Eight-vertex model

In statistical mechanics, the eight-vertex model is a generalisation of the ice-type (six-vertex) models; it was discussed by Sutherland, and Fan & Wu, and solved by Baxter in the zero-field case.

As with the ice-type models, the eight-vertex model is a square lattice model, where each state is a configuration of arrows at a vertex. The allowed vertices have an even number of arrows pointing towards the vertex; these include the six inherited from the ice-type model (1-6), and sinks and sources (7, 8).

We consider a ${\displaystyle N\times N}$ lattice, with ${\displaystyle N^{2}}$ vertices and ${\displaystyle 2N^{2}}$ edges. Imposing periodic boundary conditions requires that the states 7 and 8 occur equally often, as do states 5 and 6, and thus can be taken to have the same energy. For the zero-field case the same is true for the two other pairs of states. Each vertex ${\displaystyle j}$ has an associated energy ${\displaystyle \epsilon _{j}}$ and Boltzmann weight ${\displaystyle w_{j}=e^{-{\frac {\epsilon _{j}}{kT}}}}$, giving the partition function over the lattice as

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