Traditional holographic duality usually assumes a smooth and continuous spacetime background, where boundary entanglement strictly follows the area law. However, the real microscopic quantum gravity background may possess a non-trivial fractal structure. Within the framework of connectivity engineering, this paper investigates the regulatory mechanism of fractal networks (Sierpinski gasket) on quantum entanglement structures. Numerical results show that, unlike a one-dimensional gapped chain restricted to an extremely low area law constant, the entanglement capacity of the fractal network is significantly enhanced due to the presence of multiple branches and geometric shortcuts, approaching the upper bound of the Page curve at the bipartite symmetry point for this finite size. This topological resonance brought by self-similar hierarchical levels of the fractal demonstrates a generalized area law dominated by fractal dimension. This result proves that the fractal dimension of the underlying spacetime is the key geometric origin determining the long-range distribution of quantum information and the upper bound of entanglement capacity.
Traditional holographic duality usually assumes a smooth and continuous spacetime background, where boundary entanglement strictly follows the area law. However, the real microscopic quantum gravity background may possess a non-trivial fractal structure. Within the framework of connectivity engineering, this paper investigates the regulatory mechanism of fractal networks (Sierpinski gasket) on quantum entanglement structures. Numerical results show that, unlike a one-dimensional gapped chain restricted to an extremely low area law constant, the entanglement capacity of the fractal network is significantly enhanced due to the presence of multiple branches and geometric shortcuts, approaching the upper bound of the Page curve at the bipartite symmetry point for this finite size. This topological resonance brought by self-similar hierarchical levels of the fractal demonstrates a generalized area law dominated by fractal dimension. This result proves that the fractal dimension of the underlying spacetime is the key geometric origin determining the long-range distribution of quantum information and the upper bound of entanglement capacity.
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 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. Published by the American Physical Society 2024
Holographic duality suggests that continuous spacetime geometry can emerge from underlying quantum entanglement. Based on the connectivity engineering framework, we construct a purely geometric decoupled lattice model to explore the entanglement area law and the emergence mechanism of discrete Ryu-Takayanagi (RT)-type scaling. By tuning the nearest-neighbor coupling λ, the system evolves from a disconnected direct sum decomposition (λ=0) to a globally connected phase. Precise numerical results indicate that weak nearest-neighbor coupling can stitch the rigid fault lines in the information distance matrix, inducing the formation of a smooth and continuous effective geometry. The core result is that in the stitched connected phase, the subsystem entanglement entropy is dominated by the geometric boundary, satisfying a scaling relation primarily governed by a linear area law with logarithmic corrections. Through a hold-out test on irregular subregions, the RT-type scaling calibrated by square samples provides a controlled error estimate for rectangular subregions. The L-shaped concave subregion exhibits a systematic negative deviation, corresponding to the well-known corner correction effect in free-fermion systems, further distinguishing perimeter scaling from minimal-surface scaling. Combined with Kitaev-Preskill diagnostic analysis, the results support that the discrete RT-type scaling can emerge from a purely geometric discrete network. This model provides a clean, exactly solvable underlying benchmark for studying discrete spacetime geometry and holographic entanglement on quantum simulators.
In this paper, we discover the topological quantization and phase transition mechanisms of holographic extremal surfaces in discrete tensor networks, providing a rigorous discrete dynamical benchmark for the proposition that "gravity is entanglement". Using a one-dimensional spin network with a fixed truncated hopping step k, we derive a rigorous analytical solution for the peak position of entanglement entropy, demonstrating that it is solely determined by the underlying connectivity step k. The perturbative phase diagram further reveals that this quantized gravity ladder is strongly protected by the topological gap below a critical perturbation V_c, exhibiting unique critical scaling behavior at the transition point. The fitted critical exponent beta approx 0.07 is significantly lower than the mean-field exponent of conventional phase transitions, revealing a unique "critical rigidity" characteristic of finite-size long-range topological networks. This model not only provides a rigorous graph-theoretical benchmark for a discrete analogue of the Ryu-Takayanagi area law but also, through the "geometric decoupling" mechanism, constructs an analytically solvable microscopic dynamical sandbox for the highly debated entanglement wedge disconnection in the black hole island formula.
The core of quantum error correction lies in how to isolate local errors to prevent them from contaminating the global system. Within the framework of connectivity engineering, this paper proposes a holographic quantum error correction mechanism based on pure geometric decoupling. Numerical results show that in a locally connected network (k=1), local errors rapidly diffuse globally through the increase of entanglement entropy; while in the decoupled network (k=3), long-range hopping constructs a graph-topological barrier, strictly locking the error within the isolated sub-chain where it was injected, and the mutual entanglement entropy between A and B remains strictly at its initial zero value. Since the physical hopping between subsystems is cut off, the injected error state cannot trigger global chaos, and the purity of boundary information remains stable. This mechanism does not rely on traditional redundant encoding, but rather utilizes the intrinsic graph-topological isolation property of geometric connectivity to achieve "physical-layer" fault tolerance. It provides a minimal discrete dynamical benchmark for understanding Bulk Locality in holographic duality and points the way for designing redundancy-free quantum memories.
The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.
Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law SX=Θ(LXdb) under a uniform gap and satisfies LXdb≲SX≲LXdblnLX when the gap closes no faster than algebraically. Here LX is the subsystem linear size and db the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmix with max{SX,zXln2}⩽SXmix⩽SX+zXln2, where zX is the local zero-mode weight; an extensive zX thus makes SXmix follow the bulk dimension.