Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

In this paper, we study the quasi-periodic operators $H_{\epsilon,\omega}(x)$: $$(H_{\epsilon,\omega}(x)\vec{\psi})_n=\epsilon\sum_{k\in\mathbb{Z}}W_k\vec{\psi}_{n-k}+V(x+n\omega)\vec{\psi}_n,$$ where $$\vec{\psi}=\{\vec{\psi}_n\}\in\ell^2(\mathbb{Z},\mathbb{C}^l),\ V(x)=\text{diag}\left(v_1(x),\cdots,v_l(x)\right)$$ with $v_i$ ($1\leq i \leq l$) being real analytic functions on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $W_k$ ($k\in\mathbb{Z}$) being $l\times l$ matrices satisfying $\|W_k\|\leq C_0e^{-\rho|k|}$. Using techniques developed by Bourgain and Goldstein [\textit{{Ann. of Math. 152(3):835--879, 2000}}], we show that for $|\epsilon|\leq \epsilon_{0}(V,\rho,l,C_0)$ ( depending only on $V,\rho, l, C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{\epsilon,\omega}(x)$ exhibits Anderson localization if $\omega\in \mathcal{F}$.