Quantum Theory of Many-Particle Systems, Phys. 540 Consequences

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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 G(1)(k; E)=G(0)(k; E) Vˆ 3 1 d k! i km k!m V km k!m × − ν (2π)3 # α β| | α β$ m m ! "α β # dE! (0) (0) G (k!; E!) G (k; E) × C 2π # ↑ $ • includes extra summation over external quantum numbers to facilitate coupling to total spin etc. • 2nd order (internal labeling as convenient) 1 dE dE d3q d3k 1 Σ(2)(k; E)=( 1)i2 1 2 ! − 2 2π 2π (2π)3 (2π)3 ν m m m m ! ! ! ! α "β γ δ km k! q/2m V k qm k! + q/2m ×# α − β| | − γ δ$ (0) (0) (0) G (k! + q/2; E + E )G (k! q/2; E )G (k q; E E ) × 1 2 − 2 − − 1 k qm k! + q/2m V km k! q/2m ×# − γ δ| | α − β$

QMPT 540

Self-energy in infinite system • Exact self-energy similarly from

dE# Σ∗(γ, δ; E)= γ U δ i γµ V δν G(ν,µ; E#) −" | | #− 2π " | | # C µ,ν ! ↑ " 1 dE dE + 1 2 γµ V %ν G(%, ζ; E )G(ν, ρ; E ) 2 2π 2π " | | # 1 2 ! ! ",µ,"ν,ζ,ρ,σ G(σ,µ; E + E E) ζρ Γ(E ,E ; E,E + E E) δσ × 1 2 − " | 1 2 1 2 − | #

• to Σ(k; E)= U(k) − 3 1 d k! dE! i km k!m V km k!m G(k!; E!) − ν (2π)3 " α β| | α β# 2π m m C !α β " " ↑ 1 dE dE d3q d3k 1 i2 1 2 ! − 2 2π 2π (2π)3 (2π)3 ν m m m m " " " " α !β γ δ km k! q/2m V k qm k! + q/2m ×" α − β| | − γ δ# G(k q; E E )G(k! + q/2; E + E )G(k! q/2; E ) × − − 1 1 2 − 2 k qm k! + q/2m Γ(E,E ,E ) km k! q/2m ×" − γ δ| 1 2 | α − β#

QMPT 540 Propagator • DE 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 • only magnitude of wave− vector− needed θ(k k ) θ(k k) • Noninteracting propagator G(0)(k; E)= − F + F − E ε(k)+iη E ε(k) iη !2k2 − − − • with ε(k)= + U(k) 2m • Particle spectral function 1 Im Σ(k; E) Sp(k; E)=− π (E ε(k) Re Σ(k; E))2 + (Im Σ(k; E))2 − − • Hole spectral function 1 Im Σ(k; E) Sh(k; E)= π (E ε(k) Re Σ(k; E))2 + (Im Σ(k; E))2 − − • Dispersion relation (check) εF ∞ Sp(k; E") Sh(k; E") G(k; E)= dE" + dE" ε E E" + iη E E" iη ! F − !−∞ − − QMPT 540

Other quantities • Energy per particle ε EN ν d3k F !2k2 0 = dE + E S (k; E) N 2ρ (2π)3 2m h ! !−∞ " # νk3 • Density ρ = F but also from 6π2

ν ∞ 2 ρ = 2 dkk n(k) 2π 0 ! εF • integral over momentum distribution n(k)= dE Sh(k; E) 2 2 !−∞ T ν ∞ ! k • Kinetic energy = dkk2 n(k) N 2π2ρ 2m !0 • Pressure at T=0 from density derivative of energy per particle

2 ∂(E/N) • P = ρ = ρ [ ε F ( E/N )] using E/N = E/(ρV ) ∂ρ −

• and ε F = ∂ E/ ∂ N so at saturation E/N = ε F (H-vH theorem) QMPT 540 Self-energy in the electron gas • Second-order is not appropriate for the electron gas 1 dE dE d3q d3k 1 Σ(2)(k; E)=( 1)i2 1 2 ! − 2 2π 2π (2π)3 (2π)3 ν m m m m ! ! ! ! α "β γ δ km k! q/2m V k qm k! + q/2m ×# α − β| | − γ δ$ (0) (0) (0) G (k! + q/2; E + E )G (k! q/2; E )G (k q; E E ) × 1 2 − 2 − − 1 k qm k! + q/2m V km k! q/2m ×# − γ δ| | α − β$ • Consider direct matrix element in 2nd-order self-energy

km k! qm V k qm k!m δ δ V (q) ! α − β| | − γ δ#D ⇒ mαmγ mβ mδ 4πe2 V (q)= • with q2 • Performing E2 and k’ integration and identifying noninteracting polarization propagator (including spin summations) dE d3q Σ(2)(k; E)=2i 1 V 2(q)G(0)(k q; E E )Π(0)(q; E ) D 2π (2π)3 − − 1 1 ! ! • generates “infrared” divergence due to q-4 contribution so infinite! continues in higher order! QMPT 540

Higher-order terms • Replace Π (0) by Π (0) 2 V Π (0) 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 dE d3q ΣRPA(k; E)=2i 1 V 2(q)G(0)(k q; E E )ΠRPA(q; E ) 2π (2π)3 − − 1 1 ! !

