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

In our previous Letter [Phys. Rev. B 103, L140408 (2021)], we presented a discussion of the fundamental physical properties of the interactions parametrizing 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 nonchiral and independent of the spin-orbit interaction, and which we consequently termed "improper." In the preceding Comment by Cardias et al., Phys. Rev. B 105, 026401 (2022), 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 nonchiral 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."