Synchronization with Annealed Disorder and Higher-Harmonic Interactions in Arbitrary Dimensions: When Two Dimensions Are Special

TL;DR

This paper investigates how annealed disorder and higher-harmonic interactions influence synchronization transitions in high-dimensional Kuramoto models, revealing that annealed disorder removes odd-even dimensional effects and introduces a new correlation-driven transition, with two dimensions playing a special role.

Contribution

The study develops an arbitrary dimensional center-manifold framework for Kuramoto models with annealed disorder and higher-harmonic interactions, uncovering new universal and dimension-specific synchronization phenomena.

Findings

Annealed disorder removes the odd-even dimensional dichotomy in synchronization transitions.

Higher-harmonic interactions allow tuning between continuous and discontinuous transitions.

A new correlation-driven transition occurs, especially notable in two dimensions.

Abstract

The impact of disorder on collective phenomena depends crucially on whether it is quenched or annealed. In synchronization problems, quenched disorder in higher dimensional Kuramoto models is known to produce unconventional dimensional effects, including a striking odd even dichotomy: synchronization transitions are continuous in even dimensions and discontinuous in odd dimensions. By contrast, the impact of annealed disorder has received comparatively little attention. Here we study a D dimensional Kuramoto model with both fundamental and higher-harmonic interactions under annealed disorder, and develop an arbitrary dimensional center-manifold framework to analyze the nonlinear dynamics near the onset of collective behavior. We show that annealed disorder fundamentally alters the role of dimensionality. With fundamental coupling alone, it completely removes the odd even dichotomy, yielding continuous synchronization transitions with universal mean-field scaling in all dimensions. Higher-harmonic interactions preserve this universality while rendering the synchronization transition tunable between continuous and discontinuous. At the same time, they give rise to a novel, correlation-driven transition between a symmetry-protected incoherent phase and a symmetry broken state lacking global synchronization, which is therefore invisible to the conventional Kuramoto order parameter. This transition is continuous in two dimensions but discontinuous in higher dimensions, revealing an emergent and previously-unrecognized special role of two dimensions.