Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors

TL;DR

This paper systematically classifies atomic-scale multipoles under crystallographic point groups and explores their influence on electronic structures and linear response phenomena in condensed matter physics.

Contribution

It introduces a comprehensive classification scheme for multipoles including their effects on band structures and response functions, extending to momentum space and spin considerations.

Findings

Complete table of active multipoles for 32 point groups

Analysis of multipole effects on linear responses like magneto-electric and spin conductivity

Framework for identifying electronic order parameters and cross-correlated phenomena

Abstract

Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal, and electric toroidal multipoles, give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structures and linear responses, such as the magneto-electric effect, magneto-current (magneto-gyrotropic) effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from microscopic point of view.