QMPT 540 Ring approximation • Add Fock term: represented by • Approximation known as G(0)W (0) • Self-consistent formulation GW • 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 0 1 ∞ Im Π(q,E") 1 Im Π(q,E") Π(q,E)= dE" + dE" −π 0 E E" + iη π E E" iη ! − !−∞ − − • RPA generates discrete poles (plasmons) • Plasmon term above Fermi energy (im part) π d3q Im ΣRP A(k; E)= θ( k q k )θ(q q) p 2 (2π)3 | − | − F c − ! 1 − ∂Π(0)(q, E˜) δ (E Ep(q) ε(k q)) ˜ QMPT 540 × − − −  ∂E $ ˜  $E=Ep(q) $  $  $

Development • Algebra somewhat lengthy • When q=0 point included, integrand requires expansion yielding a logarithmic singularity (Hedin 1967)

• From delta-function: occurs at E = E p (0) + ε ( k ) (+ for k>kF) ± • Results (only plasmon)

rs =2

• Real part singular for k=kF

QMPT 540 Continuum contribution • Employ corresponding dispersion relation E+ RP A E RP A RP A 1 Im Πc (q,E!) 1 − − Im Πc (q,E!) Πc (q,E)= dE! + dE! −π E E E! + iη π E+ E E! iη ! − − !− − − • Insert in self-energy • For energies above the Fermi energy 3 E+ RP A d q 2 RP A Im Σc (k; E) = 2 3 V (q) dE! Im Πc (q, E!) θ( k q kF )δ(E E! ε(k q)) (2π) E | − | − − − − ! ! − • similarly for energies below + RPA RPA HF ∞ Im Σ (k; E#) • Real part from Re Σ (k; E)=Σ (k)+P dE# | | π E E# !−∞ − • Continuum only • Much smaller than plasmon terms

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 !2k2 E (k)= + Re ΣRPA(k; E (k)) Q 2m Q • outside these domains -> more than one delta-function solution possible with 1 ∂Re ΣRPA(k; E) − Z (k)= 1 Q − ∂E ! "E=EQ(k)# " • Almost realized for k/kF=0.2 at rs=2 " " • Peak is referred to as the “plasmaron”

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 (0) dE! (0) (0) Π (k; q,E)= G (k + q/2; E + E!)G (k q/2; E!) 2πi − • For dressed propagators! f dE! Π (k; q,E)= G(k + q/2; E + E!)G(k q/2; E!) 2πi − • Using Lehmann! for sp propagators yields εF ˜ ˜ f ∞ Sp(k + q/2; E)Sh(k q/2; E#) Π (k; q,E)= dE˜ dE˜# − ˜ ˜ εF E (E E#)+iη ! !−∞ − − εF ∞ Sh(k + q/2; E˜)Sp(k q/2; E˜#) dE˜ dE˜# − ˜ ˜ − εF E +(E# E) iη !−∞ ! − − QMPT 540 Development • For positive energies εF f ∞ Im Π (k; q,E)= π dE˜ dE˜#Sp(k + q/2; E˜)Sh(k q/2; E˜#)δ(E (E˜ E˜#)) − ε − − − ! F !−∞ E+εF = π dES˜ (k + q/2; E˜)S (k q/2; E˜ E) − p h − − !εF • By integrating over wave vector k the dressed version of the d3k Lindhard function is generated Πf (q,E)= Πf (k; q,E) (2π)3 ! • Obeys usual dispersion relation f 0 f f 1 ∞ Im Π (q,E") 1 Im Π (q,E") Π (q,E)= dE" + dE" −π 0 E E" + iη π E E" iη ! − !−∞ − − • Plot for rs = 2 (dashed)

• rs = 4 (solid)

• noninteracting q/kF = 0.25 • Huge spreading for self-consistency

QMPT 540

Polarization propagator • As usual Πf (q, E) Π(q, E)= 1 2V (q)Πf (q, E) − • and the screened Coulomb interaction becomes 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) 2 = V (q)+2V (q)!Π(q, E) "

• Demonstrating that it has the same analytical properties as Π W (q, E) V (q)+∆W (q, E) ≡ 0 1 ∞ Im W (q, E") 1 Im W (q, E") = V (q) dE" + dE" − π 0 E E" + iη π E E" iη ! − !−∞ − −

