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

Abstract

In this paper, we study the quasi-periodic operators $H_{ε,ω}(x)$: $$(H_{ε,ω}(x)\vecψ)_n=ε\sum_{k\in\mathbb{Z}}W_k\vecψ_{n-k}+V(x+nω)\vecψ_n,$$ where $$\vecψ=\{\vecψ_n\}\in\ell^2(\mathbb{Z},\mathbb{C}^l),\ V(x)=\text{diag}\left(v_1(x),\cdots,v_l(x)\right)$$ 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 $\|W_k\|\leq C_0e^{-ρ|k|}$. Using techniques developed by Bourgain and Goldstein [\textit{Ann. of Math. 152(3):835--879, 2000}], we show that for $|ε|\leq ε_{0}(V,ρ,l,C_0)$ ( depending only on $V,ρ, l, C_0$) and $x\in \mathbb{R}/\mathbb{Z}$, there is some full Lebesgue measure subset $\mathcal{F}$ of the Diophantine frequencies such that $H_{ε,ω}(x)$ exhibits Anderson localization if $ω\in \mathcal{F}$.