# A Classical Model Correspondence for G-symmetric Random Tensor Networks

## TL;DR

This paper establishes a link between the entanglement entropy of G-symmetric random tensor networks and classical Ising models, revealing an area law and topological entanglement entropy related to the symmetry group G.

## Contribution

It introduces a systematic correspondence between entanglement entropy in symmetric tensor networks and classical spin model free energies, highlighting topological features.

## Key findings

- Entanglement entropy follows an area law with topological correction.
- Topological entanglement entropy equals log|G| for simply connected regions.
- The difference in domain wall counts relates to topological entropy.

## Abstract

We consider the scaling of entanglement entropy in random Projected Entangled Pairs States (PEPS) with an internal symmetry given by a finite group G. We systematically demonstrate a correspondence between this entanglement entropy and the difference of free energies of a classical Ising model with an addition non-local term. This non-local term counts the number of domain walls in a particular configuration of the classical spin model. We argue that for that overwhelming majority of such states, this gives rise to an area law scaling with well-defined topological entanglement entropy. The topological entanglement entropy is shown to be log|G| for a simply connected region A and which manifests as a difference in the number of domain walls of ground state energies for the two spin models.