Bosonic model with Z3 fractionalization

A flurry of recent theoretical activity has produced specific model system realizations of fractionalized phases in two dimensions. Essentially all of the fractionalized states constructed so far are Z2 states. 7 On a formal level, these realizations employ the following route to Z2 fractionalization: Strong local correlations lead to a U~1! gauge theory as a low-energy description; this gauge theory is then driven into a deconfined state by a condensation of objects carrying gauge charge 2 that also appear in the low-energy description. This formal structure has been brought out very directly in Refs. 8, 5, and 6. On a more physical level, the fractionalized insulator is produced departing from a superconducting state by a condensation of double vortices. The main body of work concentrated on the Z2 states since these are expected to be the simplest to realize. However, it is clear that more complicated fractionalized states are also possible. For example, it is conceivable that in some systems the superconducting state is quantum-disordered by a condensation of triple vortices; the resulting insulator is then a Z3 fractionalized state. In this paper, we indicate how a Z3 fractionalized state can be engineered in a relatively simple bosonic model with unfrustrated nearest-neighbor hopping and short-range twobody repulsive interactions. Much of the construction parallels closely the Z2 examples of Refs. 5, 6, and 3: The lowenergy Hilbert space is selected—by stipulating particular charge interactions—in a manner that naturally admits splitting the boson charge into three pieces; this Hilbert space is protected by a large charge gap. The effective description of the fractionalized state has gapped chargons carrying electrical charge 1/3 and coupled to some special Z3 gauge theory which we analyze in detail. Our main message here is that one does not need very contrived systems to obtain more complicated fractionalization patterns.