Real-Space Renormalization: 1D Ising Decimation

Block-spin RG transformation on the 1D Ising chain. Trace how the coupling K = J/k_BT flows under repeated decimation and converge to fixed points.

Initial K = J/k_BT0.80
RG steps8
Decimation factor b2
Chain length N64
Decimation rule (b=2): Integrate out every other spin. New coupling K' = ½ ln cosh(2K).
Fixed points: K*=0 (T=∞, disordered — stable in 1D), K*=∞ (T=0, ordered — unstable in 1D).
Key result: In 1D, K'<K for all finite K, so all trajectories flow to K*=0. No phase transition in 1D (Peierls, 1936).
2D: Non-trivial fixed point K*≈0.44 exists → phase transition at T_c (Wilson-Kadanoff RG).