  Nonlinear optical (NLO) responses of noncentrosymmetric nonmagnets have drawn
a lot of attention in the past decades because of their significance in
materials characterization, green energy and device applications. However, the
magnetism-induced NLO responses have rarely been studied so far. In this paper,
we first extend the numerical calculation friendly formula by Rashkeev
$\textit{et al.}$ [Phys. Rev. B $\textbf{57}$, 3905 (1998)] for second harmonic
generation (SHG) in nonmagnetic materials to include magnetic systems and then
calculate the magnetism-induced NLO responses of BiFeO$_3$, a multiferroic that
exhibits both ferroelectricity and antiferromagnetic (AFM) ordering at room
temperature and has a band gap that falls in the visible frequency region.
First, we find that the calculated magnetism-induced SHG susceptibilities are
large and the SHG intensity is tunable with the reversal of magnetization. In
particular, we find a strong magnetic contrast of the SHG signal of
approximately 440% at SHG photon energy of 4.82 eV, thus enabling a magnetic
control of the SHG in BiFeO$_3$. Also, because of the sensitivity of the SHG
signal to the direction of the N\'eel vector, the SHG can be utilized to detect
the reversal of the N\'eel vector in the AFM materials, which is an important
issue for AFM spintronics. Second, the calculated BPVE in BiFeO$_3$ are also
strong, being larger than some well-known NLO compounds such as BaTiO$_3$,
GaAs, CdS and CdSe. Finally, we analyse the origins of the prominent features
in the NLO response spectra in terms of the calculated quantum geometric
quantities. Our interesting findings suggest that the magnetism-driven NLO
responses in BiFeO$_3$ are significant, anisotropic and tunable, and that
understanding the magnetism-driven components of both SHG and BPVE is essential
for their applications in, e.g., multiferroic-based photovoltaic devices.
