  We calculate the two-photon absorption in bulk and single layer hexagonal
boron nitride (hBN) both by an ab-initio real-time Bethe-Salpeter approach and
by a the real-space solution of the excitonic problem in tight-binding
formalism. The two-photon absorption obeys different selection rules from those
governing linear optics and therefore provides complementary information on the
electronic excitations of hBN. Combining the results from the simulations with
a symmetry analysis we show that two-photon absorption is able to probe the
lowest energy $1s$ states in the single layer hBN and the lowest dark
degenerate dark states of bulk hBN. This deviation from the "usual" selection
rules based on the continuous hydrogenic model is explained within a simple
model that accounts for the crystalline symmetry. The same model can be applied
to other two-dimensional materials with the same point-group symmetry, such as
the transition metal chalcogenides. We also discuss the selection rules related
to the inversion symmetry of the bulk layer stacking.

Materials Science;Mesoscale and Nanoscale Physics