An effective 2-band $e_g$ model of sulfur hydride H$_3$S for high-$T_{\textrm{c}}$ superconductivity

To understand high transition temperature ($T_{\textrm{c}}$) superconductivity in sulfur hydride H$_3$S, we propose an effective 2-band model having the $e_g$ symmetry as the minimal model for H$_3$S. Two $e_g$ orbitals centered on a sulfur S atom are chosen for the smallest representation of relevant bands with the van-Hove singularity around the Fermi levels except for the $\Gamma$-centered small hole pockets by the sulfur 3$p$ orbitals. By using the maximally localized Wannier functions, we derive the minimal effective model preserving the body-centered cubic (bcc) crystal symmetry of the H$_3$S phase having the highest $T_{\textrm{c}}$ ($\sim$ 203 K under pressures) among the other polymorphs of H$_3$S.