# Fracton fusion and statistics

## TL;DR

This paper develops a fusion and statistical theory for gapped excitations in Abelian fracton phases, incorporating lattice translation symmetry to characterize their unique mobility and phase distinctions.

## Contribution

It introduces a fusion theory that accounts for translation symmetry and mobility constraints, providing new descriptions for fracton models and their phase classifications.

## Key findings

- Fusion theory describes fracton and sub-dimensional excitations.
- Provides simple fusion descriptions for X-cube and checkerboard models.
- Proves distinctness of translation-invariant fracton phases in specific models.

## Abstract

We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless U(1) tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled two-dimensional toric code layers, and giving a description of the p-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and sub-dimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell.