Zigzag ordering, defects, and anomalous relaxation in antiferromagnetic Kuramoto lattices

Abstract

We investigate the nonequilibrium ordering dynamics of coupled Kuramoto oscillators with negative nearest-neighbor coupling, which induces a zigzag antiferromagnetic ordering. In one dimension, the defect density exhibits anomalously slow coarsening, decaying as $(D(t)\sim t^{-1/4})$ before saturating at a system-size-dependent time $(t_c(N)\sim N^z)$ with (z=2). The local persistence probability follows a stretched-exponential form, $(P(t)\sim \exp(-c t^\alpha))$, with $(\alpha=1/4)$. These exponents are observed are independent of the magnitude of the coupling, which merely rescales the characteristic time scale. The equality $(\alpha=\delta=1/4)$ together with (z=2) is consistent with a distinct universality class. These results demonstrate that deterministic nonlinear dynamics and geometric frustration alone are sufficient to generate slow relaxation and anomalous scaling, without quenched disorder or stochastic noise. A continuum approximation and the corresponding coarse-grained partial differential equation provide a theoretical explanation for the observed anomalous exponents, while linear stability analysis accounts for the emergence of the zigzag ordered state. In two dimensions, geometric frustration inhibits complete ordering and gives rise to long-lived metastable domain-wall structures. An initial transient defect decay is observed before crossover and saturation. These results demonstrate how frustration and continuous phase variables can fundamentally modify coarsening dynamics and generate anomalously slow relaxation in deterministic many-body systems.