Title: Quantum Entanglement on Fractal Landscapes

Abstract: We 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. Focusing on free-fermion quantum models known for their simplicity and effectiveness in studying highly entangled quantum systems, 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 divergence in the 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, we observe the emergence of a 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. Building upon the insights gained from the entanglement fractal, we explicitly elucidate the origin of the logarithmic divergence on fractals where translation symmetry is broken and the Widom conjecture is inapplicable. 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.