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]
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 x N
N2
2N2
j
\epsilonj
| ||||
w | ||||
j=e |
Z=\sum\exp\left(-
\sumjnj\epsilonj | |
kT |
\right)
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} w1=w2&=a\\ w3=w4&=b\\ w5=w6&=c\\ w7=w8&=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.
The proof relies on the fact that when
\Delta'=\Delta
\Gamma'=\Gamma
\begin{align} \Delta&= | a2+b2-c2-d2 | \\ \Gamma&= |
2(ab+cd) |
ab-cd | |
ab+cd |
\end{align}
T
T'
a
b
c
d
a'
b'
c'
d'
a:b:c:d=\operatorname{snh}(η-u):\operatorname{snh}(η+u):\operatorname{snh}(2η):k\operatorname{snh}(2η)\operatorname{snh}(η-u)\operatorname{snh}(η+u)
k
η
u
\begin{align} \operatorname{snh}(u)&=-i\operatorname{sn}(iu)=i\operatorname{sn}(-iu)\\ where\operatorname{sn}(u)&=
H(u) | |
k1/2\Theta(u) |
\end{align}
H(u)
\Theta(u)
k
T
u
u
v
T(u)T(v)=T(v)T(u).
Q(u)
The other crucial part of the solution is the existence of a nonsingular matrix-valued function
Q
u
Q(u),Q(u')
\begin{align} \zeta(u)&=[c-1H(2η)\Theta(u-η)\Theta(u+η)]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 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=\epsilon5-2kT\sum
infty | |
n=1 |
\sinh2((\tau-λ)n)(\cosh(nλ)-\cosh(n\alpha)) | |
n\sinh(2n\tau)\cosh(nλ) |
\end{align}
\begin{align} \tau&= | \piK' | \\ λ&= |
2K |
\piη | \\ \alpha&= | |
iK |
\piu | |
iK |
\end{align}
K
K'
k
k'
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=\pm1
\begin{align} \alphaij&=\sigmaij\sigmai,j+1\\ \muij&=\sigmaij\sigmai+1,j. \end{align}
The most general form of the energy for this model is
\begin{align} \epsilon&=-\sumij(Jh\muij+Jv\alphaij+J\alphaij\muij+J'\alphai+1,j\muij+J''\alphaij\alphai+1,j) \end{align}
Jh
Jv
J
J'
J''
We denote horizontal and vertical spins (arrows on edges) in the eight-vertex model
\mu
\alpha
\sigma
\mu
\alpha
\mu
\alpha
\sigma
Equating general forms of Boltzmann weights for each vertex
j
\epsilonj
Jh
Jv
J
J'
J''
\begin{align} \epsilon1&=-Jh-Jv-J-J'-J'', \epsilon2=Jh+Jv-J-J'-J''\\ \epsilon3&=-Jh+Jv+J+J'-J'', \epsilon4=Jh-Jv+J+J'-J''\\ \epsilon5&=\epsilon6=J-J'+J''\\ \epsilon7&=\epsilon8=-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
ZI=2Z8V