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