AI 中文总结
该研究推导了阿贝尔陈-西蒙斯理论中键环链态的多熵闭式表达式，发现通用四分量链态的多熵坍缩仅在特定雷尼指数和能级下成立，破缺依赖于雷尼指数与能级算术结构。
AI 中文摘要
我们研究阿贝尔 $U(1)_k$ 陈-西蒙斯理论中的真实多熵。对于键环链态（仅一个区分分量 $K$ 与其余 $\boldsymbol{q}-1$ 个分量间的环绕数非零），我们推导出了一般 $\boldsymbol{q}$、能级 $k$ 和雷尼指数 $n$ 下 $\boldsymbol{q}$ 分雷尼多熵的精确闭式表达式。当 $\boldsymbol{q}=4$ 时，这表明真实多熵 $\boldsymbol{\rm GM}^{(4)}_n$ 对所有 $n$ 都精确坍缩到三方信息 $\boldsymbol{I}_{3,n}$；当 $\boldsymbol{q}=5$ 时，对所有 $n$，其同样由三方和双方雷尼多熵的线性组合完全确定。随后我们超出键环类，研究具有任意两两环绕数的一般四分量链态。对陈-西蒙斯能级 $2\leq k\leq24$ 的数值扫描显示，在键环态上解析得到的全 $n$ 坍缩无法推广到通用链态。值得注意的是，在所有考察的能级上，$n=2$ 时坍缩保持精确；在 $n=3$ 时，在所扫描范围内，仅当 $3\mid k$ 时出现破缺，且进一步依赖于 $k$ 的 3-adic 赋值；在 $n=4$ 和 $n=5$ 时，在所考察的每个能级上均出现破缺，且破缺率随 $k$ 剧烈变化。这些结果表明，破缺并非仅由复合 $\boldsymbol{\rm Z}_k$ 的零因子结构控制，而是表现出对雷尼指数和陈-西蒙斯能级算术结构的非平凡联合依赖。
英文摘要
We study genuine multi-entropy in Abelian $U(1)_k$ Chern-Simons theory. For key-ring link states, where only the linking numbers between one distinguished component $K$ and the remaining $\mathtt{q}-1$ components are nonzero, we derive an exact closed-form expression for the $\mathtt{q}$-partite Rényi multi-entropy for general $\mathtt{q}$, level $k$, and Rényi index $n$. For $\mathtt{q}=4$, this shows that the genuine multi-entropy $\mathrm{GM}^{(4)}_n$ collapses exactly onto the tripartite information $I_{3,n}$ for all $n$, while for $\mathtt{q}=5$ it is likewise completely determined, for all $n$, by a linear combination of tripartite and bipartite Rényi multi-entropies. We then go beyond the key-ring class and study general four-component link states with arbitrary pairwise linking numbers. A numerical scan over Chern--Simons levels $2\leq k\leq24$ shows that the all-$n$ collapse found analytically for key-ring states does not survive for generic link states. Remarkably, the collapse remains exact at $n=2$ for every level examined. At $n=3$, violations occur, within the scanned range, only when $3\mid k$, with a further dependence on the $3$-adic valuation of $k$. At $n=4$ and $n=5$, violations occur for every level examined, with rates that vary strongly with $k$. These results show that the breakdown is not controlled simply by the zero-divisor structure of composite $\mathbb Z_k$, but instead exhibits a nontrivial joint dependence on the Rényi index and the arithmetic structure of the Chern--Simons level.
Comments64 pages, 3 figures
