Migdal-Eliashberg equations - the effective model for superconducting state in H3S

The high-temperature superconducting state in sulfur trihydride ($T_{C}=203$~K) has been investigated in the context of the non-adiabatic and anharmonic effects. The Migdal-Eliashberg equations and the extended Eliashberg equations, which include the lowest-order vertex corrections, have been solved numerically in the self-consistent way. For $R3m$ crystal structure, the lowest-order vertex corrections decrease the value of the Coulomb pseudopotential from $0.123$ to $0.108$. The anharmonic effects work antagonistically in relation to the vertex corrections shifting the value of $\mu^{\star}$ to $0.156$. The studies conducted for the structure $Im\overline{3}m$, where the Eliashberg function includes both the non-adiabatic and anharmonic effects, prove the even higher value of $\mu^{\star}=0.185$. Independently of the assumed method of the analysis, the nearly identical no mean-field dependence of the order parameter on the temperature was obtained: $2\Delta(0)/k_{B}T_{C}\sim 4.7$ - due to the significant strong-coupling and retardation effects: $\lambda\sim 2$ and $k_{B}T_{C}\slash \omega_{\rm \ln}\sim 0.15$-$0.19$. It means that the classical equations of Migdal-Eliashberg can be treated as a correct effective model for the superconducting state in $\rm H_{3}S$. This paper has shown that the McMillan or Allen-Dynes formulas substantially lower the value of the critical temperature in relation to the result obtained with the Eliashberg equations.