Multipartite edge modes and tensor networks

TL;DR

This paper develops a new tensor network model for low-dimensional gravity that incorporates matter and multipartite edge modes, overcoming previous limitations related to discretization and gauge invariance.

Contribution

It introduces a novel tensor network framework with a topological area operator and multipartite edge modes, advancing the modeling of gravity in tensor networks.

Findings

Constructed tensor network including matter and a topological area

Edge modes are multipartite, not bipartite entangled pairs

Area operators are non-commuting, unlike in other models

Abstract

Holographic tensor networks model AdS/CFT, but so far they have been limited by involving only systems that are very different from gravity. Unfortunately, we cannot straightforwardly discretize gravity to incorporate it, because that would break diffeomorphism invariance. In this note, we explore a resolution. In low dimensions gravity can be written as a topological gauge theory, which can be discretized without breaking gauge-invariance. However, new problems arise. Foremost, we now need a qualitatively new kind of "area operator," which has no relation to the number of links along the cut and is instead topological. Secondly, the inclusion of matter becomes trickier. We successfully construct a tensor network both including matter and with this new type of area. Notably, while this area is still related to the entanglement in "edge mode" degrees of freedom, the edge modes are no longer bipartite entangled pairs. Instead they are highly multipartite. Along the way, we calculate the entropy of novel subalgebras in a particular topological gauge theory. We also show that the multipartite nature of the edge modes gives rise to non-commuting area operators, a property that other tensor networks do not exhibit.