AI 中文总结
针对不可归一化矢量场拓扑分类难题，通过转换为(n + 1)维归一化矢量场构建通用框架，扩展同伦分类，导出拓扑不变量，为不可归一化场拓扑表征提供统一途径，助力多系统拓扑现象研究。
AI 中文摘要
传统上，物理矢量场的拓扑分类依赖于场归一化和基于同伦的不变量。然而，当场振幅消失时，归一化变得不明确，阻碍了直接的拓扑表征。在此，我们通过将具有可紧致化基空间的不可归一化n维矢量场转换为(n + 1)维归一化矢量场，引入了一个通用框架。该构造将基于同伦的分类扩展到包含振幅零点的场。我们明确展示了针对一、二和三维不可归一化矢量场的方法，并导出了相应的拓扑不变量。所得拓扑电荷在连续变形下是稳健的，仅在嵌入结构变得奇异时才会改变。我们的框架为不可归一化场的拓扑表征提供了统一途径，并为研究包括磁纹理、铁电体、电磁场和波系统在内的广泛系统中的拓扑现象打开了大门。
英文摘要
Topological classification of physical vector fields conventionally relies on field normalization and homotopy-based invariants. However, when field amplitudes vanish, normalization becomes ill-defined, preventing a direct topological characterization. Here, we introduce a general framework for the topological classification of non-normalizable $n$-dimensional vector fields with compactifiable base spaces by transforming them into $(n+1)$-dimensional normalized vector fields. This construction extends homotopy-based classification to fields containing amplitude zeros. We explicitly demonstrate the approach for one-, two-, and three-dimensional non-normalized vector fields and derive the corresponding topological invariants. The resulting topological charges are robust under continuous deformations and can change only when the embedding structure becomes singular. Our framework provides a unified route to the topological characterization of non-normalizable fields and opens the door to the study of topological phenomena in a broad range of systems, including magnetic textures, ferroelectrics, electromagnetic fields, and wave systems.
Comments8 pages, 4 figures
