Holographic Tensor Networks as Tessellations of Geometry

Abstract: Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more general geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop two holographic tensor network models: the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs, and the random PEE tensor network. In both models we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly computes the area of this surface.