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In a interaction-round-a-face model of $n^2$ particles in a lattice, a weight $W(a,b,c,d)$ is assigned to each face in the lattice based on the spins $a,b,c,d$ (listed say from the bottom-left corner in counter-clockwise fashion) of the particles at the corners of the face. Based on this, a partition function is formed $Z=\sum\prod W(a,b,c,d)$, where the the sum is over all possible spins of all particles and the product is over all the faces in the lattice. The well-known text Exactly Solved Models in Statistical Mechanics by Baxter shows how to obtain exact solutions for $Z$ in a couple cases, like the six-vertex model and the eight-vertex model, using "commuting transfer matrices." In reading Baxter's book, I am uncertain of the applicability of this method. For instance, I am working on a particular application where particles in a square lattice can have either up spin (+1) or down spin (-1) and the weight of any face $W(a,b,c,d)=1$ unless all four spins around the face alternate, in which case $W(1,-1,1,-1)=W(-1,1,-1,1)=4$. (The boundary condition here is that the all the spins on the boundary must be up.)

Does anyone know if the commuting-transfer-matrix method (or any other method) should yield an exact solution for this particular weighting of the spins? More generally, should the commuting-transfer-matrix method yield exact solutions for any weighting of the spins (even in the two-spin case)?

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