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.

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 $$N\times N$$ lattice, with $$N^2$$ vertices and $$2N^2$$ 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 $$j$$ has an associated energy $$\epsilon_j$$ and Boltzmann weight $$w_j=e^{-\frac{\epsilon_j}{kT}}$$, giving the partition function over the lattice as

Z=\sum \exp\left(-\frac{\sum_j n_j\epsilon_j}{kT}\right) $$ 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

\begin{align} w_1=w_2&=a\\ w_3=w_4&=b\\ w_5=w_6&=c\\ w_7=w_8&=d. \end{align} $$

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 $$ \Delta'=\Delta$$ and $$ \Gamma'=\Gamma$$, for quantities

\begin{align} \Delta&=\frac{a^2+b^2-c^2-d^2}{2(ab+cd)}\\ \Gamma&=\frac{ab-cd}{ab+cd} \end{align} $$ the transfer matrices $$ 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=\operatorname{snh}(\eta-u):\operatorname{snh} (\eta +u):\operatorname{snh} (2\eta): k\operatorname{snh} (2\eta)\operatorname{snh} (\eta-u)\operatorname{snh} (\eta+u) $$ for fixed modulus $$k$$ and $$\eta$$ and variable $$u$$. Here snh is the hyperbolic analogue of sn, given by

\begin{align} \operatorname {snh} (u) &=-i\operatorname {sn} (iu) = i\operatorname {sn} (-iu) \\ \text{where } \operatorname {sn} (u)&= \frac{H(u)}{k^{1/2}\Theta(u)} \end{align} $$ and $$H(u)$$ and $$\Theta(u)$$ are Theta functions of modulus $$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 $$Q(u)$$
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

\begin{align} \zeta(u)&=[c^{-1}H(2\eta)\Theta(u-\eta)\Theta(u+\eta)]^N\\ \phi(u)&=[\Theta(0)H(u)\Theta(u)]^N. \end{align} $$

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 ($$) 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

\begin{align} f=\epsilon_5-2kT\sum_{n=1}^\infty \frac{\sinh^2((\tau-\lambda)n)(\cosh(n\lambda)-\cosh(n\alpha))}{n\sinh(2n\tau)\cosh(n\lambda)} \end{align} $$ for

\begin{align} \tau&=\frac{\pi K'}{2K}\\ \lambda&=\frac{\pi \eta}{iK}\\ \alpha&=\frac{\pi u}{iK} \end{align} $$ where $$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 $$\sigma=\pm 1$$ on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces:

\begin{align} \alpha_{ij}&=\sigma_{ij}\sigma_{i,j+1}\\ \mu_{ij}&=\sigma_{ij}\sigma_{i+1,j}. \end{align} $$



The most general form of the energy for this model is

\begin{align} \epsilon&=-\sum_{ij}(J_h\mu_{ij}+J_v\alpha_{ij}+J\alpha_{ij}\mu_{ij}+J'\alpha_{i+1,j}\mu_{ij}+J''\alpha_{ij}\alpha_{i+1,j}) \end{align} $$ where $$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 $$\mu$$, $$\alpha$$ 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 $$\sigma$$ configuration then corresponds to a unique $$\mu$$, $$\alpha$$ configuration, whereas each  $$\mu$$, $$\alpha$$ configuration gives two choices of $$\sigma$$ configurations.

Equating general forms of Boltzmann weights for each vertex $$j$$, the following relations between the $$\epsilon_j$$ and $$J_h$$, $$J_v$$, $$J$$, $$J'$$, $$J''$$ define the correspondence between the lattice models:

\begin{align} \epsilon_1&=-J_h-J_v-J-J'-J,\quad \epsilon_2=J_h+J_v-J-J'-J\\ \epsilon_3&=-J_h+J_v+J+J'-J,\quad \epsilon_4=J_h-J_v+J+J'-J\\ \epsilon_5&=\epsilon_6=J-J'+J''\\ \epsilon_7&=\epsilon_8=-J+J'+J''. \end{align} $$

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 $$Z_I=2Z_{8V}$$ 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.