  In this paper we discuss the optical response due to the excitonic effect of
two types of hBN bilayers: AB and AA'. Understanding the properties of these
bilayers is of great utility to the study of twisted bilayers at arbitrary
angles, since these two configurations correspond to the limit cases of 0 and
60 degree rotation. To obtain the excitonic response we present a method to
solve a four-band Bethe-Salpeter equation, by casting it into a 1D problem,
thus greatly reducing the numerical burden of the calculation when compared
with strictly 2D methods. We find results in good agreement with ab initio
calculations already published in the literature for the AA' bilayer, and
predict the excitonic conductivity of the AB bilayer, which remains largely
unstudied. The main difference in the conductivity of these two types of
bilayers is the appearance of a small, yet well resolved, resonance between two
larger ones in the AB configuration. This resonance is due to a mainly
interlayer exciton, and is absent in the AA' bilayer. Also, the conductivity of
the AB bilayer is due to both intralayer and interlayer excitons and is
dominated by p-states, while intralayer states are the relevant ones for the
AA' configuration, like in a monolayer. The effect of introducing a bias in the
AA' bilayer is also discussed.
