Criticality of nonreciprocal phase oscillators with long-range interactions

We study noisy identical Kuramoto-Sakaguchi oscillators with phase lag $\alpha \in [0,\pi \mathrm{/}2)$, where $\alpha >0$ induces nonreciprocal interactions. Numerical phase diagrams in the $(\sigma ,\alpha )$ plane in fully-connected graphs, formed by long-range weights tuned by $\sigma$ on $d$-dimensional lattices, reveal a critical phase lag ${\alpha }_c$, below which spontaneous synchronization occurs. This critical phase lag decreases monotonically with $\sigma$. We characterize the critical behavior analytically using the dynamical renormalization group theory.