Holographic Tensor Networks as Tessellations of Geometry

TL;DR

This paper develops holographic tensor network models based on PEE tessellations of AdS space, reproducing the Ryu-Takayanagi formula and advancing the geometric understanding of holography.

Contribution

It introduces new PEE-based tensor network models, including factorized, HaPPY-like, and random variants, linking network cuts to geometric surface areas.

Findings

All models reproduce the Ryu-Takayanagi formula exactly.

Networks are constructed from PEE tessellations of AdS space.

Models include tensor product, perfect tensor, and random constructions.

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 highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.