Many-Body Theory of Synchronization by Long-Range Interactions

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r)=g_0\mathrm{/}{r}^{\alpha }$ and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of $ϵ=\alpha \mathrm{/}d-1$. For $\alpha \le d$, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For $\alpha >d$, the transition is smeared by the quenched disorder, and the macroscopic order parameter $\text{Av}\psi$ decays slowly with $g_0$ as $|\Av\psi| \propto g_0^2$.