Geometry- and topology-controlled synchronization phase transition on manifolds

Abstract

In this work, we explore how the geometry and topology of the underlying manifold shape the synchronization phase transition of a system. To do so, we extend the Kuramoto-Sakaguchi model from spheres to compact, connected, orientable, and homogeneous Riemannian manifolds of arbitrary dimension. Starting from the mean-field kinetic equation on the manifold, we derive a local response equation for the order parameter near the incoherent state and separate the geometric and topological contributions to the phase transition out of the incoherent state. The manifold geometry determines a coefficient $κ\left(M\right)$ to control the critical coupling for the linear loss of stability of the incoherent state. The manifold topology constrains the cubic term of the response equation through the Euler characteristic $χ\left(M\right)$. Under a local sign condition on the cubic term, topology does not allow a generic continuous or tricritical synchronization phase transition to occur when $χ\left(M\right)\neq 0$, and it imposes a non-zero net defect charge on the incipient ordered texture. When an additional local stabilization condition holds in that nonzero-Euler class, topology further selects a discontinuous phase transition. When $χ\left(M\right)=0$, topology does not impose that obstruction, so continuous, discontinuous, and tricritical local branches are all allowed. We verify these findings on representative families including hyperspheres, equal even-sphere products, complex Grassmannians, complex projective spaces, flat tori, real Stiefel manifolds, rotation groups, and unitary groups. Our framework recovers the classical hyperspherical parity law and extends it to a broad class of non-spherical state spaces.

Figures (1)

• Figure 1: Schematic summary of the present work. Starting from the generalized Kuramoto-Sakaguchi dynamics on a compact homogeneous manifold, we derive the response equation of the order parameter near the incoherent state. The linear term carries the geometric coefficient $\kappa\left(M\right)$, which fixes the critical coupling $K_{c}$ for the linear loss of stability of the incoherent state. The nonlinear term carries the topological constraint through the Euler characteristic $\chi\left(M\right)$. When $\chi\left(M\right)\neq 0$, topology does not allow generic continuous or tricritical local phase transitions to occur and enforces a non-zero net defect charge on the incipient ordered texture. If the additional local stabilization condition $\Lambda_{5}>0$ also holds in that non-zero-Euler class, topology further selects a discontinuous phase transition. When $\chi\left(M\right)=0$, topology removes that obstruction, and continuous, discontinuous, and tricritical local phase transitions all remain allowed.