Renormalization Group: Applications in Statistical Physics I-II

Scaling relations
◮ Critical isotherm: τ-dependence in ˆf ′ ± must cancel prefactor,
ˆf ′ ±(x) ∝ x (2−α−∆)/∆ as x → ∞; hence
φ(0, h) ∝ h (2−α−∆)/∆ = h 1/δ , δ = ∆/β .
◮ Isothermal susceptibility:
( )
χ τ ∂φ
V = = χ ± |τ| −γ , γ = α + 2(∆ − 1) .
∂h
τ, h=0
Eliminate ∆ ⇒ scaling relations:
∆ = βδ , α + β(1 + δ) = 2 = α + 2β + γ , γ = β(δ − 1) ;
⇒ only two independent (static) critical exponents.
Mean-field: α = 0, β = 1 2 , γ = 1, δ = 3, ∆ = 3 2
(dim. analysis).
Experimental exponent values different, but still universal:
depend only on symmetry, dimension . . ., not microscopic details.