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Disclinations in the Geometric Theory of Defects

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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 \(2\pi\).

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 19-11-50067.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    M. O. Katanaev

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Correspondence to M. O. Katanaev.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 313, pp. 87–108 https://doi.org/10.4213/tm4158.

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Katanaev, M.O. Disclinations in the Geometric Theory of Defects. Proc. Steklov Inst. Math. 313, 78–98 (2021). https://doi.org/10.1134/S0081543821020097

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