The relationship between magnetic order and the second harmonic generation (SHG) effect is a fundamental area of study in condensed matter physics with significant practical implications. In order to gain a clearer understanding of this intricate relation, this study presents a comprehensive classification scheme for the SHG effect in magnetically ordered materials. This framework offers a straightforward approach to connecting magnetic order and the SHG effect. The characteristics of the SHG tensors in all magnetic point groups are studied using the isomorphic group method, followed by a comprehensive SHG effect classification scheme that includes seven types based on the symmetries of the magnetic phases and their corresponding parent phases. In addition, a tensor dictionary containing the SHG and linear magneto-optic (LMO) effect is established. Furthermore, an extensive SHG database of magnetically ordered materials is also built up. This classification strategy exposes an anomalous SHG effect with even characteristics (i.e., invariant) under time-reversal symmetry, which is solely contributed by magnetic structure. Moreover, the proposed classification scheme facilitates the determination of magnetic structures through the SHG effect.

磁序与二次谐波产生（SHG）效应之间的关系是凝聚态物理研究的一个基本领域，具有重要的实际意义。为了更清楚地理解这种复杂的关系，本研究提出了磁有序材料中倍频效应的综合分类方案。该框架提供了一种连接磁序和倍频效应的简单方法。利用同构群方法研究了所有磁点组中的倍频张量的特征，然后根据磁相及其相应母相的对称性，提出了包括七种类型的综合倍频效应分类方案。此外，还建立了包含SHG和线性磁光（LMO）效应的张量字典。此外，还建立了广泛的磁有序材料的二次谐波数据库。这种分类策略揭示了在时间反转对称性下具有均匀特性（即不变）的异常倍频效应，该效应仅由磁结构贡献。此外，所提出的分类方案有助于通过倍频效应确定磁性结构。