Six-vertex model

Example

Let $\Lambda$ be the $N \times N$ square grid on a torus, with $N^2$ vertices and $2N^2$ edges, and let $a, b, c > 0$. A configuration orients every edge so that each vertex has exactly two incoming and two outgoing edges, the ice rule. There are $\binom{4}{2} = 6$ ways to orient the four edges at a vertex under this rule. They come in three pairs related by reversing all arrows. In types 1 and 2 the arrows point right and up, or left and down. In types 3 and 4 they point right and down, or left and up. In types 5 and 6 both horizontal arrows point into the vertex and both vertical arrows point out, or the reverse. In the symmetric zero-field case these pairs get the weights $a$, $b$ and $c$. The six-vertex model is the probability measure on such orientations proportional to the product of the vertex weights. Its partition function is $$Z_N = \sum_{\text{ice configurations}} a^{n_1 + n_2} b^{n_3 + n_4} c^{n_5 + n_6},$$ where $n_k$ is the number of vertices of type $k$.