Migdal–Kadanoff Block Spin RG

Real-space renormalization: K' = f(K) on Ising model; fixed points & RG flow

Ising spin lattice — click to step through RG levels
RG flow: K' vs K — fixed points and flow arrows
Current K
K' after decimation
K* (fixed point)
0
RG iterations
Migdal-Kadanoff (1976): Bond-moving approximation — move bonds before decimating. For 2D Ising with b=2:
K' = ½ ln[cosh(2K²)]  (1D chain decimation after bond moving)
Fixed points: K*=0 (high-T, disordered), K*=∞ (low-T, ordered), K*_c (critical — unstable). MK gives K*_c ≈ 0.305 for d=2, b=2 (exact: 0.4407).
RG flow: K<K*_c flows to 0 (disordered); K>K*_c flows to ∞ (ordered). The eigenvalue at K*_c gives ν = 1/log_b(λ).