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).

Eightvertex2

We consider a ${\displaystyle N\times N}$ lattice, with ${\displaystyle N^{2}}$ vertices and ${\displaystyle 2N^{2}}$ 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 ${\displaystyle j}$ has an associated energy ${\displaystyle \epsilon _{j}}$ and Boltzmann weight ${\displaystyle w_{j}=e^{-{\frac {\epsilon _{j}}{kT}}}}$, giving the partition function over the lattice as

${\displaystyle Z=\sum \exp \left(-{\frac {\sum _{j}n_{j}\epsilon _{j}}{kT}}\right)}$

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

{\displaystyle {\begin{aligned}w_{1}=w_{2}&=a\\w_{3}=w_{4}&=b\\w_{5}=w_{6}&=c\\w_{7}=w_{8}&=d.\end{aligned}}}

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 ${\displaystyle \Delta '=\Delta }$ and ${\displaystyle \Gamma '=\Gamma }$, for quantities

{\displaystyle {\begin{aligned}\Delta &={\frac {a^{2}+b^{2}-c^{2}-d^{2}}{2(ab+cd)}}\\\Gamma &={\frac {ab-cd}{ab+cd}}\end{aligned}}}

the transfer matrices ${\displaystyle T}$ and ${\displaystyle T'}$ (associated with the weights ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, ${\displaystyle d}$ and ${\displaystyle a'}$, ${\displaystyle b'}$, ${\displaystyle c'}$, ${\displaystyle d'}$) commute. Using the star-triangle relation, Baxter reformulated this condition as equivalent to a parametrisation of the weights given as

${\displaystyle a:b:c:d=\operatorname {snh} (\eta -u):\operatorname {snh} (\eta +u):\operatorname {snh} (2\eta ):k\operatorname {snh} (2\eta )\operatorname {snh} (\eta -u)\operatorname {snh} (\eta +u)}$

for fixed modulus ${\displaystyle k}$ and ${\displaystyle \eta }$ and variable ${\displaystyle u}$. Here snh is the hyperbolic analogue of sn, given by

{\displaystyle {\begin{aligned}\operatorname {snh} (u)&=-i\operatorname {snh} (iu)\\{\text{where }}\operatorname {snh} (u)&={\frac {H(u)}{k^{1/2}\Theta (u)}}\end{aligned}}}
${\displaystyle T(u)T(v)=T(v)T(u).}$

The matrix function ${\displaystyle Q(u)}$

The other crucial part of the solution is the existence of a nonsingular matrix-valued function ${\displaystyle Q}$, such that for all complex ${\displaystyle u}$ the matrices ${\displaystyle Q(u),Q(u')}$ commute with each other and the transfer matrices, and satisfy Template:NumBlk where

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

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

{\displaystyle {\begin{aligned}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{aligned}}}

for

{\displaystyle {\begin{aligned}\tau &={\frac {\pi K'}{2K}}\\\lambda &={\frac {\pi \eta }{iK}}\\\alpha &={\frac {\pi u}{iK}}\end{aligned}}}

where ${\displaystyle K}$ and ${\displaystyle K'}$ are the complete elliptic integrals of moduli ${\displaystyle k}$ and ${\displaystyle k'}$. 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 ${\displaystyle \sigma =\pm 1}$ on faces of a square lattice. The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces:

{\displaystyle {\begin{aligned}\alpha _{ij}&=\sigma _{ij}\sigma _{i,j+1}\\\mu _{ij}&=\sigma _{ij}\sigma _{i+1,j}.\end{aligned}}}

The most general form of the energy for this model is

{\displaystyle {\begin{aligned}\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{aligned}}}

where ${\displaystyle J_{h}}$, ${\displaystyle J_{v}}$, ${\displaystyle J}$, ${\displaystyle J'}$ describe the horizontal, vertical and two diagonal 2-spin interactions, and ${\displaystyle J''}$ 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 ${\displaystyle \mu }$, ${\displaystyle \alpha }$ 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 ${\displaystyle \sigma }$ configuration then corresponds to a unique ${\displaystyle \mu }$, ${\displaystyle \alpha }$ configuration, whereas each ${\displaystyle \mu }$, ${\displaystyle \alpha }$ configuration gives two choices of ${\displaystyle \sigma }$ configurations.

Equating general forms of Boltzmann weights for each vertex ${\displaystyle j}$, the following relations between the ${\displaystyle \epsilon _{j}}$ and ${\displaystyle J_{h}}$, ${\displaystyle J_{v}}$, ${\displaystyle J}$, ${\displaystyle J'}$, ${\displaystyle J''}$ define the correspondence between the lattice models:

{\displaystyle {\begin{aligned}\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 _{2}=J_{h}-J_{v}+J+J'-J''\\\epsilon _{5}&=\epsilon _{6}=J-J'+J''\\\epsilon _{7}&=\epsilon _{8}=-J+J'+J''.\end{aligned}}}

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 ${\displaystyle Z_{I}=2Z_{8V}}$ 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.