Orbital Fluctuation Theory for LaFeAsO$_{1-x}$H$_x$: Pnictogen Height Instability and Superconductivity due to Orbital Fluctuations

The isostructural transition in the tetragonal ($C_4$) phase, with sizable change in the As-height, is realized in heavily H-doped LaFeAsO, Pr-doped CaFe$_2$As$_2$, and Na-doped BaFe$_2$As$_2$. Here, we study the mechanism of spin-fluctuation-driven structure transition in LaFeAsO$_{1-x}$H$_x$ by using the self-consistent vertex correction (SC-VC) method. In heavily-doped case ($x\sim0.5$), the non-nematic orbital order is caused by the VC due to $d_{xy}$-orbital spin fluctuations, and triggers the $C_4$ isostructural transition. In lightly-doped case ($x\sim0$), the orthorhombic phase is realized by the orbital-nematic order, which originates from the VC due to ($d_{xz},d_{yz}$)-orbital spin fluctuations. Both nematic and non-nematic orbital fluctuations contribute in realizing higher-$T_{\rm c}$ superconductivity.