Disclinations in the geometric theory of defects

TL;DR

This paper explores the geometric theory of defects in media with spin structure, interpreting disclinations and dislocations through Riemann--Cartan geometry and topological solitons like monopoles.

Contribution

It introduces a novel interpretation of the 't Hooft--Polyakov monopole as a defect in solid state physics and applies Chern--Simons action to model disclinations.

Findings

Monopoles correspond to dislocation and disclination distributions.

Chern--Simons action effectively describes single disclinations.

Examples include spherical and linear disclinations with specific topological properties.

Abstract

In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of elastic deformations but with nontrivial ${\mathbb S}{\mathbb O}(3)$-connection which produces nontrivial curvature and torsion tensors. We show that the 't Hooft--Polyakov monopole has physical interpretation in solid state physics describing media with continuous distribution of dislocations and disclinations. The Chern--Simons action is used for the description of single disclinations. Two examples of point disclinations are considered: spherically symmetric point "hedgehog" disclination and the point disclination for which the $n$-field has a fixed value at infinity and essential singularity at the origin. The example of linear disclinations with the Franc vector divisible by $2\pi$ is considered.