  It is well-known that exciton effects are determinant to understand the
optical absorption spectrum of low-dimensional materials. However, the role of
excitons in nonlinear optical responses has been much less investigated at an
experimental level. Additionally, computational methods to calculate nonlinear
conductivities in real materials are still not widespread, particularly taking
into account excitonic interactions. We present a methodology to calculate the
excitonic second-order optical responses in 2D materials relying on: (i) ab
initio tight-binding Hamiltonians obtained by Wannier interpolation and (ii)
the Bethe-Salpeter equation with effective electron-hole interactions. Here, in
particular, we explore the role of excitons in the shift current of monolayer
materials. Focusing on MoS$_2$ and GeS monolayer systems, our results show that
$2p$-like excitons, which are dark in the linear response regime, yield a
contribution to the photocurrent comparable to that of $1s$-like excitons.
Under radiation with intensity $\sim 10^{4} $W/cm$^2$, the excitonic theory
predicts in-gap photogalvanic currents of almost $\sim 10$ nA in sufficiently
clean samples, which is typically one order of magnitude higher than the value
predicted by independent-particle theory near the band edge.
