DISTRIBUTION PATTERNS OF TWO-DIMENSIONAL VECTOR FIELDS AND CORRESPONDING TOPOLOGICAL CHARGES

Topology-related phenomena are at the forefront of contemporary physics, and the topological charges of defects constitute one of the central topics in this field. Such charges are often represented by the winding numbers of associated vector-field distributions. Therefore, distinguishing vector-field patterns and their corresponding winding numbers provides an intuitive way to identify the topological charges of defects. It also suggests a route to designing specific vector-field distributions that can generate defect structures carrying prescribed topological charges. In this regard, it is essential for researchers to develop a clear understanding of vector fields with different winding numbers, namely different topological charges. In this paper, we first clarify the relationship between the topological charge of defects arising from a Schrödinger wave-function field in quantum mechanics and the winding number of the corresponding two-dimensional vector-field distribution. We then present in detail the vector-field patterns associated with arbitrary positive and negative integer winding numbers in two dimensions. Finally, we provide a quantitative calculation of the vector orientation at an arbitrary point in the two-dimensional plane. This paper may serve as a useful reference for junior graduate students and senior undergraduate students who are beginning to study topological physics.