Comprehensive classification of second harmonic generation effect in magnetic materials

Studying of the relationship between magnetism and second harmonic generation effect (SHG) is not only a fundamental topic in condensed matter physics, but also has practical significance in experiments. In this work, we construct a complete and exclusive classification of SHG effect in magnetic materials, which offers a straightforward approach to inspect this intriguing connection and reveals the symmetries and origins of SHG. First, we study the symmetries of SHG tensors in all magnetic point groups with isomorphic group method, then we propose a comprehensive classification of SHG effect containing seven types by considering the symmetries of the magnetic phases and the corresponding parent phases. A tensor dictionary containing SHG and linear magneto-optic effect (LMO) is created, and a SHG database of magnetic materials is also established. Particularly, we find an anomalous SHG effect which is even and contributed by magnetism merely through our classification. Besides, the classification is conducive to the determination of magnetic structures via SHG.