Superselection sectors in the 3d Toric Code

TL;DR

This paper rigorously classifies superselection sectors in the 3D Toric Code, revealing how infinite flux strings and their configurations influence the ground state structure on an infinite lattice.

Contribution

It introduces a rigorous framework for defining and classifying superselection sectors in the 3D Toric Code, including the role of infinite flux strings and geometric conditions.

Findings

Single infinite flux string states must be monotonic to be ground states

States with more than 3 infinite flux strings are not in ground state sectors

A necessary and sufficient condition for multiple flux strings involves 'infinity directions'

Abstract

We rigorously define superselection sectors in the 3d (spatial dimensions) Toric Code Model on the infinite lattice $\mathbb{Z}^3$. We begin by constructing automorphisms that correspond to infinite flux strings, a phenomenon that's only possible in open manifolds. We then classify all ground state superselection sectors containing infinite flux strings, and find a rich structure that depends on the geometry and number of strings in the configuration. In particular, for a single infinite flux string configuration to be a ground state, it must be monotonic. For configurations containing multiple infinite flux strings, we define "infinity directions" and use that to establish a necessary and sufficient condition for a state to be in a ground state superselection sector. Notably, we also find that if a state contains more than 3 infinite flux strings, then it is not in a ground state superselection sector.