Renormalization Group

1D Ising Decimation · RG Flow Diagram · Fixed Points

K* = 0 (T=∞)
Disordered FP
K* = ∞ (T=0)
Ordered FP
K* = K_c
Critical FP
0.50
8
32
Kadanoff block spin / Decimation RG: The 1D Ising model (H = −J Σσᵢσᵢ₊₁) is exactly solvable under decimation. Integrating out every other spin gives a new effective coupling: K' = ½ ln(cosh 2K). The flow K → K' always flows to K=0 (disordered) — consistent with the fact that the 1D model has no phase transition at finite temperature. In 2D, the Ising model has a critical fixed point K_c ≈ 0.441 where the RG flow is unstable — this is the critical point. The RG philosophy: physics at different length scales is connected by these flow equations; critical phenomena are controlled by fixed points.