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The spin-3/2 Heisenberg antiferromagnet on the bilayer honeycomb lattice is aminimal model to describe the magnetic behavior ofBi$_3$Mn$_4$O$_{12}$(NO$_3$). We study this model with frustrating inter-layersecond-neighbor couplings, taking into account quantum and thermalfluctuations. We use a path integral formulation in terms of coherent states todescribe the low energy physics of the model. We show that for a particularpoint in the parameter space, close to the experimental estimated couplings, acontinuum classical degeneracy is lifted by both quantum and thermalfluctuations, and a collinear state is then selected by an order by disordermechanism. Our results provide a global perspective in the understanding of theexperimental observations.

An effective field theory approach for the $S = 3/2$ bilayer honeycomb antiferromagnet