从Wigner-6j重耦合到局域O(2)框架的完整O(3)相互作用
Through the Looking-Glass: Efficient Parity-Complete Learning via Local $O(2)$ Frames
- City University of Hong Kong（香港城市大学）
- Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对现有等变原子模型无法同时处理极张量与赝张量的问题，提出基于局域O(2)框架的完整O(3)卷积框架，将CGTP复杂度从O(L^6)降至O(L^3)，构建了包含O2Linear等的闭算子系统。
AI中文摘要：
等变原子模型通常使用Clebsch-Gordan张量积（CGTP）或边对齐SO(2)操作。CGTP支持一般O(3)表示，但在高阶角动量时计算成本高昂；现有局域SO(2)框架方法针对SO(3)或O(3)的自然宇称子集设计，因此当极张量和赝张量同时出现时，无法提供完整的局域表示。本研究针对非共线磁性、电场和磁场引入广义Wigner-6j卷积，提供可整合任意节点等变特征的O(3)下通用卷积方案；进一步开发基于局域O(2)框架的完整O(3)卷积框架，将CGTP的计算复杂度从O(L^6)降至O(L^3)；建立全局O(3)不可约表示（irrep）与局域O(2) irrep的对应关系，并用该对应关系构建包含O2Linear、O2TensorProduct和O2Gate的闭算子系统。
英文摘要：
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.
