Disclinations in the Geometric Theory of Defects

M. O. Katanaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 87-108

Résumé

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 $.