Localization in a random $x-y$ model with the long-range interaction: Intermediate case between single particle and many-body problems

Many-body localization in an $XY$ model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power law distance dependent hopping $V(R) \propto R^{-\alpha}$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\alpha < 3d/2$ where $d$ is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R) \propto R^{-2\alpha}$ of "extended" resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in $XY$ model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\alpha \geq 2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.