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Diverse Exotic Orders and Fermiology in Fe-based Superconductors: A Unified Mechanism for $B_{1g}/B_{2g}$ Nematicity in FeSe/(Cs,Rb)Fe$_2$As$_2$ and Smectic Order in BaFe$_2$As$_2$

A rich variety of nematic/smectic orders in Fe-based superconductors is an important unsolved problem in strongly correlated electron systems. A unified understanding of these orders has been investigated for the last decade. In this article, we explain the $B_{1g}$ symmetry nematic transition in FeSe$_{1-x}$Te$_x$, the $B_{2g}$ symmetry nematicity in AFe$_2$As$_2$ (A=Cs, Rb), and the smectic state in BaFe$_2$As$_2$ based on the same framework. We investigate the quantum interference mechanism between spin fluctuations by developing the density wave equation. The observed rich variety of nematic/smectic orders is naturally understood in this mechanism. The nematic/smectic orders depend on the characteristic shape and topology of the Fermi surface (FS) of each compound. (i) In FeSe$_{1-x}$Te$_x$, each FS is very small and the dxy-orbital hole pocket is below the Fermi level. In this case, the small spin fluctuations on three dxz, dyz, and dxy orbitals cooperatively lead to the $B_{1g}$ nematic order. The experimental Lifshitz transition below the nematic transition temperature $(T_S)$ is naturally reproduced. (ii) In BaFe$_2$As$_2$, the dxy-orbital hole pocket emerges around M point, and each FS is relatively large. The strong spin fluctuations due to the dxy-orbital nesting give rise to the $B_{1g}$ nematic order and the smectic order, and the latter transition temperature ($T^*$) exceeds the former one $T_S$. (iii) In heavily hole-doped AFe$_2$As$_2$, the large dxy-orbital hole pocket and the four tiny Dirac pockets appear due to the hole-doping. The $B_{2g}$ nematic bond order emerges on the dxy-orbital hole pocket due to the same interference mechanism. The present paramagnon interference mechanism provides a unified explanation of why the variety of nematic/smectic orders in Fe-based superconductors is so rich, based on the well-established fermiology of Fe-based superconductors.