Sorting topological stabilizer models in three dimensions

TL;DR

This paper introduces bulk commutation quantities for 3D topological stabilizer models, enabling classification into distinct phase types such as TQFT, foliated, fractal, or type-II models, extending understanding beyond 2D cases.

Contribution

It develops new bulk commutation invariants for 3D models, providing a coarse classification scheme for different topological phases including fracton orders.

Findings

Classifies 3D topological stabilizer models into distinct phase types.

Identifies invariants that distinguish TQFT, foliated, fractal, and type-II phases.

Extends 2D S-matrix concepts to three dimensions.

Abstract

The S-matrix invariant is known to be complete for translation invariant topological stabilizer models in two spatial dimensions, as such models are phase equivalent to some number of copies of toric code. In three dimensions, much less is understood about translation invariant topological stabilizer models due to the existence of fracton topological order. Here we introduce bulk commutation quantities inspired by the 2D S-matrix invariant that can be employed to coarsely sort 3D topological stabilizer models into qualitatively distinct types of phases: topological quantum field theories, foliated or fractal type-I models with rigid string operators, or type-II models with no string operators.