Renormalization Group: Applications in Statistical Physics I-II

Vertex functions
⇒ connected Feynman diagrams:
u
u
Dyson equation:
+ +
⇒ propagator self-energy:
u
u
u
+
= + Σ + Σ Σ +...
= + Σ
C(q) −1 = C 0 (q) −1 − Σ(q).
Generating functional for vertex functions, Φ α = δ ln Z[h]/δh α :
∫
Γ[Φ] = − ln Z[h] + d d x ∑ h α Φ α , Γ (N)
{α i } = ∏
N δΓ[Φ]
δΦ α ∣ ;
i h=0
α
i
〈 4∏
4∏
⇒ Γ (2) (q) = C(q) −1 , S(q i )
〉c = − C(q i ) Γ (4) ({q i })
i=1
⇒ one-particle irreducible Feynman graphs.
Perturbation series in nonlinear coupling u ⇔ loop expansion.
i=1
u
u