Long-Range Order in Coupled $D$-dimensional Kuramoto Oscillators

We show that the long-range order (LRO) strikingly emerges in systems of locally coupled $D$-dimensional vector Kuramoto oscillators on low-dimensional lattices ($d=1,2$), but only for odd $D$. This parity-dependent effect is traced to two-oscillator dynamics, where odd-$D$ units synchronize for any coupling, while even-$D$ pairs require a finite threshold. This fundamental difference selectively seeds collective order in large-scale systems, a phenomenon demonstrated by our numerical simulations. A renormalization group analysis reveals a RG flow to a weak-coupling fixed point for $d \le 2$. In this limit, odd-$D$ systems effectively map to a ferromagnetic model, developing an ordered ``hemisphere" phase, whereas even-$D$ systems remain disordered. Our findings further reveal orientational LRO emerges in both $d=1$ and $d=2$, but frequency LRO requires $d=2$. We contrast these results with the established behavior of models possessing continuous symmetry, highlighting how quenched disorder provides a fundamentally new route to order.