Many-Body Theory of Synchronization by Long-Range Interactions

TL;DR

This paper develops a many-body theoretical framework to analyze synchronization in long-range coupled oscillators on a lattice, revealing how the transition behavior depends on the decay exponent of the coupling.

Contribution

It introduces a systematic perturbation theory to compute the order parameter and correlation functions for oscillators with power-law interactions, extending understanding beyond mean-field approximations.

Findings

For $oldsymbol{ ext{alpha} extless d}$, the system shows a sharp synchronization transition.

For $oldsymbol{ ext{alpha} extgreater d}$, the transition is smeared by quenched disorder.

The order parameter decays as $|oldsymbol{ ext{average} ext{ } ext{ ext{psi}}}| ext{ ext{ }} ext{ ext{proportional to}} ext{ ext{ }} g_0^2$ for $ ext{ ext{alpha} extgreater d}$.

Abstract

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r) = g_0/r^\alpha$ 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 $\epsilon = \alpha/d-1$. 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 $\Av\psi$ decays slowly with $g_0$ as $|\Av\psi| \propto g_0^2$.