  One of interesting open questions for the high transition temperature (Tc)
superconductivity in sulfur hydrides is why high pressure phases of H3S have
extremely high Tc's. Recently, it has been pointed out that the presence of the
van Hove singularities (vHs) around the Fermi level is crucial. However, while
there have been quantitative estimates of Tc based on the Migdal-Eliashberg
theory, the energy dependence of the density of states (DOS) has been neglected
to simplify the Eliashberg equation. In this study, we go beyond the constant
DOS approximation and explicitly consider the electronic structure over 40eV
around the Fermi level. In contrast with the previous conventional
calculations, this approach with a sufficiently large number of Matsubara
frequencies enables us to calculate Tc without introducing the empirical pseudo
Coulomb potential. We show that while H3S has much higher Tc than H2S for which
the vHs is absent, the constant DOS approximation employed so far seriously
overestimates (underestimates) Tc by ~ 60K (~ 10K) for H3S (H2S). We then
discuss the impact of the strong electron-phonon coupling on the electronic
structure with and without the vHs and how it affects the superconductivity.
Especially, we focus on (1) the feedback effect in the self-consistent
calculation of the self-energy, (2) the effect of the energy shift due to the
zero-point motion, and (3) the effect of the changes in the phonon frequencies
due to strong anharmonicity. We show that the effect of (1)-(3) on Tc is about
10-30K for both H3S and H2S. Eventually, Tc is estimated to be 181K for H3S at
250GPa and 34K for H2S at 140GPa, which explains the pressure dependence of Tc
observed in the experiment. In addition, we evaluate the lowest order vertex
correction beyond the Migdal-Eliashberg theory and discuss the validity of the
Migdal approximation for sulfur hydrides.
