Eight-vertex model

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

File:Eightvertex2.png

We consider a $N\backslash times\; N$ 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 $\backslash epsilon\_j$ and Boltzmann weight

:$$

w_j=\exp\left(-\frac{\epsilon_j}{k_\mathrm{B}T}\right)

giving the partition function over the lattice as

:$$

Z=\sum \exp\left(-\frac{\sum_j n_j\epsilon_j}{k_\mathrm{B}T}\right)

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 states 1 and 2, and 3 and 4, consequently must occur as pairs. The vertices may 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 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.

The proof relies on the fact that when $\backslash Delta\text{'}=\backslash Delta$ and $\backslash Gamma\text{'}=\backslash Gamma$, 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 $T\text{'}$ (associated with the weights $a$, $b$, $c$, $d$ and $a\text{'}$, $b\text{'}$, $c\text{'}$, $d\text{'}$) commute. Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrization 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 $\backslash eta$ 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)}{\sqrt{k} \Theta(u)}

\end{align}

and $H(u)$ and $\backslash Theta(u)$ 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 other crucial part of the solution is the existence of a nonsingular matrix-valued function $Q$, such that for all complex $u$ the matrices $Q(u),\; Q(u\text{'})$ commute with each other and the transfer matrices, and satisfy

{{NumBlk|:|$\backslash zeta(u)T(u)Q(u)=\backslash phi(u-\backslash eta)Q(u+2\backslash eta)+\backslash phi(u+\backslash eta)Q(u-2\backslash eta)$|{{EquationRef|1}}}}

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.

The commutation of matrices in ({{EquationNote|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

:$$

\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 $K\text{'}$ are the complete elliptic integrals of moduli $k$ and $k\text{'}$.

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 states of this model are spins $\backslash sigma=\backslash pm\; 1$ 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}

Isingduallattice

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 $J\_h$, $J\_v$, $J$, $J\text{'}$ describe the horizontal, vertical and two diagonal 2-spin interactions, and $J\text{'}\text{'}$ describes the 4-spin interaction between four faces at a vertex; the sum is over the whole lattice.

Isinginteractions

We denote horizontal and vertical spins (arrows on edges) in the eight-vertex model $\backslash mu$, $\backslash alpha$ 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 $\backslash sigma$ configuration then corresponds to a unique $\backslash mu$, $\backslash alpha$ configuration, whereas each $\backslash mu$, $\backslash alpha$ configuration gives two choices of $\backslash sigma$ configurations.

Equating general forms of Boltzmann weights for each vertex $j$, the following relations between the $\backslash epsilon\_j$ and $J\_h$, $J\_v$, $J$, $J\text{'}$, $J\text{'}\text{'}$ 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.

• Six-vertex model
• Transfer-matrix method
• Ising model

{{Reflist}}

• {{Citation | last1=Baxter | first1=Rodney J. | title=Exactly solved models in statistical mechanics | url=http://physics.anu.edu.au/theophys/_files/Exactly.pdf | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | isbn=978-0-12-083180-7 | mr=690578 | year=1982 | access-date=2012-08-12 | archive-date=2021-04-14 | archive-url=https://web.archive.org/web/20210414063635/https://physics.anu.edu.au/theophys/_files/Exactly.pdf | url-status=dead }}

Category:Exactly solvable models

Category:Statistical mechanics

Category:Lattice models