Bilayers of chiral spin states

We study the behavior of two planes of a quantum Heisenberg antiferromagnet in the regime in which a chiral spin liquid is stabilized in each plane. The planes are coupled by an exchange interaction of strength J₃. We show that in the regime of small J₃ (for both ferromagnetic and antiferromagnetic coupling), the system dynamically selects an antiferromagnetic ordering of the ground state chiralities of the planes. For the case of an antiferromagnetic interaction between the planes, we find that, at some critical value J₃^c of the interlayer coupling, there is a phase transition to a valence-bond state on the interlayer links. We derive an effective Landau-Ginzburg theory for this phase transition. It contains two U(1) gauge fields coupled to the order parameter field. We study the low-energy spectrum of each phase. In the condensed phase an "anti-Higgs-Anderson" mechanism occurs. It effectively restores time-reversal invariance by rendering massless one of the gauge fields while the other field locks the chiral degrees of freedom locally. There is no phase transition for ferromagnetic couplings.