低维超立方晶格中的扭曲态
Twisted states in low-dimensional hypercubic lattices
摘要：
在环中观察到由正弦耦合的全同振子组成的非零缠绕数的扭曲态。在这些状态下，每个振荡器的相位不断地移动，顺时针方向跟随其前一个相邻振荡器，并且在$2π$上围绕环的这种相移的总和表征每个状态的绕组数。在这项工作中，我们考虑有限大小的$d$ -维超立方晶格，即正方形（$d=2$）和立方（$d=3$）晶格周期边界条件。对于相同的振荡器，我们观察到新的状态，其中属于$d=2$（$d=3$）的每条线（平面）的振荡器是相位同步的非零绕组数沿垂直方向。如果我们把相位同步振荡器的每个子集看作一个单独的振荡器，这些态可以在具有相同绕组数的环中被简化为扭曲态。对于非均匀固有频率的非全同振子，我们观察到类似的模式，在每条线$(d=2)$和平面$(d=3)$中具有轻微的非均匀相位。我们发现，这些国家一般出现随机配置时，全球耦合强度大于临界值的状态。
英文摘要：
Twisted states with non-zero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2π$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators belonging to each line (plane) for $d=2$ ($d=3$) are phase synchronized with non-zero winding numbers along the perpendicular direction. These states can be reduced into twisted states in a ring with the same winding number if we regard each subset of phase-synchronized oscillators as one single oscillator. For nonidentical oscillators with heterogeneous natural frequencies, we observe similar patterns with slightly heterogeneous phases in each line $(d=2)$ and plane $(d=3)$. We show that these states generally appear for random configurations when the global coupling strength is larger than the critical values for the states.
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