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 $.

Bibliographie

[1] Azevedo S., “Charged particle with magnetic moment in the background of line topological defect”, Phys. Lett. A, 307:1 (2003), 65–68 | DOI | MR | Zbl

[2] Bakke K., Furtado C., “Abelian geometric phase due to the presence of an edge dislocation”, Phys. Rev. A, 87:1 (2013), 012130 | DOI

[3] Bilby B.A., Bullough R., Smith E., “Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry”, Proc. R. Soc. London A, 231 (1955), 263–273 | DOI | MR

[4] E. B. Bogomol'nyi, “The stability of classical solutions”, Sov. J. Nucl. Phys., 24:4 (1976), 449–454 | MR

[5] Böhmer C.G., Obukhov Yu.N., “A gauge-theoretical approach to elasticity with microrotations”, Proc. R. Soc. London A, 468 (2012), 1391–1407 | MR | Zbl

[6] Cartan E., “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion”, C. r. Acad. sci. Paris, 174 (1922), 593–595 | Zbl

[7] Chern S.-S., Simons J., “Characteristic forms and geometric invariants”, Ann. Math. Ser. 2, 99:1 (1974), 48–69 | DOI | MR | Zbl

[8] Ciappina M.F., Iorio A., Pais P., Zampeli A., “Torsion in quantum field theory through time-loops on Dirac materials”, Phys. Rev. D, 101:3 (2020), 036021 | DOI | MR

[9] De Berredo-Peixoto G., Katanaev M.O., “Inside the BTZ black hole”, Phys. Rev. D, 75:2 (2007), 024004 | DOI | MR

[10] De Berredo-Peixoto G., Katanaev M.O., “Tube dislocations in gravity”, J. Math. Phys., 50:4 (2009), 042501 | DOI | MR | Zbl

[11] De Berredo-Peixoto G., Katanaev M.O., Konstantinova E., Shapiro I.L., “Schrödinder equation in the space with cylindrical geometric defect and possible application to multi-wall nanotubes”, Nuovo Cimento B, 125:8 (2010), 915–931

[12] De Padua A., Parisio-Filho F., Moraes F., “Geodesics around line defects in elastic solids”, Phys. Lett. A, 238:2–3 (1998), 153–158 | DOI | MR

[13] B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry: Methods and Applications, v. I, The Geometry of Surfaces, Transformation Groups, and Fields, Editorial URSS, Moscow, 1998 | DOI | MR | MR

[14] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry: Methods and Applications bookinfo Part I: The Geometry of Surfaces, Transformation Groups, and Fields, Grad. Texts Math., 93, Springer, New York, 1992 | DOI | MR | MR

[15] Dzyaloshinskii I.E., Volovik G.E., “On the concept of local invariance in the theory of spin glasses”, J. Phys., 39:6 (1978), 693–700 | DOI

[16] Dzyaloshinskii I.E., Volovik G.E., “Poisson brackets in condensed matter physics”, Ann. Phys., 125:1 (1980), 67–97 | DOI | MR

[17] Frank F.C., “On the theory of liquid crystals”, Discuss. Faraday Soc., 25 (1958), 19–28 | DOI

[18] Furtado C., Moraes F., de M. Carvalho A.M., “Geometric phases in graphitic cones”, Phys. Lett. A, 372:32 (2008), 5368–5371 | DOI | Zbl

[19] Hehl F.W., McCrea J.D., Mielke E.W., Ne'eman Y., “Metric-affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance”, Phys. Rep., 258:1–2 (1995), 1–171 | DOI | MR

[20] Hertz J.A., “Gauge models for spin-glasses”, Phys. Rev. B, 18:9 (1978), 4875–4885 | DOI

[21] 't Hooft G., “Magnetic monopoles in unified gauge theories”, Nucl. Phys. B, 79:2 (1974), 276–284 | DOI | MR

[22] A. Kadić and D. G. B. Edelen, A Gauge Theory of Dislocations and Disclinations, Lect. Notes Phys., 174, Springer, Berlin, 1983 | DOI | MR | Zbl

[23] M. O. Katanaev, “Wedge dislocation in the geometric theory of defects”, Theor. Math. Phys., 135:2 (2003), 733–744 | DOI | MR | MR | Zbl

[24] M. O. Katanaev, “One-dimensional topologically nontrivial solutions in the Skyrme model”, Theor. Math. Phys., 138:2 (2004), 163–176 | DOI | MR | MR | Zbl

[25] Katanaev M., “Torsion and Burgers vector of a tube dislocation”, Proc. Sci., 127 (CNCFG2010) (2011), 022

[26] Katanaev M.O., “Rotational elastic waves in double wall tube”, Phys. Lett. A, 379:24–25 (2015), 1544–1548 | DOI | MR | Zbl

[27] Katanaev M.O., “Rotational elastic waves in a cylindrical waveguide with wedge dislocation”, J. Phys. A: Math. Theor., 49:8 (2016), 085202 | DOI | MR | Zbl

[28] M. O. Katanaev, “Description of disclinations and dislocations by the Chern–Simons action for $\mathbb {SO}(3)$ connection”, Phys. Part. Nucl., 49:5 (2018), 890–893 | DOI

