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 >= 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).