Ising Model Duality

Kramers-Wannier duality: high-T ↔ low-T Ising model via dual lattice

Kramers-Wannier (1941): the 2D Ising partition function satisfies Z(K) ∝ Z(K*) where tanh(K*) = e^{−2K} — mapping high-temperature to low-temperature. The self-dual point kTc/J = ½ln(1+√2) ≈ 1.134 (exact critical temperature). Dual lattice vertices sit at original lattice face centers; dual bonds cross original bonds.