Emergent inductors in nonhelical magnets

The emergent inductor, which is a concept of an inductor employing quantum mechanics on helical magnets, has been studied actively from both theoretical and experimental aspects. Interestingly, finite inductance has been observed not only in spiral magnetic phases but also across various other magnetic phases, although the underlying mechanism behind emergent inductance in nonhelical magnets remains unresolved. In this paper, we broaden the concept of emergent inductance to encompass nonhelical magnets and establish a comprehensive formalism of the emergent inductance generated by magnetization dynamics. Using a diagrammatic approach, we derive the linear response of current density mediated by magnetization dynamics and formulate impedance and inductance for general magnetic materials. We reveal that the emergent inductance is composed of two distinct components, each contributing positively or negatively. Notably, the positive inductance arises from Ohmic dissipation including the effect of the emergent electric field (Berry phase in real space) while the negative inductance arises from the polarization of itinerant electrons (Berry curvature in $k\ensuremath{-}t$ space). We also apply the present method to quasi-1D models and show the numerical results of the inductance, the spin dynamics, the impedance, and the $Q$ value.