  Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z} $ with $q_1\in \mathbb{Z}_+$
and $q_2\in\mathbb{Z}_+$. Let $\Delta+X$ be the discrete periodic Schr\"odinger
operator on $\mathbb{Z}^2$, where $\Delta$ is the discrete Laplacian and
$X:\mathbb{Z}^2\to \mathbb{C}$ is $\Gamma$-periodic. In this paper, we develop
tools from complex analysis to study the isospectrality of discrete periodic
Schr\"odinger operators. We prove that if two $\Gamma$-periodic potentials $X$
and $Y$ are Fermi isospectral and both $X=X_1\oplus X_2$ and $Y= Y_1\oplus Y_2$
are separable functions, then, up to a constant, one dimensional potentials
$X_j$ and $Y_j$ are Floquet isospectral, $j=1,2$. This allows us to prove that
for any non-constant separable real-valued $\Gamma$-periodic potential, the
Fermi variety $F_{\lambda}(V)/\mathbb{Z}^2$ is irreducible for any $\lambda\in
\mathbb{C}$, which partially confirms a conjecture of Gieseker, Kn\"{o}rrer and
Trubowitz in the early 1990s.
