Spectral dimension determines criticality in nonreciprocal phase oscillators

Abstract

Spectral dimension is a key determinant of critical phenomena, but its role in nonreciprocal systems remains unexplored. We study noisy identical Kuramoto-Sakaguchi oscillators with phase lag $α\in [0,π/2)$, where $α>0$ induces nonreciprocal interactions. Numerical phase diagrams in the $(d_s,α)$ plane in complex networks, where $d_s$ denotes the spectral dimension, reveal a critical phase lag $α_c$, below which spontaneous synchronization occurs. This critical phase lag appears only for $d_s > d_s^c$, where $d_s^c = 2$ is the critical spectral dimension, and increases monotonically with $d_s$. We analytically derive the criticality using the dynamical renormalization group method.