Order parameter scaling of chirality in structural phase transitions
Chirality, defined by the absence of mirror and inversion symmetries, has attracted considerable attention owing to its unique physical phenomena, including cross-correlated responses such as current-induced magnetization (CIM) and chiral phonons. Recently, it has been established that chirality is characterized by electric toroidal (ET) multipoles: the ET monopole $G_0$ in cubic systems and the ET quadrupole $G_u$ in noncubic systems. In this paper, we investigate achiral-to-chiral (AtC) structural phase transitions driven by atomic displacements and construct $G_{0,u}$ as explicit functions of the displacement order parameter $η$ based on a group-theoretical approach. We show that the leading-order dependence of $G_{0,u}(η)$ is determined by the symmetry of the parent structure and the character of the displacive mode, providing a symmetry-based classification of AtC transitions beyond a binary distinction between achiral and chiral phases. We also demonstrate that $G_{0,u}(η)$ is directly reflected in observable quantities such as CIM and chiral phonon splitting (CPS), both of which scale consistently with $G_{0,u}(η)$. We further clarify the microscopic mechanism by which AtC transitions give rise to chiral phonons and CPS through the coupling between $G_{0,u}(η)$ and phonon degrees of freedom.