标题： 纠缠分形化 显示英文标题

标题： Entanglement Fractalization

摘要： 我们通过分析冯·诺依曼熵（即纠缠熵）和纠缠轮廓，在标度极限中数值研究了分形几何与量子纠缠的相互作用。 采用西尔皮斯基地毯上的二次费米子模型，我们发现了引人注目的结果。 对于在化学势处表现出有限态密度的无能隙基态，我们揭示了一种由面积定律中的对数修正所表征的超面积定律。 这扩展了在平移不变的欧几里得晶格上已建立的超面积定律，其中关于托普利茨矩阵渐近行为的Gioev-Klich-Widom猜想具有重要影响。 此外，不同于晶格的分形结构，我们在接近标度极限时，在纠缠轮廓数据中观察到了一种新颖的自相似且普遍的模式，称为“纠缠分形”。 值得注意的是，这种模式类似于复杂的中国剪纸设计。 我们提供了生成这种分形的一般规则，为纠缠熵的普遍标度提供了见解。 同时，作为纠缠分形的直接结果，并超越平移不变系统中纠缠轮廓的单一标度行为，我们在分形系统的纠缠轮廓中识别出两种不同的标度行为。 对于有能隙的基态，我们观察到纠缠熵遵循广义面积定律，其依赖于互补子系统边界之间的豪斯多夫维数。 显示英文摘要

摘要： We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a novel self-similar and universal pattern termed an ``entanglement fractal'' in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems.