: Monte Carlo calculations may modify this

assessment

• Also: for other observables no such statements can be made

• Still nuclear matter saturation problem!

QMPT 540

Possible solutions

• Include three-body interactions: inevitable on account of isobar

– Simplest diagram: space of nucleons -> 3-body force

– Inclusion in nuclear matter still requires phenomenology to get

saturation right

– Also needed for few-body nuclei; there is some incompatibility

• Include aspects of relativity

– Dirac-BHF approach: ad hoc adaptation of BHF to nucleon spinors

– Physical effect: coupling to scalar-isoscalar meson reduced with density

– Antiparticles? Dirac sea? Three-body correlations?

– Spin-orbit splitting in nuclei OK

– Nucleons less correlated with higher density? (compare liquid 3He)

QMPT 540

Personal perspective

• Based on results from (e,e’p) reactions as discussed here

– nucleons are dressed (substantially) and this should be included in the

description of nuclear matter (depletion, high-momentum components in

the ground state, propagation w.r.t. correlated ground state <--> BHF?)

– SRC dominate actual value of saturation density

• from 208Pb charge density: 0.16 nucleons/fm3

• determined from s-shell proton occupancy at small radius

• occupancy determined mostly by SRC (see next chapter)

– Result for SCGF of ladders

• Ghent discrete approach

• St. Louis gaussians (Libby Roth)

• ccBHF --> SCGF closer to box

• do not include LRC