Renormalization Group: Applications in Statistical Physics I-II

Scaling hypothesis for correlation function
Scaling ansatz, defines Fisher exponent η and correlation length ξ:
C(τ, q) = |q| −2+η Ĉ ± (qξ) , ξ = ξ ± |τ| −ν .
◮ Thermodynamic susceptibility:
χ(τ, q = 0) ∝ ξ 2−η ∝ |τ| −ν(2−η) = |τ| −γ , γ = ν(2 − η) .
◮ Spatial correlations for x → ∞:
C(τ, x) = |x| −(d−2+η) ˜C± (x/ξ) ∝ ξ −(d−2+η) ∝ |τ| ν(d−2+η) .
〈S(x)S(0)〉 → 〈S〉 2 = φ 2 ∝ (−τ) 2β
⇒ hyperscaling relations:
β = ν (d − 2 + η) , 2 − α = dν .
2
Mean-field values: ν = 1 2 , η = 0 (Ornstein–Zernicke).
Diverging spatial correlations induce thermodynamic singularities !