# 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,^{[1]} and Fan & Wu,^{[2]} and solved by Baxter in the zero-field case.^{[3]}

## Description

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 lattice, with vertices and 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 has an associated energy and Boltzmann weight , giving the partition function over the lattice as

where the summation is over all allowed configurations of vertices in the lattice. In this general form the partition function remains unsolved.

## Solution in the zero-field case

The zero-field case of the model corresponds physically to the absence of external electric fields. Hence, the model remains unchanged under the reversal of all arrows; the states 1 and 2, and 3 and 4, consequently must occur as pairs. The vertices can be assigned arbitrary weights

The solution is based on the observation that rows in transfer matrices commute, for a certain parametrisation of these four Boltzmann weights. This came about as a modification of an alternate solution for the six-vertex model; it makes use of elliptic theta functions.

### Commuting transfer matrices

The proof relies on the fact that when and , for quantities

the transfer matrices and (associated with the weights , , , and , , , ) commute. Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrisation of the weights given as

for fixed modulus and and variable . Here snh is the hyperbolic analogue of sn, given by

and and are Jacobi elliptic functions of modulus . The associated transfer matrix thus is a function of alone; for all ,

### The matrix function

The other crucial part of the solution is the existence of a nonsingular matrix-valued function , such that for all complex the matrices commute with each other and the transfer matrices, and satisfy Template:NumBlk where

The existence and commutation relations of such a function are demonstrated by considering pair propagations through a vertex, and periodicity relations of the theta functions, in a similar way to the six-vertex model.

### Explicit solution

The commutation of matrices in (Template:EquationNote) allow them to be diagonalised, and thus eigenvalues can be found. The partition function is calculated from the maximal eigenvalue, resulting in a free energy per site of

for

where and are the complete elliptic integrals of moduli and . The eight vertex model was also solved in quasicrystals.

## Equivalence with an Ising model

There is a natural correspondence between the eight-vertex model, and the Ising model with 2-spin and 4-spin nearest neighbour interactions. The states of this model are spins on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces:

The most general form of the energy for this model is

where , , , describe the horizontal, vertical and two diagonal 2-spin interactions, and describes the 4-spin interaction between four faces at a vertex; the sum is over the whole lattice.

We denote horizontal and vertical spins (arrows on edges) in the eight-vertex model , respectively, and define up and right as positive directions. The restriction on vertex states is that the product of four edges at a vertex is 1; this automatically holds for Ising 'edges'. Each configuration then corresponds to a unique , configuration, whereas each , configuration gives two choices of configurations.

Equating general forms of Boltzmann weights for each vertex , the following relations between the and , , , , define the correspondence between the lattice models:

It follows that in the zero-field case of the eight-vertex model, the horizontal and vertical interactions in the corresponding Ising model vanish.

These relations gives the equivalence between the partition functions of the eight-vertex model, and the 2,4-spin Ising model. Consequently a solution in either model would lead immediately to a solution in the other.

## See also

## Notes

## References

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