Abstract

In the geometric theory of defects, media with a spin structure (for example, ferromagnets) are regarded as manifolds with given Riemann–Cartan geometry. We consider the case with the Euclidean metric, which corresponds to the absence of elastic deformations, but with nontrivial \(\mathbb{SO}(3)\) connection, which produces nontrivial curvature and torsion tensors. We show that the ’t Hooft–Polyakov monopole has a physical interpretation; namely, in solid state physics it describes media with continuous distribution of dislocations and disclinations. To describe single disclinations, we use the Chern–Simons action. We give two examples of point disclinations: a spherically symmetric point “hedgehog” disclination and a point disclination for which the \(n\)-field takes a fixed value at infinity and has an essential singularity at the origin. We also construct an example of linear disclinations with Frank vector divisible by \(2\pi\).

中文翻译：

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缺陷几何理论中的向错

摘要

在缺陷的几何理论中，具有自旋结构的介质（例如，铁磁体）被视为具有给定 Riemann-Cartan 几何的流形。我们考虑欧几里德度量的情况，它对应于没有弹性变形，但具有非平凡的\(\mathbb{SO}(3)\)连接，它产生非平凡的曲率和扭转张量。我们证明了 't Hooft-Polyakov 单极子具有物理解释；即，在固态物理学中，它描述了具有连续分布的位错和向错的介质。为了描述单向错，我们使用陈-西蒙斯行动。我们给出了两个点向错的例子：一个球对称点“刺猬”向错和一个点向错，其中\(n\)-field 在无穷远处取一个固定值，并且在原点有一个本质奇点。我们还构建了一个线性向错的例子，其中 Frank 向量可被\(2\pi\)整除。