Partition function of the eight-vertex model with domain wall boundary condition

We derive the recursive relations of the partition function for the eight-vertex model on an $N\times N$ square lattice with domain wall boundary condition. Solving the recursive relations, we obtain the explicit expression of the domain wall partition function of the model. In the trigonometric/rational limit, our results recover the corresponding ones for the six-vertex model.

💡 Research Summary

The paper addresses the long‑standing problem of obtaining an exact expression for the partition function of the eight‑vertex model on an $N\times N$ square lattice when the domain wall boundary condition (DWBC) is imposed. The eight‑vertex model is a natural extension of the six‑vertex (or ice‑type) model: each vertex can be in one of eight possible arrow configurations, and the Boltzmann weights are expressed in terms of elliptic (theta) functions depending on two spectral parameters $u$ and $v$ and a model parameter $\eta$. The DWBC fixes all arrows on the left and top boundaries to point inward and all arrows on the right and bottom boundaries to point outward. This special boundary condition, first introduced for the six‑vertex model, leads to remarkable algebraic simplifications and allows the partition function to be written in a determinant form (the Izergin‑Korepin determinant).

The authors begin by formulating the model precisely, writing the vertex weights $a(u,v), b(u,v), c(u,v), d(u,v)$ in terms of the elliptic sigma function $\sigma(z)$. They verify that these weights satisfy the Yang‑Baxter equation, guaranteeing integrability. With the DWBC specified, they derive a recursion relation for the partition function $Z_N({u},{v})$ that connects the $N$‑size lattice to the $(N-1)$‑size lattice. The recursion reads

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