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. ζ ( u ) T ( u ) Q ( u ) = ϕ ( u − η ) Q ( u + 2 η ) + ϕ ( u + η ) Q ( u − 2 η ) {displaystyle zeta (u)T(u)Q(u)=phi (u-eta )Q(u+2eta )+phi (u+eta )Q(u-2eta )}     (1) 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 N × N {displaystyle N imes N} lattice, with N 2 {displaystyle N^{2}} vertices and 2 N 2 {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 j {displaystyle j} has an associated energy ϵ j {displaystyle epsilon _{j}} and Boltzmann weight w j = e − ϵ j k T {displaystyle w_{j}=e^{-{frac {epsilon _{j}}{kT}}}} , giving the partition function over the lattice as