Renormalization Group: Applications in Statistical Physics I-II

Momentum shell RG: general structure
General choice: ζ = d−2+η
2
⇒ τ ′ = b 1/ν τ, h ′ = b (d+2−η)/2 h.
◮ Only two relevant parameters τ and h.
◮ Few marginal couplings u i → u
i ′ = ui ∗ + b −x i
u i , x i > 0.
◮ Other couplings irrelevant: v i → v
i ′ = b −y i
v i , y i > 0.
After single RG transformation:
{
f sing (τ, h, {u i }, {v i }) = b −d f sing
(b 1/ν τ, b d−ζ h, ui ∗ + u } {
i vi
})
b x ,
i b y .
i
After sufficiently many l ≫ 1 RG transformations:
)
f sing (τ, h, {u i }, {v i }) =b −ld f sing
(b l/ν τ, b l(d+2−η)/2 h, {ui ∗ }, {0} .
Choose matching condition b l |τ| ν = 1 ⇒ scaling form:
f sing (τ, h) = |τ| dν ˆf ±
(h/|τ| ν(d+2−η)/2) .
Correlation function scaling law: use b l = ξ/ξ ± ⇒
(
C(τ, x, {u i }, {v i }) = b −2lζ C b l/ν τ, x )
b l , {u∗ i }, {0}
→ ˜C ± (x/ξ)
|x| d−2+η .