Spectral Curve of Periodic Fisher Graphs

We study the spectral curves of dimer models on periodic Fisher graphs, obtained from a ferromagnetic Ising model on $\mathbb{Z}^2$. The spectral curve is defined by the zero locus of the determinant of a modified weighted adjacency matrix. We prove that either they are disjoint from the unit torus ($\mathbb{T}^2=\{(z,w):|z|=1,|w|=1\}$) or they intersect $\mathbb{T}^2$ at a single real point.