Long-Range Order in Coupled $D$-dimensional Kuramoto Oscillators
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908996770725888 |
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| author | Qiu, Zhongpu Wu, Tianyi Zhang, Linkai Fang, Sheng Meng, Jun Fan, Jingfang Chaté, Hugues |
| author_facet | Qiu, Zhongpu Wu, Tianyi Zhang, Linkai Fang, Sheng Meng, Jun Fan, Jingfang Chaté, Hugues |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22151 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-Range Order in Coupled $D$-dimensional Kuramoto Oscillators Qiu, Zhongpu Wu, Tianyi Zhang, Linkai Fang, Sheng Meng, Jun Fan, Jingfang Chaté, Hugues Statistical Mechanics 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. |
| title | Long-Range Order in Coupled $D$-dimensional Kuramoto Oscillators |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2604.22151 |