The 2D Ising model has a remarkable self-duality. High-temperature expansion of one lattice maps onto the low-temperature expansion of the dual lattice.
The duality relation: sinh(2J/k_BT) · sinh(2J/k_BT*) = 1
If the model has a unique critical point T_c, it must be the fixed point of this map: sinh(2J/k_BT_c) = 1, giving T_c = 2J/ln(1+√2) ≈ 2.269J/k_B.
Left: original lattice with spins ±1. Right: dual lattice with the dual temperature T*. At T_c both are identical (self-dual point).
Magnetization order parameter M(T) maps to disorder parameter μ(T*). The duality exchanges ordered and disordered phases — a deep symmetry that Onsager used to solve the model exactly in 1944.