A pedagogical overview on 2D and 3D Toric Codes and the origin of their topological orders

TL;DR

This paper provides a pedagogical overview of 2D and 3D Toric Codes, explaining how topological order and ground state degeneracy emerge from lattice discretization of a three-dimensional torus.

Contribution

It offers a clear, educational comparison of 2D and 3D Toric Codes, highlighting the role of topology in their topological order and ground state degeneracy.

Findings

Topological order in 3D Toric Code arises from lattice discretization of a 3D torus.

Quasiparticle behavior in 2D and 3D Toric Codes is conceptually similar.

Topology determines the ground state degeneracy in the 3D model.

Abstract

In this work, we will show how the topological order of the Toric Code appears when the lattice on which it is defined discretizes a three-dimensional torus. In order to do this, we will present a pedagogical review of the traditional two-dimensional Toric Code, with an emphasis on how its quasiparticles are conceived and transported. With that, we want to make clear not only how all these same quasiparticle conception and transportation fit into this three-dimensional model, but to make it clear how topology controls the degeneracy of ground state in this new situation.