Doping effects on electronic states in electron-doped FeSe: Impact of self-energy and vertex corrections

The pairing glue of high-$T_{\rm c}$ superconductivity in heavily electron-doped (e-doped) FeSe, in which hole-pockets are absent, has been an important unsolved problem. Here, we focus on a heavily e-doped bulk superconductor Li$_{1-x}$Fe$_x$OHFeSe ($T_{\rm c} \sim 40$K). We construct a multiorbital model beyond the rigid band approximation and analyze the spin and orbital fluctuations by taking both vertex corrections (VCs) and self-energy into consideration. Without e-doping ($x=0$), the ferro-orbital order without magnetism in FeSe is reproduced by the VCs.The orbital order quickly disappears when the hole-pocket vanishes at $x \sim 0.03$. With increasing $x$ further, the spin fluctuations remain small, whereas orbital fluctuations gradually increase with $x$ due to the VCs. The negative feedback due to the self-energy is crucial to explain experimental phase diagram. Thanks to both vertex and self-energy corrections, the orbital-fluctuation-mediated $s_{++}$-wave state appears for a wide doping range, consistent with experiments.