Eight-vertex model

In statistical mechanics, the eight-vertex model is a generalization of the ice-type (six-vertex) models.

It was discussed by Sutherland[1] and Fan & Wu,[2] and solved by Rodney Baxter in the zero-field case.

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), sinks (7), and sources (8).

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.

and Boltzmann weight giving the partition function over the lattice as where the outer summation is over all allowed configurations of vertices in the lattice.

In this general form the partition function remains unsolved.

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 vertices may be assigned arbitrary weights The solution is based on the observation that rows in transfer matrices commute, for a certain parametrization of these four Boltzmann weights.

This came about as a modification of an alternate solution for the six-vertex model which makes use of elliptic theta functions.

, for quantities the transfer matrices

Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrization of the weights given as for fixed modulus

Here snh is the hyperbolic analogue of sn, given by and

are theta functions of modulus

The other crucial part of the solution is the existence of a nonsingular matrix-valued function

commute with each other and the transfer matrices, and satisfy 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.

The commutation of matrices in (1) 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

are the complete elliptic integrals of moduli

The eight vertex model was also solved in quasicrystals.

There is a natural correspondence between the eight-vertex model, and the Ising model with 2-spin and 4-spin nearest neighbor interactions.

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

The restriction on vertex states is that the product of four edges at a vertex is 1; this automatically holds for Ising "edges."

Equating general forms of Boltzmann weights for each vertex

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.