Through the Looking-Glass: Efficient Parity-Complete Learning via Local $O(2)$ Frames

Abstract

Making all symmetry-allowed tensor representations computationally accessible remains a central challenge in equivariant atomistic learning. Clebsch--Gordan tensor products (CGTPs) are computationally demanding, whereas efficient $SO(2)$-based alternatives lack a unified treatment of natural- and unnatural-parity representations. We first introduce a generalized Wigner-$6j$ convolution that exactly recouples interactions involving additional node representations, replacing the first edge-level tensor-product intermediate with reusable node features. This reduces edge computation and storage but leaves the $\mathcal{O}(L^5)$ angular scaling of the sparse CGTPs unchanged. To address this remaining bottleneck, we develop a local $O(2)$ framework whose convolutions scale as $\mathcal{O}(L^3)$. The framework explicitly accounts for both rotations and reflections. Its linear maps, tensor products, and gated nonlinearities, combined with frame transformations, ensure global $O(3)$ equivariance for both spatial parities. Incorporating time-reversal labels extends this construction to $O(3)\times {\mathbb{Z}}_2^{\mathcal{T}}$. We apply the framework to magnetic TACE (mTACE), using distinct interactions for systems with and without spin--orbit coupling. On collinear CrN and noncollinear Fe benchmarks, mTACE achieves substantial reductions in atomic- and magnetic-force errors. We also provide EquivariantX, a library for global and local $O(2)$-equivariant computation.