Renormalization Group — Block-Spin Decimation (1D Ising)

Block-spin RG on the 1D Ising model: integrate out every other spin, get effective coupling K′ = ½ ln(cosh 2K). The RG flow has two fixed points: K*=0 (disordered) and K*=∞ (ordered). The critical point of the 2D model is visible in higher-D generalizations.

Init coupling K 0.80 Temp (β) 1.00

Left: RG flow K → K′ = ½ ln(cosh 2K). Right: block-spin coarse-graining of spin chain. The 1D Ising critical point is at K=∞ (T=0); for T>0 the system is always disordered. Each RG step halves the chain length.