Exploring the Tensor Networks/AdS Correspondence

TL;DR

This paper investigates the tensor networks/AdS correspondence, revealing how near-perfect tensors and Coxeter groups can model gravitational features, correlation functions, and black hole analogues in a discrete tensor network framework.

Contribution

It introduces a Coxeter group-based approach to tensor networks in AdS space, models semi-classical correlations, and constructs a tensor network version of the BTZ black hole.

Findings

Coxeter groups effectively describe tensor networks in negatively curved space.

Perturbations of perfect tensors produce geodesic-related correlation functions.

Constructed tensor network analog of the BTZ black hole with horizon features.

Abstract

In this paper we study the recently proposed tensor networks/AdS correspondence. We found that the Coxeter group is a useful tool to describe tensor networks in a negatively curved space. Studying generic tensor network populated by perfect tensors, we find that the physical wave function generically do not admit any connected correlation functions of local operators. To remedy the problem, we assume that wavefunctions admitting such semi-classical gravitational interpretation are composed of tensors close to, but not exactly perfect tensors. Computing corrections to the connected two point correlation functions, we find that the leading contribution is given by structures related to geodesics connecting the operators inserted at the boundary physical dofs. Such considerations admit generalizations at least to three point functions. This is highly suggestive of the emergence of the analogues of Witten diagrams in the tensor network. The perturbations alone however do not give the right entanglement spectrum. Using the Coxeter construction, we also constructed the tensor network counterpart of the BTZ black hole, by orbifolding the discrete lattice on which the network resides. We found that the construction naturally reproduces some of the salient features of the BTZ black hole, such as the appearance of RT surfaces that could wrap the horizon, depending on the size of the entanglement region A.