Z₂ topological order near the Neel state of the square lattice antiferromagnet

We classify quantum states proximate to the Neel state of the spin S=1/2 square lattice antiferromagnet with two-spin near-neighbor and four-spin ring exchange interactions. Motivated by recent experiments on the cuprates and the iridates, we examine states with Z₂ topological order, an order which is not present in the semiclassical limit. Some of the states break one or more of reflection, time-reversal, and lattice rotation symmetries, and can account for the observations. Such states can also be generalized to metals at finite doping, and may have implications for the pseudogap phase of the cuprates.