Fractionalization, Topological Order, and Quasiparticle Statistics

We argue, based on general principles, that topological order is essential to realize fractionalization in gapped insulating phases in dimensions $d\ge 2$. In $d=2$ with genus $g$, we derive the existence of the minimum topological degeneracy $q^g$ if the charge is fractionalized in units of $1/q$, irrespective of microscopic model or effective theory. Furthermore, if the quasiparticle is either boson or fermion, it must be at least $q^{2g}$.