Fracton fusion and statistics*

In this work, we describe fusion and statistical processes in Abelian fracton phases in terms of their gapped excitations. The restricted mobility of fractonic excitations implies that statistical processes do not take the form of familiar braiding processes. Also, the number of distinct excitation types in fracton phases is infinite, in contrast to conventional phases with intrinsic topological order. Moreover, if one considers excitations supported in a region with linear size $L$, the number of excitation types supported in the region grows exponentially with $L$. To build a manageable theory that incorporates these features, we consider lattice translation symmetry. Without translation symmetry, the fusion of excitations in an Abelian fracton phase is described by an infinite Abelian group, whose elements correspond to distinct excitation types. Translation symmetry acts on this Abelian group, giving it more structure and making it a more manageable object to work with. Moreover, this action allows us to describe the mobility of excitations at the level of the fusion theory, which then forms the basis for a description of statistical processes.