Multipoles as quantitative order parameters for altermagnetic spin splitting

TL;DR

This paper establishes a quantitative link between altermagnetic spin splitting and higher order multipoles of charge and magnetization density, using first-principles calculations on perovskite materials to identify multipoles as potential order parameters.

Contribution

It introduces a quantitative relation between spin splitting and local multipoles, showing the superposition of multiple multipoles influences altermagnetism, advancing the understanding of order parameters in this field.

Findings

Spin splitting correlates with multiple multipoles, not just the lowest order.

Structural distortions can modulate multipole magnitudes and spin splitting.

A multi-component order parameter is necessary to describe altermagnetism.

Abstract

We establish a quantitative relation between the altermagnetic spin-splitting and different higher order multipoles of the charge and magnetization density around the magnetic atoms. Magnetic multipoles such as octupoles or triakontadipoles have been suggested as potential ferroic order parameters for d- and g-wave altermagnetism, respectively, based mainly on qualitative symmetry arguments. We use first-principles-based electronic structure calculations to establish a clear quantitative relation between the strength of the altermagnetic spin splitting and the magnitude of certain local multipoles. We vary the magnitude of these multipoles either by applying an appropriate constraint on the charge density or by varying a corresponding structural distortion mode, using two simple perovskite materials, SrCrO3 and LaVO3, as model systems. Our analysis indicates that in general the altermagnetic spin splitting is not exclusively determined by the lowest order nonzero magnetic multipole, but results from a superposition of contributions from different multipoles with comparable strength, suggesting the need for a multi-component order parameter to describe altermagnetism. We also discuss different measures to quantify the overall spin-splitting of a material, without relying on features that might be specific to only individual bands.