Title: Classification of second harmonic generation effect in magnetically ordered materials

Abstract: 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 connect magnetic order and 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 characteristic under time-reversal symmetry, which is solely contributed by magnetic structure. Moreover, the proposed classification scheme facilitates the determination of magnetic structures through SHG effect.