This is an ongoing project.  Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+v_0,$$

where $-\Delta$ is the discrete Laplacian and $v_0$ is periodic in the sense that it is well defined on  $\Z^d/q_1\Z\oplus q_2 \Z\oplus\cdots\oplus q_d\Z$. For $d=2$, we tentatively proved that the Fermi variety $F_{\lambda}(v_0)/\Z^2$ is irreducible except for one value of  $\lambda$. We also construct a non-constant periodic function $v_0$ such that its Fermi variety is reducible for  some $\lambda$, which disproves a conjecture by  Gieseker, Kn\"orrer and Trubowitz.

Under some assumptions of irreducibility of Fermi variety $F_{\lambda}(v_0)/\Z^d$, we show that $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays  exponentially. The assumptions are conjectured to be true for any periodic function $v_0$. As an application, we show that when $d=2$, $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays exponentially.