Reply to 'Comment on "Proper and improper chiral magnetic interactions" '

TL;DR

This paper clarifies the distinction between proper and improper magnetic interactions in atomistic spin models, addressing misconceptions raised in a comment and emphasizing the importance of a missing step in the local approach to first-principles mapping.

Contribution

It provides a detailed correction to the interpretation of Dzyaloshinskii-Moriya interactions, highlighting the significance of the missing step in the local approach to ensure physical consistency.

Findings

Identifies the missing step in the local approach to magnetic energy mapping.

Explains how ignoring this step leads to non-physical interactions.

Reaffirms the distinction between proper and improper magnetic interactions.

Abstract

In our previous Letter [Phys. Rev. B 103, L140408 (2021)], we presented a discussion of the fundamental physical properties of the interactions parameterizing atomistic spin models in connection to first-principles approaches that enable their calculation for a given material. This explained how some of those approaches can apparently lead to magnetic interactions that do not comply with the expected physical properties, such as Dzyaloshinskii-Moriya interactions which are non-chiral and independent of the spin-orbit interaction, and which we consequently termed `improper'. In the preceding Comment [Phys. Rev. B 105, 026401], the authors present arguments based on the distinction between global and local approaches to the mapping of the magnetic energy using first-principles calculations to support their proposed non-chiral Dzyaloshinskii-Moriya interactions and their dismissal of our distinction between `proper' and `improper' magnetic interactions. In this Reply, we identify the missing step in the local approach to the mapping and explain how ignoring this step leads to the identification of magnetic interactions which do not comply with established physical principles and that we have previously termed `improper'.