In this article, we review the recent significant progress in the theoretical studies of the electronic states by mainly focusing on Fe-based and cuprate superconductors. These superconductors are “unconventional” in that strong electron-electron correlation mediates the pairing; they are different from conventional phonon-mediated BCS superconductors. To seek the high-$T_{\mathrm{c}}$ pairing mechanism, many scientists have focused on the mysterious spontaneous rotational symmetry breaking above $T_{\mathrm{c}}$, such as nematic order at $\mathit{q}=\mathbf{0}$ and smectic order at $\mathit{q}\ne \mathbf{0}$. Such exotic correlation-driven symmetry breaking in metals has become a central issue in condensed matter physics. We demonstrate the emergence of the nematic and smectic orders due to orbital polarization ($n_{xz}\ne n_{yz}$) and the symmetry breaking in the correlated inter-site hopping (= bond order $\delta t_{i,j}$) in Fe-based and cuprate superconductors. In addition, we discuss exotic spontaneous loop current orders driven by the pure imaginary $\delta t_{i,j}$. These interesting “unconventional density-waves” originate from the quantum interference between different spin fluctuations that is described by the vertex correction (VC) in the field theory. In the next stage, we discuss electron-correlation-driven superconductivity due to the fluctuations of unconventional density-waves. For this purpose, we suggest the beyond-Migdal–Eliashberg gap equation by including the VCs into the equation. In Fe-based superconductors, high-$T_{\mathrm{c}}$ s-wave superconductivity can be mediated by nematic and smectic fluctuations because the pairing interaction is magnified by the VCs. We also discuss the multipolar fluctuation pairing mechanism in heavy fermion systems, owing to the cooperation between the strong spin-orbit interaction and the strong electron correlation. To summarize, we suggest that the quantum interference mechanism described by the VCs plays essential roles in not only various unconventional density-waves but also exotic superconducting states in many strongly correlated metals. We finally discuss some interesting future issues with respect to the quantum interference mechanism.

在本文中，我们主要关注铁基和铜酸盐超导体，回顾了近年来电子态理论研究的重大进展。这些超导体是“非常规的”，因为强电子-电子相关性介导了配对；它们不同于传统的声子介导的 BCS 超导体。寻求高${吨}_{\mathrm{C}}$配对机制，许多科学家都将注意力集中在打破上面的神秘自发旋转对称性上${吨}_{\mathrm{C}}$，例如向列顺序$\mathit{q}=\mathbf{0}$和近晶序$\mathit{q}\ne \mathbf{0}$. 金属中这种奇异的相关性驱动的对称性破缺已成为凝聚态物理学的核心问题。我们证明了由于轨道极化导致的向列和近晶阶的出现（$n_{Xz}\ne n_{是z}$）和相关站点间跳跃中的对称性破坏（=债券顺序$\delta {吨}_{我,j}$）在铁基和铜酸盐超导体中。此外，我们讨论了由纯虚数驱动的奇异自发环路电流阶数$\delta {吨}_{我,j}$. 这些有趣的“非常规密度波”源于场论中顶点校正（VC）所描述的不同自旋涨落之间的量子干涉。在下一阶段，我们将讨论由于非常规密度波的波动而导致的电子关联驱动的超导性。为此，我们通过将 VC 包含在方程中来建议 beyond-Migdal-Eliashberg 间隙方程。在铁基超导体中，高${吨}_{\mathrm{C}}$ s波超导性可以通过向列和近晶涨落来调节，因为配对相互作用被 VC 放大了。我们还讨论了重费米子系统中的多极涨落配对机制，这是由于强自旋轨道相互作用和强电子相关性之间的协作。总而言之，我们认为 VC 描述的量子干涉机制不仅在各种非常规密度波中发挥重要作用，而且在许多强相关金属的奇异超导态中也起着重要作用。我们最后讨论了一些关于量子干涉机制的有趣的未来问题。