QMPT 540 More development • Imaginary part of screened Coulomb can be written as Im W (q, E)=2V 2(q)Im Π(q, E) Im Πf (q, E) =2V 2(q) (1 2V (q) Re Πf (q, E))2 + (2V (q) Im Πf (q, E))2 − • Since ∆ W =2 V 2 Π the self-energy can be written as dE d3q Σ (k; E)=i 1 G(k q; E E )∆W (q; E ) ∆W 2π (2π)3 − − 1 1 ! ! • Insert spectral representation and perform energy integration 3 ˜ ˜ 1 d q ∞ ˜ ∞ ˜ Sp(k q; E)Im W (q, E") Σ∆W (k; E) 3 dE" dE − − π (2π) 0 ε E E˜ E˜ + iη ! ! ! F − " − 3 0 εF 1 d q Sh(k q; E˜)Im W (q, E˜") dE˜" dE˜ − 3 ˜ ˜ − π (2π) E E" E iη ! !−∞ !−∞ − − − • First term (particle domain) 3 E εF d q − Im Σ (k; E)= dE˜"S (k q; E E˜")Im W (q, E˜") ∆W (2π)3 p − − ! !0 • Hole domain 3 0 d q ˜ ˜ ˜ Im Σ∆W (k; E)= 3 dE"Sh(k q; E E")Im W (q, E") − (2π) E εF − − QMPT 540 ! ! −

Final steps • Still need generalization of HF term 3 d k! Σ (k)= V (k k!)n(k!) V − (2π)3 − ! • So GW self-energy becomes εF 1 ∞ Im Σ∆W (k; E") 1 Im Σ∆W (k; E") ΣGW (k; E)=ΣV (k) dE" + dE" − π ε E E" + iη π E E" iη ! F − !−∞ − − • with DE self-consistency loop complete • Holm and von Barth (1998) calculation employed sum of Gaussians W (k) (E E (k))2 S(k; E)= n exp − n √ − 2Γ2 (k) n 2πΓn(k) n ! " # • 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

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)

rs 1 2 4 5 10 20 QMC 0.5180 0.2742 0.1466 0.1198 0.0644 0.0344 0.5144 0.2729 0.1474 0.1199 0.0641 0.0344 GW 0.5160(2) 0.2727(5) 0.1450(5) 0.1185(5) 0.0620(9) 0.032(1) 0.2741 0.1465 GW (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 E /N 0.4582 0.2291 0.1145 0.0916 0.0458 0.0229 − x

• 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 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

f ∞ ∞ Sp(q + K/2; E")Sp(K/2 q; E"") G (K, q; E)= dE" dE"" − pphh E E E + iη !εF !εF − " − "" εF εF Sh(q + K/2; E")Sh(K/2 q; E"") dE" dE"" − − E E" E"" iη !−∞ !−∞ − − −

QMPT 540

Lehmann representation • Dispersion relation (arrows for location of poles in energy plane) km m ∆Γ (K,E) k!m m ! α α! | pphh | β β! " 1 ∞ Im kmαmα! ∆Γpphh(K,E!) k!mβmβ! = − dE! ! | | " π E E + iη !2εF − ! 2εF 1 Im kmαmα! ∆Γpphh(K,E!) k!mβmβ! + dE! ! | | " π E E! iη !−∞ − − kmαmα ∆Γ (K,E) k!mβmβ + kmαmα ∆Γ (K,E) k!mβmβ ≡! ! | ↓ | ! " ! ! | ↑ | ! " • Corresponding self-energy 3 1 d k! dE! Σ (k; E)= i G(k!; E!) ∆Γ − ν (2π)3 2π mαm !α! " " 1 (k k!)m m ∆Γ (K,E+ E!) 1 (k k!)m m ×#2 − α α! | pphh | 2 − α α! $ • Energy integral can be performed (note arrows) 1 d3k εF ! 1 1 Σ∆Γ(k; E)= dE! (k k!)mαmα ∆Γ (E + E!) (k k!)mαmα Sh(k!,E!) ν (2π)3 ! 2 − ! | ↓ | 2 − ! # mαm !α! " "−∞ 3 1 d k ∞ ! 1 1 3 dE! 2 (k k!)mαmα! ∆Γ (E + E!) 2 (k k!)mαmα! Sp(k!,E!) − ν (2π) ε ! − | ↑ | − # mαm F !α! " " ∆Σ (k; E)+∆Σ (k; E) QMPT 540 ≡ ↓ ↑ Final steps and implementation • Total self-energy εF 1 ∞ Im Σ(k; E") 1 Im Σ(k; E") Σ(k; E)=ΣV (k) dE" + dE" − π ε E E" + iη π E E" iη ! F − !−∞ − − = ΣV (k)+∆Σ (k; E)+∆Σ (k; E) ↓ ↑ • includes HF-like term 3 d k! 1 Σ (k)= 1 (k k!)m m V 1 (k k!)m m n(k!) V (2π)3 ν ! 2 − α α! | | 2 − α α! # mαm ! "α! • Practical calculations first done with mean-field sp propagators in scattering equation 1 d3k (0) ! 1 (0) 1 Σ∆Γ(k; E)= 3 2 (k k!)mαmα! ∆Γ (E + ε(k!)) 2 (k k!)mαmα! θ(kF k!) ν (2π) ! − | ↓ | − # − mαm !α! " 1 d3k ! 1 (0) 1 3 2 (k k!)mαmα! ∆Γ (E + ε(k!)) 2 (k k!)mαmα! θ(k! kF ) − ν (2π) ! − | ↑ | − # − mαm !α! "