[29] M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133 | DOI | MR | Zbl

[30] M. O. Katanaev, “Gauge parameterization of the $n$-field”, Proc. Steklov Inst. Math., 306 (2019), 127–134 | DOI | MR | Zbl

[31] Katanaev M.O., “The 't Hooft–Polyakov monopole in the geometric theory of defects”, Mod. Phys. Lett. B, 34:12 (2020), 2050126 | DOI | MR

[32] M. O. Katanaev and I. G. Mannanov, “Wedge dislocations, three-dimensional gravity, and the Riemann–Hilbert problem”, Phys. Part. Nucl., 43:5 (2012), 639–643 | DOI | MR

[33] Katanaev M.O., Mannanov I.G., “Wedge dislocations and three-dimensional gravity”, $p$-Adic Numbers Ultrametric Anal. Appl., 4:1 (2012), 5–19 | DOI | MR | Zbl

[34] Katanaev M.O., Volkov B.O., “Point disclinations in the Chern–Simons geometric theory of defects”, Mod. Phys. Lett. B, 34:Supp01 (2020), 2150012 | DOI | MR

[35] Katanaev M.O., Volovich I.V., “Theory of defects in solids and three-dimensional gravity”, Ann. Phys., 216:1 (1992), 1–28 | DOI | MR | Zbl

[36] Katanaev M.O., Volovich I.V., “Scattering on dislocations and cosmic strings in the geometric theory of defects”, Ann. Phys., 271:2 (1999), 203–232 | DOI | MR | Zbl

[37] Kleinert H., Gauge fields in condensed matter, v. 1, 2, World Scientific, Singapore, 1989 | MR | Zbl

[38] Kléman M., “The general theory of dislocations”, Dislocations in solids, v. 5, ed. by F.R.N. Nabarro, North-Holland, Amsterdam, 1980, 243–297

[39] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, v. 1, Interscience Publ., New York, 1963 | MR | MR | Zbl

[40] Kondo K., “On the geometrical and physical foundations of the theory of yielding”, Proc. 2nd Japan Natl. Congr. for Applied Mechanics (1952), Japan Natl. Committee Theor. Appl. Mech. (Sci. Counc. Japan), Tokyo, 1953, 41–47 | MR

[41] Kosevich A.M., Fizicheskaya mekhanika realnykh kristallov, Nauk. dumka, Kiev, 1981

[42] Kröner E., Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, Berlin, 1958 | MR | Zbl

[43] Kröner E., “Continuum theory of defects”, Physics of defects: Les Houches Summer Sch., Sess. 35, 1980, ed. by R. Balian et al., North-Holland, Amsterdam, 1981, 215–315

[44] Kunin I.A., Kunin B.I., “Gauge theories in mechanics”, Trends in application of pure mathematics to mechanics, Lect. Notes Phys., 249, Springer, Berlin, 1986, 246–249 | DOI | MR

[45] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, 7, Pergamon, Oxford, 1970 | MR

[46] Lazar M., Hehl F.W., “Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations”, Found. Phys., 40:9 (2010), 1298–1325 | DOI | MR | Zbl

[47] M. Monastyrsky, Topology of Gauge Fields and Condensed Matter, Springer, New York, 1993 | MR

[48] Moraes F., “Geodesics around a dislocation”, Phys. Lett. A, 214:3–4 (1996), 189–192 | DOI | MR | Zbl

[49] W. Nowacki, Teoria sprȩżystości, PWN, Warszawa, 1970

[50] Nye J.F., “Some geometrical relations in dislocated crystals”, Acta metall., 1 (1953), 153–162 | DOI

[51] A. M. Polyakov, “Particle spectrum in quantum field theory”, JETP Lett., 20:6 (1974), 194–195

[52] Prasad M.K., Sommerfield C.M., “Exact classical solution for the 't Hooft monopole and the Julia–Zee dyon”, Phys. Rev. Lett., 35:12 (1975), 760–762 | DOI

[53] Randono A., Hughes T.L., “Torsional monopoles and torqued geometries in gravity and condensed matter”, Phys. Rev. Lett., 106:16 (2011), 161102 | DOI

[54] Rivier N., Duffy D.M., “Line defects and tunnelling modes in glasses”, J. Phys., 43:2 (1982), 293–306 | DOI | MR

[55] Rubakov V.A., Klassicheskie kalibrovochnye polya: Bozonnye teorii, 2-e izd., KomKniga, M., 2005; V. A. Rubakov, Classical Theory of Gauge Fields, Princeton Univ. Press, Princeton, NJ, 2002 ; URSS, Moscow, 1999 | MR | Zbl

[56] Sedov L.I., Berditchevski V.L., “A dynamic theory of continual dislocations”, Mechanics of generalized continua, Proc. UITAM Symp. (Freudenstadt, Stuttgart, 1967), ed. by E. Kröner, Springer, Heidelberg, 1967, 214–238

[57] Shnir Ya.M., Magnetic monopoles, Springer, Berlin, 2005 | MR | Zbl

[58] V. S. Vladimirov, Equations of Mathematical Physics, M. Dekker, New York, 1971 | MR | MR | Zbl