  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.
