Supplemental material: Photoinduced Enhancement of Excitonic Order

In the main text, we have considered a single phonon branch which is coupled to the transfer integral between the electron bands. In a more realistic scenario, the system is coupled to multiple phonon branches, and the question we want to address here is whether or not this has a qualitative effect on the mechanism discussed in the main text. In the following, we first introduce another type of electron-phonon (el-ph) coupling, which has been pointed out in the recent LDA calculation of Ref. [1]. Then we demonstrate that, in order to see the enhancement of the EI order and the gap, one needs at least one phonon mode that is coupled to the transfer integral between the electron bands. Hence, the number of phonon modes does not influence the physics we discussed in the main text.