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[08]Emergence of Discrete RT-type Scaling and Area Law in Geometrically Decoupled Lattices

Wenjun Luo

Published Oct 25, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23634326

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arXiv:2006.10089•Journal of High Energy Physics•Dec 1, 2020•48 citations
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.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Abstract\n : \nRecent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain ‘diagonal approximation’ will hold. In terms of the bulk Newton constant G, this yields a correction of order O(G−1/2) near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis. \nWe illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS3 vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the O(G−1/2) correction found Penington et al. for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.
Inspired by the ER=EPR conjecture in holographic physics, we construct a minimal and exactly solvable geometrically decoupled spin network benchmark within the framework of connectivity engineering. The system consists of two geometrically decoupled subsystems (Universes A and B) connected by a throat C. By constructing a directed boundary using complex hopping phases to introduce a synthetic gauge field, we measure the effective light-cone information propagation speed to be approximately 0.51; this propagation speed can be tuned by the geometric topology of the throat, satisfying the bandwidth scaling relation where v_arrive is proportional to W_C. Furthermore, we numerically extract an empirical dual bound where I_max times v_arrive is less than or equal to K, revealing a fundamental trade-off limit between entanglement generation capacity and information propagation speed. This work abandons gravitational metric degrees of freedom while retaining the core geometric-topological structures of holographic duality, providing a tunable and deterministic discrete numerical benchmark for holographic information transport.
Abstract Recent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain ‘diagonal approximation’ will hold. In terms of the bulk Newton constant G, this yields a correction of order O(G−1/2) near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis. We illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS3 vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the O(G−1/2) correction found Penington et al. for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.