Exploring the Tensor Networks/AdS Correspondence

TL;DR

<3-5 sentence high-level summary>We study how tensor networks can encode AdS/CFT using hyperbolic Coxeter tessellations, identifying the perfect-tensor construction’s shortcoming of a flat entanglement spectrum and lack of connected local correlators. By introducing small perturbations away from perfect tensors and exploiting geodesic structures, we recover geodesic-sourced 2- and 3-point functions and draw parallels to Witten diagrams, while maintaining RT-based entanglement features to leading order. We further realize a BTZ black hole within this discrete framework via orbifolding the Coxeter lattice, obtaining horizon-anchored RT surfaces and a thermal-like entanglement structure, and we compare to random-tensor models to illuminate domain-wall pictures in the Renyi entropy. Overall, the work provides a symmetry-guided, discrete-gravity framework that links boundary correlators, bulk geodesics, and black-hole horizons in tensor-network models, while noting the need for additional ingredients (e.g., edge weights, time evolution) to fully reproduce holographic spectra.

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.

Figures (13)

• Figure 1: This is an illustration of a tessellation specified by the triple $[p=6, q=4]$.
• Figure 2: This is an illustration of a tessellation specified by $[p=6, q=4]$. The numbers are labels of the 6 legs of the tensor. These tensors are generally not symmetric under permutation of the legs.
• Figure 3: In this picture, the numbers label the layer to which the hexagon belong to. The hexagons apart from 2 are truncated and we are only illustrating here the different tensor types we discussed in the text. It can be seen that at layer 2 there are 3 tensors of type $\tau_{2}(2)$ and at layer 3 there are 3 tensors of type $\tau_2(3)$ and 4 tensors of type $\tau_1(3)$. Finally there are two type $g(4)$ tensors, and 8 type $\tau_1(4)$ tensors.
• Figure 4: This is an illustration of how the action of $\sigma_x$ on an external leg is replaced by products of other Pauli matrices in the interior leg in the hexagon code.
• Figure 5: We reproduce here the figure in HAPPY. Operators with support in region $B$ and $D$ are inserted. The dotted lines correspond to geodesics forming the boundaries of the causal wedges of each boundary region. The regions colored red denote residual regions outside of the union of the four causal wedges. Connected correlation functions could feature when the residual region itself is connected.