Spectral and dynamical localization at large disorder for the nonlocal random operators on Zd

In this paper, we study the nonlocal random operators Hω = ɛSϕ + Vω(n) acting on l2(Zd), whose free part Sϕ is a translation invariant operator with off diagonal elements decaying as e−logρ(‖n−m‖+1) with ρ > 1 and n,m∈Zd. Under the assumption that the probability distribution of Vω(n) is absolutely continuous, we prove that at large disorder (0 < ɛ ≪ 1), the spectrum of Hω is pure point for typical ω, the corresponding eigenfunctions exhibit the same decay rate as the hopping term. This provides a partial resolution to a conjecture proposed by [Yeung and Oono, Europhys. Lett. 4(9), 1061–1065 (1987)]. Moreover, the subexponential dynamical localization for Hω is also established, which implies the absence of quantum transport. Our proof is based on the fractional moment method (FMM) and Simon–Wolff criterion.