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Synchronization with annealed disorder and higher-harmonic interactions in arbitrary dimensions: When two dimensions are special

Rupak Majumder* and Shamik Gupta†

  • *Contact author: rupak.majumder@tifr.res.in
  • †Contact author: shamik.gupta@theory.tifr.res.in
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APS Open Sci. 1, 000069 – Published 21 July, 2026

DOI: https://doi.org/10.1103/cqwb-j1xy

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. In 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 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.

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APS Open Sci. 1, 000069– Published 21 July, 2026
Vol. 1
  • Received 15 January 2026
  • Accepted 26 June 2026
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