Phase diagram for hole-doped Kitaev systems on the honeycomb lattice

In extended Kitaev models on the honeycomb lattice, off-diagonal interactions (e.g., $\mathrm{\ensuremath{\Gamma}},\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ terms) can give rise to non-Kitaev quantum spin liquids and several magnetically ordered phases. present paper, we dope holes system study resultant $t\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model using mean-field theory. The interplay between charge degrees of freedom results in a rich phase diagram. Similar doped cuprates, superconductors, pseudogap phases, Fermi liquid, strange metal paramagnetic are generated. What is different that, obtain more than one superconducting (including topological one) no matter what original state is. Chern number superconductor either $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}2$ or $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}1$, depending ratio $\mathrm{\ensuremath{\Gamma}}/|K|$ channel. We further find that an intermediate in-plane magnetic field slightly enlarge size phase.