Renormalization Group: Applications in Statistical Physics I-II

Method of characteristics
Susceptibility χ(q) = Γ (2) (q) −1 :
χ R (µ, τ R , u R , q) −1 = µ 2 ˆχ R (τ R , u R , q/µ) −1 .
solve RG equation: method of characteristics
µ → µ(l) = µ l ,
[∫ l
χ R (l) −1 = χ R (1) −1 l 2 exp
1
γ S (l ′ ) dl′
l ′ ]
,
u(l)
u(1)
τ(1)
with running couplings, initial values ˜τ(1) = τ R , ũ(1) = u R :
τ(l)
l d ˜τ(l)
dl
= ˜τ(l) γ τ (l) , l dũ(l)
dl
= β u (l) .
Near infrared-stable RG fixed point: β u (u ∗ ) = 0, β ′ u(u ∗ ) > 0
˜τ(l) ≈ τ R l γ∗ τ
, χ R (τ R , q) −1 ≈ µ 2 l 2+γ∗ S ˆχR (τ R l γ∗ τ
, u ∗ , q/µ l) −1 ,
matching l = |q|/µ ⇒ scaling form with η = −γ ∗ S , ν = −1/γ∗ τ .