In the **six-vertex model**, edges in a square lattice are oriented so that the in-degree of each vertex is exactly two. This gives six types of allowable vertices:
$$\begin{array}{cccccc}
\begin{array}{ccc}
& \uparrow & \\
\leftarrow & \bullet & \leftarrow \\
& \uparrow & \\
\end{array}
& \begin{array}{ccc}
& \downarrow & \\
\rightarrow & \bullet & \rightarrow \\
& \downarrow & \\
\end{array}
& \begin{array}{ccc}
& \uparrow & \\
\rightarrow & \bullet & \rightarrow \\
& \uparrow & \\
\end{array}
& \begin{array}{ccc}
& \downarrow & \\
\leftarrow & \bullet & \leftarrow \\
& \downarrow & \\
\end{array}
& \begin{array}{ccc}
& \downarrow & \\
\leftarrow & \bullet & \rightarrow \\
& \uparrow & \\
\end{array}
&\begin{array}{ccc}
& \uparrow & \\
\rightarrow & \bullet &\leftarrow \\
& \downarrow & \\
\end{array} \\
[z]&[z]&[az]&[az]&z&z^{-1}
\end{array}$$
If a vertex has label $z$ (along with a global parameter $a$), the vertex is assigned the weight listed above, where $[z]$ means $\frac{z-z^{-1}}{a-a^{-1}}$. As usual, the partition function $Z$ of a lattice is defined to be the sum (over all allowable orientations of the edges) of the product of the individual weights of all the vertices. In a $n\times n$ lattice with column parameters $x_1,\dots,x_n$ and row parameters $y_1,\dots,y_n$, the label of the vertex in row $i$ and column $j$ is $z_{ij}=x_i/y_j$. Then, the Izergin-Korepin determinant gives an exact formula for the partition function of a lattice in the six-vertex model
$$Z=\frac{\textstyle\prod_{i=1}^n x_i/y_i \prod_{1\le i,j\le n}[z_{ij}][az_{ij}]}{\textstyle \prod_{1\le i<j\le n}[x_i/x_j][y_j/y_i]}\det\left([z_{ij}]^{-1}[az_{ij}]^{-1}\right).$$ This assumes domain wall boundary conditions, where edges on the left and right are all in and edges are the top and bottom are all out. Kuperberg (1996) famously used the Izergin-Korepin determinant to prove the alternating sign matrix conjecture.

In the **eight-vertex model**, vertices with in-degree 0 or 4 are also allowed.

$$ \begin{array}{cc} \begin{array}{ccc} & \uparrow & \\ \leftarrow & \bullet & \rightarrow \\ & \downarrow & \\ \end{array} & \begin{array}{ccc} & \downarrow& \\ \rightarrow& \bullet & \leftarrow \\ & \uparrow & \\ \end{array} \\ \text{source} & \text{sink} \end{array} $$ Question: Is there an assignment for the weights of a sink and source (perhaps with other global parameters) with any type of boundary conditions so that the partition function for a lattice in the eight-vertex model can be similarly represented by a determinant? or, for that matter, any exact formula?