Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

Let $H_0$ be a discrete periodic Schr\"odinger operator on $\ell^2(\mathbb{Z}^d)$: $$H_0=-\Delta+V,$$ where $\Delta$ is the Laplacian and $V: \mathbb{Z}^d\to \mathbb{C}$ periodic. We prove that for any $d\geq3$, Fermi variety at every energy level irreducible (modulo periodicity). For $d=2$, we except average of potential periodicity) has most two components This sharp since $d=2$ constant $V$, $V$-level exactly also Bloch $d\geq 2$. As applications, when $V$ real-valued function, set extrema spectral band functions, edges in particular, dimension $d-2$ 3$, finite cardinality $d=2$. show $H=-\Delta +V+v$ does not have embedded eigenvalues provided $v$ decays super-exponentially.