Bilayers of Chiral Spin States

Abstract

We study the behavior of two planes of 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 $J_3$. We show that in the regime of small $J_3$ (for both ferromagnetic {\it and} antiferromagnetic coupling), the system dynamically selects an \underline{antiferromagnetic} ordering of the ground state {\it chiralities} of the planes. For the case of an antiferromagnetic interaction between the planes, we find that, at some critical value $J_3^c$ of the inter-layer 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.