2D Ising spins (black=↑, white=↓)
Tc = 2/ln(1+√2) ≈ 2.269
ξ ≈ computing...
Correlation function:
C(r) = ⟨σᵢσⱼ⟩ − ⟨σ⟩²
At Tc: C(r) ~ r⁻(d-2+η)
η ≈ 0.25 (2D Ising)
Away from Tc:
C(r) ~ e⁻ʳ/ξ / r^(d-1)/2
ξ ~ |T-Tc|⁻ν, ν=1
Structure factor:
S(k) = Σ C(r)e^(ikr)
S(k) ~ 1/(k²+ξ⁻²)
(Ornstein-Zernike)
C(r) = ⟨σᵢσⱼ⟩ − ⟨σ⟩²
At Tc: C(r) ~ r⁻(d-2+η)
η ≈ 0.25 (2D Ising)
Away from Tc:
C(r) ~ e⁻ʳ/ξ / r^(d-1)/2
ξ ~ |T-Tc|⁻ν, ν=1
Structure factor:
S(k) = Σ C(r)e^(ikr)
S(k) ~ 1/(k²+ξ⁻²)
(Ornstein-Zernike)