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.

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.

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 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.

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.

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.