Kramers–Wannier duality

Remarks

The proof compares two expansions of the partition function on a large planar graph and its dual graph. The high-temperature expansion writes $Z(K) = 2^{N} (\cosh K)^{E} \sum_{G} (\tanh K)^{\abs{G}}$, summing over the edge sets $G$ in which every vertex has even degree; $N$ and $E$ are the numbers of vertices and edges. The low-temperature expansion on the dual lattice writes $Z^*(K^*) = 2 e^{K^* E} \sum_{G} e^{-2K^* \abs{G}}$ over the same sets $G$, now seen as domain walls between spins on the dual sites. The two sums agree when $\tanh K = e^{-2K^*}$; the square lattice is self-dual and has $E \approx 2N$, which gives the identity after taking logarithms and the limit. The map $K \mapsto K^*$ is a decreasing involution that exchanges high and low temperature, with a single fixed point $K_c$ satisfying $\sinh(2K_c) = 1$, that is $K_c = \frac12 \ln(1 + \sqrt 2)$.

Going deeper

For a student

Onsager's exact solution later confirmed that the singularity is indeed at the self-dual point. The same idea works for the Potts model on the square lattice, giving the self-dual point $e^{K} = 1 + \sqrt q$, and it extends to abelian lattice models on general planar graphs. In modern language it is a Fourier transform on the configuration space, and it relates order and disorder operators. For the Ising model on the triangular lattice the dual is the honeycomb lattice, and combining duality with the star–triangle relation still locates the critical point.