Duality among competing orders in antiferromagnets and topological insulators: nonlinear sigma model approach

We revisit our framework for studying competing orders in quantum antiferro-magnets (A. Tanaka and X. Hu, Phy. Rev. Lett. 95 (2005) 036402; Phys. Rev. B 74 (2006) 140407(R)), in which we showed that when two or more ordering tendencies are organized into a single composite order parameter, the corresponding effective sigma model action can generally host a new topological term. It has since been argued by several authors that such terms are indicative of novel critical behaviors. Here we reinforce this assertion by searching for nonva-nishing fermionic bilinears (Golstone-Wilczek-type currents) associated with the same models. We arrive at a duality relation among the competing orders (e.g. a direct relation between the topological charge of the antiferromagnetic order sector and a Noether current associated with the rotational symmetry in the valence bond solid order sector), which becomes relevant upon approaching criticality. We also discuss the physical implications of this duality in the context of topological insulators.