AI 中文总结
研究三维手性向列模型在递归格上的有限态离散近似，分析不同离散化方案的调制吸引子结构，揭示非零手性参数驱动的临界模式选择。
AI 中文摘要
我们研究了手性向列晶格模型的有限态近似，其局部指向矢嵌入三维取向空间。本文分析的模型以旋转四极相互作用表述，该相互作用提供了直接的角度手性参数$\Delta$，并对于限制在xy平面内的指向矢简化为通常的平面手性向列形式。在根植Cayley树上，统计问题被写成分支配分函数的非线性递归。在无穷配位数极限下，固定$rJ_Q$，递归变为状态概率的softmax映射。我们分析了笛卡尔三态、平面四态和非平面四态离散化。前者支持有序和周期二吸引子，而平面四态离散化展示了一系列相称和较长周期的调制吸引子。其无序不动点在$t=9/16$处通过相位$q=\pm 2\Delta$的复共轭对失去稳定性，因此对于每个非零手性选择调制临界模式。非平面四态集本质上是各向异性的，当直接重建所述模型时，支持显著的周期三以及周期二和更高周期的循环。由于吸引子稳定性本身并不建立全局热力学稳定性，这里报告的图被解释为吸引子/稳定性图，而非平衡相图。
英文摘要
We study finite-state approximations to a chiral nematic lattice model whose local directors are embedded in three-dimensional orientational space. The model analyzed here is formulated in terms of a rotated quadrupolar interaction, which provides a direct angular chirality parameter $Δ$ and reduces to the usual planar chiral-nematic form for directors confined to the xy plane. On a rooted Cayley tree, the statistical problem is written as a nonlinear recursion for branch partition functions. In the infinite-coordination limit, with $rJ_Q$ fixed, the recursion becomes a softmax map for the state probabilities. We analyze Cartesian three-state, planar four-state, and non-planar four-state discretizations. The first supports ordered and period-two attractors, whereas the planar four-state discretization displays a sequence of commensurate and longer-period modulated attractors. Its disordered fixed point loses stability at $t=9/16$ through a complex-conjugate pair with phase $q=\pm 2Δ$, so a modulated critical mode is selected for every nonzero chirality. The non-planar four-state set is intrinsically anisotropic and, when the stated model is reconstructed directly, supports prominent period-three as well as period-two and higher-period cycles. Because attractor stability does not by itself establish global thermodynamic stability, the diagrams reported here are interpreted as attractor/stability diagrams rather than equilibrium phase diagrams.
Comments9 pages, 3 figures
