经典环面码模型拓扑序的Rényi熵观点
A Rényi entropy perspective on topological order in classical toric code models
摘要：
信息论的概念越来越多地用于描述凝聚态系统中的集体现象，例如使用纠缠熵来识别相互作用的量子多体系统中的涌现拓扑顺序。在这里，我们采用这些概念的经典变体，特别是雷尼熵及其相关的互信息，以确定在经典系统的拓扑顺序。与它们的量子对应物一样，拓扑序的存在可以在这样的经典系统中通过对经典雷尼熵的普遍体积和边界定律的普遍的、次要的贡献来识别。我们证明了一个额外的子领导$O(1)$贡献一般会出现所有Rényi熵$S^{(n)}$与$n \geq 2$时，驱动系统的相变，例如到一个传统的有序相。这个额外的子引导项，我们称之为连通性贡献，可以追溯到部分子系统排序，并且与给定二分区中连通部分的数量成比例。值得注意的是，Levin-Wen求和方案-通常用于提取对Rényi熵的拓扑贡献-在这个经典背景下并没有完全消除这种额外的连接性贡献。这表明从Rényi熵中提取拓扑序需要额外的审查级别，以区分拓扑和非拓扑$O(1)$贡献。这也是量子系统的情况下，我们讨论的熵是敏感的这些连接的贡献。我们展示了这些发现的一个经典的变体的复曲面代码模型，我们研究的稳定性的拓扑秩序的存在下，一个磁场，并在有限的温度下从雷尼熵的角度进行了广泛的数值模拟。
英文摘要：
Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here we employ classical variants of these concepts, in particular Rényi entropies and their associated mutual information, to identify topological order in classical systems. Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical Rényi entropies. We demonstrate that an additional subleading $O(1)$ contribution generically arises for all Rényi entropies $S^{(n)}$ with $n \geq 2$ when driving the system towards a phase transition, e.g. into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme -- typically used to extract the topological contribution to the Rényi entropies -- does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from Rényi entropies requires an additional level of scrutiny to distinguish topological from non-topological $O(1)$ contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a Rényi entropy perspective.
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