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.
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.
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 the long-standing quest to reconcile gravity with quantum mechanics, profound connections have been unveiled between concepts traditionally pertaining to a quantum information theory, such as entanglement, and constitutive features of gravity, like holography. Developing and promoting these connections from the conceptual to the operational level unlock access to a powerful set of tools which can be pivotal toward the formulation of a consistent theory of quantum gravity. Here, we review recent progress on the role and applications of quantum informational methods, in particular tensor networks, for quantum gravity models. We focus on spin network states dual to finite regions of space, represented as entanglement graphs in the group field theory approach to quantum gravity, and illustrate how techniques from random tensor networks can be exploited to investigate their holographic properties. In particular, spin network states can be interpreted as maps from bulk to boundary, whose holographic behavior increases with the inhomogeneity of their geometric data (up to becoming proper quantum channels). The entanglement entropy of boundary states, which are obtained by feeding such maps with suitable bulk states, is then proved to follow a bulk area law with corrections due to the entanglement of the bulk state. We further review how exceeding a certain threshold of bulk entanglement leads to the emergence of a black hole-like region, revealing intriguing perspectives for quantum cosmology.
This paper encodes the subsystem entanglement structure of boundary conformal field theory into a weighted graph and defines the spectral entanglemententropy via the spectral theory of the Dirichlet-to-Neumann map. Within the constraint network framework, a continuum limit and a mechanism for dimensionalemergence are introduced to establish a rigorous transition from discrete spectrato continuous holographic geometry. It is proved that, in the rigid limit, the spectral entanglement entropy strictly decomposes into the sum of a geometric areaterm and a topological correction term: the area term arises from a strictly positive lower bound on the spectral gap of the Dirichlet-to-Neumann map and ispinned by the curvature of the extremal surface; the topological correction termoriginates from the nullity of the coarse-grained rigid kernel, whose value is proportional to the absolute value of the Euler characteristic. It is further proved thatthe stability of a quantum extremal surface is equivalent to the algebraic connectivity of the quotient graph being strictly positive, a condition characterized bythe Fiedler value. This yields three testable predictions, encompassing a universal difference in the topological correction, multi-surface transitions for non-convexregions, and the finite-temperature spectral-gap scaling, and reveals a deep isomorphism with holographic quantum error correction. This framework unifies thegeometric origin, quantum corrections, and the recoverability of error correction ofthe Ryu-Takayanagi formula within a single spectral flow structure.
Extending the holographic principle to de Sitter (dS) spacetimes remains one of the most vital open frontiers in quantum gravity, where a microscopic, bottom-up tensor-network framework that relates boundary quantum data to emergent de Sitter spacetime is still lacking. In this work, we first show the emergence of de Sitter spacetime from boundary entanglement by formulating a non-unitary continuous multi-scale entanglement renormalization ansatz (cMERA) for a concrete non-Hermitian critical fermion chain. Within this emergent spacetime, we analyze the associated geodesics and show that they act as extremal Ryu-Takayanagi (RT) surfaces undergoing a smooth timelike-to-null transition. Remarkably, we demonstrate that this continuum trajectory suggests a distinct tensor-network architecture in which the bond-counting contribution naturally truncates at the discrete timelike-to-null transition toward the deep infrared. In the resulting architecture, the null ray along the horizon is represented by zero-cost links, since the associated cut severs no tensor legs. This network structure successfully reproduces the logarithmic scaling of non-unitary critical entanglement entropy, offering a bond-counting picture for the de Sitter RT formula. Our results provide the long-sought dS/(c)MERA correspondence at the level of both emergent spacetime and discrete holographic entanglement.
Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.
Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.