By considering an extended double-exchange model with spin-orbit coupling (SOC), we derive a general form of the Berry phase $\gamma$ that electrons pick up when moving around a closed loop. This form generalizes the well-known result valid for SU(2) invariant systems, $\gamma =\mathrm{\Omega }/2$, where $\mathrm{\Omega }$ is the solid angle subtended by the local magnetic moments enclosed by the loop. The general form of $\gamma$ demonstrates that collinear and coplanar magnetic textures can also induce a Berry phase different from 0 or $\pi$, smoothly connecting the result for SU(2) invariant systems with the well-known result of Karplus and Luttinger for collinear ferromagnets with finite SOC. By taking the continuum limit of the theory, we also derive the corresponding generalized form of the real-space Berry curvature. The new expression is a generalization of the scalar spin chirality, which is presented in an explicitly covariant form. We finally show how these simple concepts can be used to understand the origin of the spontaneous topological Hall effect that has been recently reported in collinear and coplanar antiferromagnetic phases of correlated materials.

中文翻译：

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自旋轨道相互作用的迭代电子系统的实空间贝里曲率

通过考虑具有自旋轨道耦合（SOC）的扩展双交换模型，我们导出了Berry相的一般形式 $\gamma$当电子在闭环中运动时电子会吸收。这种形式概括了对SU（2）不变系统有效的众所周知的结果，$\gamma =\mathrm{\Omega }/2$，在哪里 $\mathrm{\Omega }$是由环围起来的局部磁矩对角的立体角。一般形式$\gamma$ 证明共线和共面的磁织构也可以诱发不同于0或 $\pi$，将SU（2）不变系统的结果与Karplus和Luttinger的有限SOC共线铁磁体的结果平稳地联系起来。通过采用该理论的连续极限，我们还导出了实空间贝里曲率的相应广义形式。新的表达式是标量自旋手性的概括，以明确的协变形式表示。最后，我们将展示如何使用这些简单的概念来理解自发的拓扑霍尔效应的起源，该效应最近已在相关材料的共线和共面反铁磁相中报道。