  Based on the Bethe-Salpeter equation eigenstates, we present a
first-principles many-body formalism for calculating the two-photon absorption
(TPA) coefficient of semiconductors. We apply this formalism to calculate the
TPA spectra of MoS2 and WS2 monolayers. The all-electron full-potential
linearised augmented-plane wave based functions are used for solving the
Bethe-Salpeter equation. The calculated spectra are in good agreement with the
available experimental ones for WS2 monolayer. The calculated TPA spectra
exhibit significant excitonic effects when compared to those based on the
independent particle approximation. The physical origin of TPA excitonic
transitions of MoS2 and WS2 monolayers are revealed by tracing the
sum-over-states process. We show that the spin-orbit coupling effect leads to
characteristic double peaks with an interval of half spin-orbit splitting
energy. These double peaks mainly originate from the transitions at the
vicinity of K point.
