Complete Multipole Basis Set for Single-Centered Electron Systems

TL;DR

This paper introduces a comprehensive multipole basis set for single-centered electron systems, enabling detailed analysis of electronic states, phase transitions, and spectroscopic phenomena with new formulas and visualization techniques.

Contribution

It provides a complete basis set for four types of multipoles and a compact formula for matrix elements, advancing the analysis of electron systems and related phenomena.

Findings

Demonstrated the basis set with monopole and toroidal dipole orderings

Analyzed relationships among magnetic dipoles in spin-orbit systems

Proposed a visualization method for multipole-characterized states

Abstract

A whole series of expressions for four species of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal) is provided as a complete basis set to describe arbitrary single-centered spinful electron systems. A compact formula to calculate matrix elements of these multipoles is also derived. A visualization method of an electronic state characterized in terms of multipoles is proposed. The complete basis set is useful to narrow down a candidate order parameter of electron systems in phase transition, to describe a property of cross-correlated phenomena, to analyze spectra of x-ray scattering in magnetically ordered state, and so on. We demonstrate a usage of the complete basis set by taking monopole and toroidal dipole orderings, and the mutual relationship among three distinct magnetic dipoles (orbital, spin angular momenta and anisotropic dipole) in a spin-orbit coupled system as prime examples.