QMPT 540

Results for mean-field input • Realistic interactions generate pairing instabilities • Avoid with “gap” is sp spectrum according to dE E S (k, E) ε(k)= h k < k dE S (k, E) F ! h dE E S (k, E) ε(k)= ! Q = E (k) k > k dE S (k, E) Q F ! Q • 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

QMPT 540 Self-energy below Fermi energy • Wave vectors 0.1 (solid), 0.51 (dotted), and 2.1 fm-1 (dashed) -1 • kF = 1.36 fm • Im part

!2k2 • Spectral functions with peaks at EQ(k)= + ReΣ(k; EQ(k)) 2m 1 Z2 (k) W (k) • QP spectral function S (k, E)= Q | | Q π (E E (k))2 +(Z (k)W (k))2 − Q Q • with W ( k ) = Im Σ ( k ; E Q ( k )) and 1 ∂Re Σ(k; E) − Z (k)= 1 Q − ∂E ! " #E=EQ(k)$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

ε • Energy sum rule EN ν d3k F !2k2 0 = dE + E S (k; E) N 2ρ (2π)3 2m h ! !−∞ " # • Contribution from high momenta after energy integral for different densities for Reid potential

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

km m G(K,E) k!m m = km m V k!m m ! α α! | | β β! " ! α α! | | β β! " 3 1 d q θ( q + K/2 kF ) θ( K/2 q kF ) + km m V qm m | | − | − | − qm m G(K,E) k!m m 2 (2π)3 ! α α! | | γ γ! " E ε(q + K/2) ε(K/2 q)+iη ! γ γ! | | β β! " mγ m !γ"! − − − • Brueckner G-matrix but Bethe-Goldstone equation… • Dispersion relation 1 ∞ Im kmαmα! ∆G(K,E!) k!mβmβ! km m G(K,E) k!m m = km m V k!m m dE! ! | | " ! α α! | | β β! " ! α α! | | β β! "−π E E + iη !2εF − ! kmαmα V k!mβmβ + kmαmα ∆G (K,E) k!mβmβ ≡! ! | | ! " ! ! | ↓ | ! " • Include HF term in “BHF” self-energy 3 d k! 1 Σ (k; E)) = θ(k k!) 1 (k k!)m m G(k + k!; E + ε(k!) 1 (k k!) m m BHF (2π)3 ν F − " 2 − α α! | | 2 − α α! # mαm ! "α! • Below Fermi energy: no imaginary part

QMPT 540 BHF

• DE for k < kF yields solutions at !2k2 ε (k)= + Σ (k; ε (k)) BHF 2m BHF BHF • 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 U s ( k )= Σ BHF ( k ; ε BHF ( k )) only for k < kF (0 above)

– Continuous U c ( k )= Σ BHF ( k ; ε BHF ( k )) all k • Only one calculation of G-matrix for standard choice • Iterations for continuous choice

QMPT 540

BHF θ(k kF ) θ(kF k) • Propagator GBHF (k; E)= − + − E εBHF (k)+iη E εBHF (k) iη • Energy − − − EA ν d3k !2k2 0 = + ε θ(k k ) A 2ρ (2π)3 2m BHF − F ! " # • Rewrite using on-shell self-energy A 3 2 2 3 3 E0 4 d k ! k 1 d k d k! = + θ(k k)θ(k k!) A ρ (2π)3 2m 2ρ (2π)3 (2π)3 F − F − mαm ! ! ! "α! 1 (k k!) m m G(k + k!; ε (k)+ε (k!)) 1 (k k!) m m " 2 − α α! | BHF BHF | 2 − α α! # • First term: kinetic energy free Fermi gas 2 HF 1 p HF • Compare E = θ(pF p) + ε (p) 2 − 2m p ! " # 1 = T + θ(p p)θ(p p!) pp! V pp! FG 2 F − F − " | | # !pp! • so BHF obtained by replacing V by G

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 -> Location of minimum determined Coester band 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

QMPT 540