  Bilayer antiferromagnets CrX$_{3}$ (X $=$ Cl, Br, and I) are promising
materials for spintronics and optoelectronics that are rooted in their peculiar
electronic structures. However, their bands are often hybridized from the
interlayer antiferromagnetic ordering, which are difficult to disentangle by
traditional methods. In this work, we theoretically show that nonlinear
harmonic spectra can differentiate subtle differences in their electronic
states. In contrast to prior nonlinear optical studies which often use one or
two photon energies, we systematically study the wavelength-dependent nonlinear
harmonic spectra realized by hundreds of individual dynamical simulations under
changed photon energies. Through turning on and off some excitation channels,
we can pinpoint every dipole-allowed transition that largely contributes to the
second and third harmonics. With the help of momentum matrix elements, highly
entangled resonance peaks at a higher energy above the band edge can be
assigned to specific transitions between the valence bands and three separate
regions of conduction bands. Our findings demonstrate a feasible means to
detect very complex electronic structures in an important family of
two-dimensional antiferromagnets.
