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 $\eta$ based on a group-theoretical approach. We show that the leading-order dependence of $G_{0,u}(\eta)$ 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 $G_{0,u}(\eta)$ is directly reflected in observable quantities such as CIM and chiral phonon splitting (CPS), both of which scale consistently with $G_{0,u}(\eta)$. We further clarify the microscopic mechanism by which AtC transitions give rise to chiral phonons and CPS through the coupling between $G_{0,u}(\eta)$ and phonon degrees of freedom.