T = — | T* = — | Self-dual at sinh(2J/kT)·sinh(2J/kT*)=1
Kramers-Wannier duality (1941): the 2D Ising partition function at temperature T equals (up to a factor) the partition function at a dual temperature T* satisfying:
sinh(2J/kT) · sinh(2J/kT*) = 1
The duality maps each bond of the original lattice to a site of the dual lattice, transforming low-T ordered spins into high-T disordered domain walls. The self-dual point T = T* gives sinh(2J/kT_c) = 1, so kT_c/J = 2/ln(1+√2) ≈ 2.269 — exact critical temperature.
This elegant argument predates the exact Onsager solution and requires only the assumption that there is a unique transition.
The duality maps each bond of the original lattice to a site of the dual lattice, transforming low-T ordered spins into high-T disordered domain walls. The self-dual point T = T* gives sinh(2J/kT_c) = 1, so kT_c/J = 2/ln(1+√2) ≈ 2.269 — exact critical temperature.
This elegant argument predates the exact Onsager solution and requires only the assumption that there is a unique transition.