Real-Space RG — 1D Ising Decimation Flow

Real-space renormalization group (decimation) for the 1D Ising model: sum over every other spin (block decimation). The exact RG transformation is K' = ½ ln(cosh(2K)), where K = J/kT is the dimensionless coupling. This is a contraction mapping — every initial K flows toward K*=0 (the high-T, disordered fixed point). Only K=∞ (T=0) is a trivial fixed point. This is why the 1D Ising model has no phase transition at finite T: there is no unstable fixed point at finite K. The RG flow diagram shows the successive images of K under repeated decimation, always flowing toward 0. Contrast with 2D Ising where there IS an unstable fixed point at K_c ≈ 0.44 (Onsager).

Renormalization group1D IsingDecimationFixed pointsUniversality