有序随机轨迹的非阿贝尔自旋计数：重入有限时间陈数
Non-Abelian Spin Counting of Ordered Stochastic Trajectories: Reentrant Finite-Time Chern Numbers
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中文总结 AI 辅助
该研究针对有序随机轨迹提出非阿贝尔自旋计数构造，建立陈-对应关系，在五态非平衡八字形网络中观测到重入有限时间陈数序列，揭示有限时间可将随机过程组织为不同拓扑扇区。
中文摘要 AI 辅助
常规全计数统计为积分随机流分配可交换相位，因此仅能解析净输运，通常无法区分与不同循环相关的事件时间顺序。我们引入一种非阿贝尔计数构造，其中两个基本循环的交叉会使辅助自旋绕不同轴旋转。由此得到的有序轨迹统计量其一阶矩具有精确的有限维演化方程。两个旋转角构成一个计数环面，当平均自旋非零时，其归一化方向定义了一个从二维环面（$\boldsymbol{\rm T}^2$）到二维球面（$\boldsymbol{\rm S}^2$）的映射，等价于具有陈数的复本征线丛。我们的主要分析结果是陈-对应关系：反射对称为该本征线赋予实结构，并通过反射固定圆上的第一斯蒂费尔-惠特尼类来表达 $C_T \bmod 2$。当横向极化在反射固定圆上无额外零点时，这些类在四个高对称计数点处简化为普通电流-奇偶统计。对于一个五态非平衡八字形网络，仅改变观测时间会产生四个极化间隙闭合，以及重入序列 $C_T=0 \to -1 \to 0 \to -1 \to 0$。每次转变都发生在 $(\boldsymbol{\rm \boldsymbol{\text{π}}}, \boldsymbol{\rm \boldsymbol{\text{π}}})$ 处，且与 $\boldsymbol{\rm \boldsymbol{\text{E}}}[(-1)^{Q_1+Q_2}]$ 的符号反转一致，而完整整数陈数可从二维自旋纹理独立获得。因此，有限观测时间可将固定随机过程组织为其有序路径系综的不同拓扑扇区。
英文摘要
Conventional full counting statistics assigns commuting phases to integrated stochastic currents and therefore resolves net transport but not, in general, the temporal ordering of events associated with different cycles. We introduce a non-Abelian counting construction in which crossings of two fundamental cycles rotate an auxiliary spin about different axes. The resulting ordered trajectory statistic has an exact finite-dimensional evolution equation for its first moment. The two rotation angles form a counting torus, and whenever the mean spin is nonzero its normalized direction defines a map $\mathbb T^2\to\mathbb S^2$, equivalently a complex eigenline bundle with a Chern number. Our main analytical result is a Chern--parity correspondence. Reflection symmetry equips this eigenline with a real structure and expresses $C_T\bmod2$ through first Stiefel--Whitney classes on the circles fixed by reflection. When the transverse polarization has no additional zeros along the reflection-fixed circles, these classes reduce to ordinary current-parity statistics at the four high-symmetry counting points. For a five-state nonequilibrium figure-eight network, varying only the observation time produces four polarization-gap closings and the reentrant sequence $C_T=0\to-1\to0\to-1\to0$. Every transition occurs at $(π,π)$ and coincides with a sign reversal of $\mathbb E[(-1)^{Q_1+Q_2}]$, while the full integer Chern number is obtained independently from the two-dimensional spin texture. Finite observation time can therefore organize a fixed stochastic process into distinct topological sectors of its ordered path ensemble.
