Orbital currents in insulating and doped antiferromagnets

We describe square lattice spin liquids, which break time-reversal symmetry, while preserving translational symmetry. The states are distinguished by the manner in which they transform under mirror symmetries. All the states have nonzero scalar spin chirality, which implies the appearance of spontaneous orbital charge currents in the bulk (even in the insulator); but in some cases, orbital currents are nonzero only in a formulation with three orbitals per unit cell. The states are formulated using both the bosonic and fermionic spinon approaches. We describe states with Z(2) and U(1) bulk topological order, and the chiral spin liquid with semionic excitations. The chiral spin liquid has no orbital currents in the one-band formulation, but does have orbital currents in the three-band formulation. We discuss application to the cuprate superconductors, after postulating that the broken time-reversal and mirror symmetries persist into confining phases, which may also break other symmetries. In particular, the broken symmetries of the chiral spin liquid could persist into the Neel state.