标题： 纠缠分形化 显示英文标题

标题： Entanglement Fractalization

摘要： 我们研究了在分形格子上具有非整数维度和破坏平移对称性的自由费米子模型的量子纠缠。 对于在化学势处具有有限态密度的无能隙系统，我们发现纠缠熵（EE）具有普遍的标度关系$S_{A} \sim L_{A}^{d_{s}-1} \log L_{A}$，该标度关系与分割方案无关，其中$d_s$是分形所嵌入的空间维度，$L_A$是子系统$A$的线性尺寸。 这种标度扩展了平移不变系统的 Widom 猜想到分形格子。 我们还研究了纠缠轮廓（EC）作为实空间纠缠的“断层扫描”。 EC 数据显示了一种自相似且普遍的模式，称为“纠缠分形”（EF），它类似于中国剪纸，并对不同的分割方案保持不变，从而导致 EE 标度的鲁棒性。 我们提出了一组规则来人工生成 EF 模式，该模式在标度极限下与数值结果相匹配。 对于有能隙系统，我们观察到$A$边界分形特性影响 EE 的标度为$ S_{A} \sim L_{A}^{d_{\rm bf}}$，其中$ d_{\rm bf}$是$A$边界的豪斯多夫维数，推广了面积定律。 同时，EC 主要局域在$A$的边界。 我们的研究揭示了分形几何如何与自由费米子的纠缠相互作用。 讨论了来自物理和数学的未来方向，例如实验验证和分形上的拉普拉斯算子。 显示英文摘要

摘要： We investigate the quantum entanglement of free-fermion models on fractal lattices with non-integer dimension and broken translation symmetry. For gapless systems with finite density-of-state at the chemical potential, we find a universal scaling of entanglement entropy (EE) as $S_{A} \sim L_{A}^{d_{s}-1} \log L_{A}$ that is independent of the partition scheme, where $d_s$ is the space dimension where fractals are embedded, and $L_A$ is the linear size of the subsystem $A$. This scaling extends the Widom conjecture of translation-invariant systems to fractal lattices. We also study entanglement contour (EC) as a real-space entanglement ``tomography''. The EC data show a self-similar and universal pattern called ``entanglement fractal'' (EF), which resembles Chinese papercutting and keeps invariant for different partition schemes, leading to the EE scaling robustness. We propose a set of rules to artificially generate the EF pattern, which matches the numerical results at the scaling limit. For gapped systems, we observe that the fractal feature of $A$'s boundary affects the EE scaling as $ S_{A} \sim L_{A}^{d_{\rm bf}}$, where $ d_{\rm bf}$ is the Hausdorff dimension of $A$'s boundary, generalizing the area law. Meanwhile, the EC mainly localizes at $A$'s boundary. Our study reveals how fractal geometry interacts with the entanglement of free fermions. Future directions from physics and mathematics are discussed, e.g., experimental verification and Laplacian on fractals.