The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, M(x), based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate the skyrmion-induced Hall conductivity and confirm that it depends on the shape of skyrmions, such as Bloch-type or Néel-type skyrmions. The results of this study can be applied not only to conventional skyrmion systems but also to more general magnetic structures.

中文翻译：

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拓扑绝缘子表面磁纹理诱导的异常霍尔效应的理论

霍尔效应异常是由诸如天蝎子之类的磁性纹理引起的。我们推导了一个拓扑绝缘子表面上的霍尔电导率的解析公式，其磁化强度高达三阶，M（x），基于微扰方法。我们确定了磁织构，这些磁织构对霍尔电导率的贡献在磁化强度方面高达三阶，而在空间差异方面则高达二阶。我们根据线性响应理论将磁化强度视为一种扰动，以计算每个磁性纹理的霍尔电导率。此外，我们估计了天敌离子诱导的霍尔电导率，并确认它取决于天敌离子的形状，例如Bloch型或Néel型天敌离子。这项研究的结果不仅可以应用于传统的天体离子系统，而且可以应用于更一般的磁性结构。