Representation of excited states and topological order of the toric code in MERA

TL;DR

This paper explores the representation of excited states and topological order in the toric code using MERA, linking tensor networks to holographic duality and providing methods to compute topological entanglement entropy.

Contribution

It analytically constructs the MERA network for the toric code, including excited states, and introduces a way to calculate topological entanglement entropy from MERA geometry.

Findings

MERA can be explicitly constructed for the toric code.

Excited states can be incorporated into the MERA framework.

Topological entanglement entropy can be derived from MERA geometry.

Abstract

The holographic duality relates a field theory to a theory of (quantum) gravity in one dimension more. The extra dimension represents the scale of the RG transformation in the field theory. It has been conjectured that the tensor networks which arise during the real space renormalization procedure like the multi-scale entanglement renormalization ansatz (MERA) are a discretized version of the background of the gravity theory. We strive to contribute to make this conjecture testable by considering an explicit and tractable example, namely the dual network of the toric code, for which MERA can be performed analytically. We examine how this construction can be extended to include excited states. Furthermore, we show how to calculate topological entanglement entropy from the geometry of MERA. This method is expected to generalize to systems with generic entanglement structure.