发表机构
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学前沿科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对不可约笛卡尔张量分解与耦合缺乏通用构造的问题，本文提出仅依赖秩与对称性的可复用算子，并实现于开源包natto，应用于弹性张量及等变机器学习。
AI 中文摘要
分子和材料性质，从极化率到弹性常数，都由张量描述。当张量被分解为独立变换的不可约部分时，其在旋转下的行为变得明确。在笛卡尔形式中，这些部分是对称且无迹的不可约笛卡尔张量（ICTs），其分解与耦合构成了选择规则、取向平均和对称性应用的基础。在秩为二时，分解是教科书级的结果，但更高秩以及物理张量的内禀对称性使其变得非平凡。与已成熟建立的球谐形式不同，目前缺乏针对给定内禀对称性的通用构造，以及可复用的算子来提取ICTs并精确重建原始张量。在此，我们开发了这样一种构造，并显式地获得了这些算子。这些算子仅依赖于秩和对称性，因此每个算子只需构建一次，即可应用于该类的任何张量。在此基础上，我们进一步获得了矢量的笛卡尔谐波以及将两个ICTs耦合为第三个ICTs的算子，这两者对于等变机器学习都至关重要。该构造在弹性张量上得到了演示，包括其秩为四的二阶形式和秩为六的三阶形式。秩为四张量的ICTs还定义了一种旋转和尺度不变的各向异性度量，我们利用Materials Project中晶体材料的第一性原理弹性张量对其进行了评估。该构造已在开源软件包natto中实现，该软件包以精确符号和数值形式生成算子。
英文摘要
Molecular and material properties, from the polarizability to the elastic constants, are described by tensors. Their behavior under rotations is made explicit when a tensor is decomposed into irreducible parts that transform independently. In Cartesian form these parts are the symmetric and traceless irreducible Cartesian tensors (ICTs), whose decomposition and coupling underlie selection rules, orientational averages, and the use of symmetry. The decomposition is a textbook result at rank two, but higher rank and the intrinsic symmetry of physical tensors make it nontrivial. What has been lacking, unlike in the well-established spherical formalism, is a general construction for a given intrinsic symmetry, together with reusable operators that extract the ICTs and rebuild the original tensor exactly. Here, we develop such a construction and obtain these operators explicitly. These operators depend on rank and symmetry alone, and therefore each need only be built once and then applied to any tensor of that class. Building on them, we further obtain the Cartesian harmonics of a vector and the operators that couple two ICTs into a third, both central to equivariant machine learning. The construction is demonstrated on the elastic tensor, in both its second-order form of rank four and its third-order form of rank six. The ICTs of the rank-4 tensor also define a rotation- and scale-invariant measure of anisotropy, which we evaluate across the first-principles elastic tensors of crystalline materials from the Materials Project. The construction is implemented in the open-source package natto, which produces the operators in both exact symbolic and numerical form.
