 # Eight vertex model

Updated on 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.

## 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 N × N lattice, with N 2 vertices and 2 N 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 has an associated energy ϵ j and Boltzmann weight w j = e ϵ j k T , giving the partition function over the lattice as

Z = exp ( j n j ϵ j k T )

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

w 1 = w 2 = a w 3 = w 4 = b w 5 = w 6 = c w 7 = w 8 = d .

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 Δ = Δ and Γ = Γ , for quantities

Δ = a 2 + b 2 c 2 d 2 2 ( a b + c d ) Γ = a b c d a b + c d

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

a : b : c : d = snh ( η u ) : snh ( η + u ) : snh ( 2 η ) : k snh ( 2 η ) snh ( η u ) snh ( η + u )

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

snh ( u ) = i snh ( i u ) where  snh ( u ) = H ( u ) k 1 / 2 Θ ( u )

and H ( u ) and Θ ( u ) are Jacobi elliptic functions of modulus k . The associated transfer matrix T thus is a function of u alone; for all u , v

T ( u ) T ( v ) = T ( v ) T ( u ) .

## The matrix function Q ( u ) {displaystyle Q(u)}

The other crucial part of the solution is the existence of a nonsingular matrix-valued function Q , such that for all complex u the matrices Q ( u ) , Q ( u ) commute with each other and the transfer matrices, and satisfy

where

ζ ( u ) = [ c 1 H ( 2 η ) Θ ( u η ) Θ ( u + η ) ] N ϕ ( u ) = [ Θ ( 0 ) H ( u ) Θ ( u ) ] N .

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 (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

f = ϵ 5 2 k T n = 1 sinh 2 ( ( τ λ ) n ) ( cosh ( n λ ) cosh ( n α ) ) n sinh ( 2 n τ ) cosh ( n λ )

for

τ = π K 2 K λ = π η i K α = π u i K

where K and K are the complete elliptic integrals of moduli k and k . 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 σ = ± 1 on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces:

α i j = σ i j σ i , j + 1 μ i j = σ i j σ i + 1 , j .

The most general form of the energy for this model is

ϵ = i j ( J h μ i j + J v α i j + J α i j μ i j + J α i + 1 , j μ i j + J α i j α i + 1 , j )

where J h , J v , J , J describe the horizontal, vertical and two diagonal 2-spin interactions, and J 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 μ , α 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 σ configuration then corresponds to a unique μ , α configuration, whereas each μ , α configuration gives two choices of σ configurations.

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

ϵ 1 = J h J v J J J , ϵ 2 = J h + J v J J J ϵ 3 = J h + J v + J + J J , ϵ 2 = J h J v + J + J J ϵ 5 = ϵ 6 = J J + J ϵ 7 = ϵ 8 = J + J + J .

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 Z I = 2 Z 8 V 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.

## References

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