Dual-circular Raman optical activity of axial multipolar order

Abstract

Multipolar order, such as octupolar order, is a key concept in condensed matter physics, particularly in light of elusive hidden orders. However, its experimental identification remains challenging due to the absence of direct coupling to conventional external stimuli. In this study, we propose that dual-circular Raman scattering serves as a probe of multipolar anisotropies. By combining symmetry analysis with microscopic calculations, we demonstrate that both time-reversal-even ($θ$-even) and time-reversal-odd ($θ$-odd) axial multipolar phases exhibit the Raman optical activity as a direct consequence of mirror symmetry breaking. Furthermore, we demonstrated that a multipolar phonon, a three-dimensional and alternating displacement resembling the chiral phonon, plays a vital role in the proposed optical phenomena. Our findings open a pathway for identifying multipolar orders in various materials through dual-circular Raman spectroscopy as a sensitive and versatile probe.

Figures (3)

• Figure 1: (a) $xyz$-type axial octupolar order. Faces of the octahedron are colored according to the penetrating direction of axial vectors (orange arrows) on each surface. (b) Mirror operation $m_\perp$ perpendicular to the octahedron's faces, illustrated by the green-colored mirror plane, flips the polarization of axial vectors, leading to the reversal of octupolarization. (c,d) Cross-circular Raman optical activity (ROA) of the axial multipolar systems. The conversion rate of the circular polarization of light differs between the right-to-left ($I_\text{RL}$) and left-to-right ($I_\text{LR}$) changes, depending on the axial polarization, which corresponds to the facets. Cross-circular ROA ($I_\text{RL} - I_\text{LR}$) for (c) the $\left\{ 111\right\}$ incidence and (d) the $\left\{ \bar{1}11\right\}$ incidence show the opposite signs.
• Figure 2: Hopping from $d_{z^2}$ to the perturbed $d_{yz}$ state, defined in the coordinate system with $X = [1\bar{1}0]$ and $Z = [111]$ in (a) and with $X' = [110]$ and $Z' = [\bar{1}11]$ in (b). The hopping process is decomposed into $d_{z^2} \to d_{yz}$ and $d_{z^2} \to \pm d_{zx}$, which result from the original ($H_0$) and axial octupolar ($H_\text{ax}$) terms, respectively. The hybridized $\ket{d_{zx}}$ state stems from the $\pm \pi/2$ rotation of the original $\ket{d_{yz}}$ state mediated by the face-dependent axial dipoles.
• Figure 3: Dependence on the incident light frequency $\omega$ of (a) nonlinear susceptibilities $|\chi_1|^2$ and $|\chi_2|^2$ for the $A_{2g}^+$ octupolar system and (b) indicator for cross-circular ROA CC$_\chi \equiv \left( | \chi_1|^2 - | \chi_2|^2 \right) / \left( | \chi_1|^2 + | \chi_2|^2 \right)$. The two polarities of the $A_{2g}^+$ octupolar order are considered in (b). (c) $\omega$ dependence of CC$_\chi$ for the $A_{2g}^-$ octupolar systems with opposite polarities. $\delta \omega = 0.1$ and $t_\text{ax} = \pm 0.1$ are used.