Symmetry Rules on Multipole Interactions under Crystallographic Point Groups and Application to Multiple-$Q$ Multipole States

TL;DR

This paper systematically establishes symmetry rules for multipole interactions under crystallographic point groups and explores their role in stabilizing exotic multiple-Q multipole states, with implications for unconventional electronic phases.

Contribution

It provides a comprehensive symmetry-based framework for understanding multipole interactions and demonstrates their potential to induce novel quantum states like triple-Q quadrupole order.

Findings

Symmetry conditions for antisymmetric and symmetric multipole interactions derived.

Extension of Dzyaloshinskii-Moriya and compass interactions to multipoles shown.

Triple-Q quadrupole state identified on a triangular lattice.

Abstract

Multipole degrees of freedom describe the mutual interplay among the charge, spin, and orbital degrees of freedom in electrons, which provides a microscopic understanding of unconventional electronic orderings and their associated physical phenomena. We here show the symmetry rules on multipole interactions under crystallographic point groups in a systematic manner. Depending on the bond symmetries, we show the necessary symmetry conditions of the antisymmetric multipole interactions, which correspond to the extension of the Dzyaloshinskii-Moriya interaction, as well as the symmetric ones, which correspond to the extension of the compasslike interaction. Furthermore, we demonstrate that the symmetry-allowed multipole interactions can become a source of exotic multiple-$Q$ multipole orderings. As a specific example, we analyze the effective model with the antisymmetric quadrupole interaction on a triangular lattice and show the emergence of the triple-$Q$ quadrupole state. Our results indicate that multipole interactions that often arise from the heavy-fermion, frustrated, and nematic systems can potentially induce further unconventional quantum states of matter.