# Symmetry Analysis of Spin-Dependent Electric Dipole and Its Application to Magnetoelectric Effects

## TL;DR

This paper uses group theory to classify spin-dependent electric dipoles, revealing how symmetry influences multiferroic effects and electric polarization in magnetic systems with or without inversion symmetry.

## Contribution

It provides a comprehensive symmetry-based classification of spin-dependent electric dipoles, including new insights into their behavior in systems with various symmetries and their relation to multiferroic phenomena.

## Key findings

- Electric dipoles from single spins are quadrupole-like, not dipoles.
- Inversion symmetry prevents single-spin electric dipoles but allows two-spin induced dipoles.
- Symmetry analysis explains multiferroic properties in spin dimer systems.

## Abstract

Spin-dependent electric dipole operators are investigated group-theoretically for the emergence of an electric dipole induced by a single spin or by two spins, where the spin dependences are completely classified up to the quadratic order. For a single spin, a product of spin operators behaves as an even-parity electric quadrupole operator, which differs from an odd-parity electric dipole. The lack of the inversion symmetry allows the even- and odd-parity mixing, which leads to the electric dipole described by the electric quadruple operators. Point-group tables are given for classification of the possible spin-dependent electric dipoles and for the qualitative analysis of multiferroic properties, such as an emergent electric dipole moment coexisting with a magnetic moment, electromagnon excitation, and directional dichroism. The results can be applied to a magnetic ion in crystals or embedded in molecules at a site without the inversion symmetry. In the presence of an inversion symmetry, the electric dipole does not appear for a single spin. This is not the case for the electric dipole induced by two spins with antisymmetric spin dependence, which is known as vector spin chirality, in the presence of the inversion center between the two spins. In the absence of the inversion center, symmetric spin-dependent electric dipoles are also relevant. The detailed analysis of various symmetries of two-spin states is applied to spin dimer systems and the related multiferroic properties.