Using Equivariant Neural Networks to Learn Multipolar Order Parameters

Magnetoelectric multipoles are crucial in describing order parameters and characterizing complex magnetic materials. Higher order multipoles beyond local magnetic dipole moments provide a classification of order in materials with net zero magnetization and time reversal symmetry breaking (such as altermagnets) [1]. However, these order parameters are often hidden, meaning that they are not easily identifiable in conventional experiments and can be cumbersome to calculate using traditional symmetry analysis. We present an automated framework for identifying magnetoelectric multipoles using equivariant neural networks (ENNs) that incorporate O(3) and time-reversal symmetry. As shown in [2], an external input parameter to an equivariant model can be used to break symmetry in a physically interpretable way and to learn symmetry-implied order parameters. We apply this approach to learn local atomic-site magnetoelectric multipoles in altermagnetic and magnetoelectric materials, generalizing to both collinear and non-collinear magnetic systems.  

[1] McClarty, P. A., Rau, J.G. (2024). Landau Theory of Altermagnetism. Physical Review Letters, 132, 176702.

[2] Smidt, T. E., Geiger, M., & Miller, B. K. (2021). Finding symmetry breaking order parameters with Euclidean neural networks. Physical Review Research, 3(1), L012002.