Abstract
In Chapter 3, we have considered the notion of order parameter, its amplitude and phase. The order parameter is a continuous field (scalar, vector, tensor, etc.) describing the state of the system at each point. Generally, it is a function of coordinates, ψ(r). Distortions of ψ(r) can be of two types: those containing singularities and those without singularities. At singularities, ψ is not defined. For a 3D medium, the singular regions might be either zero-dimensional (points), one-dimensional (lines), or two-dimensional (walls). These are the defects. Whenever a nonhomogeneous state cannot be eliminated by continuous variations of the order parameter (i.e., one cannot arrive at the homogeneous state), it is called topologically stable, or simply, a topological defect. If the inhomogeneous state does not contain singularities, but nevertheless is not deformable continuously into a homogeneous state, one says that the system contains a topological configuration (or soliton).
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Further Reading
General Courses
M. Kleman, Points, Lines, and Walls in Liquid Crystals, Magnetic Systems, and Various Ordered Media, John Wiley & Sons, Chichester, 1983.
T. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press, 1997.
DJ. Thouless, Topological Quantum Numbers in Nonrelativistic Physics, World Scientific, Singapore, 1998.
Knots and Strips
Louis H. Kauffman, Knots and Physics, 3rd Edition, World Scientific, Singapore 2001.
W F. Polh, J. Math. Mech. 17, 975 (1968).
J. H. White, Am. J. Math. 91, 693 (1969).
F. B. Fuller, Proc. Nat. Acad. Sci. USA 68, 815 (1981).
DNA Topology and Its Biological Effects, Edited by N. Cozzarelli and J.C. Wang, Cold Spring Harbor Laboratory Press, 1990.
Geometry and Algebraic Topology
D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing, New York, 1952.
N. Steenrod, Topology of Fiber Bundles, Princeton University Press, 1951.
W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, Inc., New York, 1967.
Reviews of Homotopy Theory Applied to Defects in Ordered Media
N.D. Mermin, Rev. Mod. Phys. 51, 591 (1979).
L. Michel, Rev. Mod. Phys. 52, 617 (1980).
V.P. Mineev, Sov. Sci. Rev. Sect. A, Vol. 2, Edited by I.M. Khalatnikov, Chur, London, Harwood Academic, New York, 1980.
H.-R. Trebin, Adv. Phys. 31, 194 (1982).
Topological Stability
G. Toulouse and M. Kleman, J. Phys. Lett. (Paris) 37, L–149 (1976); M. Kleman, J. Phys. Lett. (Paris) 38, L-199 (1977).
G.E. Volovik and V.P. Mineev, Pis’ma Zh. Eksp. Teor. Fiz. 24, 605 (1976) [JETP Lett. 24, 595 (1976)]; Zh. Eksp. Teor. Fiz. 72, 2256 (1977) [Sov. Phys. JEPT 46, 1186 (1977)].
Y. Bouligand, Physics of Defects, Edited by R. Balian, M. Kleman, and J.-P. Poirier, North-Holland, Amsterdam, 1981, p. 665.
M. Kleman and L. Michel, Phys. Rev. Lett. 40, 1387 (1978).
Y Bouligand, B. Derrida, V Poénaru, Y Pomeau, and G. Toulouse, J. Phys. (Paris) 39, 863 (1978).
Comparison with Experiments in Liquid Crystals
M.V. Kurik and O.D. Lavrentovich, Usp. Fiz. Nauk 154, 381 (1988)/Sov. Phys. Usp. 31, 196 (1988).
Biaxial Nematics
G. Toulouse, J. Phys. Lett. (Paris) 38, L–67 (1977).
Defects in Anisotropic Superfluids
G.E. Volovik, Exotic Properties of Superfluid 3He, World Scientific, Singapore, 1992.
Defects in Ferromagnets
M. Kleman, Dislocations, disclinations, and magnetism, in “Dislocations in Solids,” 5, Edited by F.R.N. Nabarro, North-Holland, Amsterdam, 1980.
M. Kleman, Magnetization processes in ferromagnets, in “Magnetism of Metals and Alloys,” Edited by M. Cyrot, North-Holland, Amstredam, 1980.
A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials, Academic Press, New York, 1979.
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(2003). Topological Theory of Defects. In: Kleman, M., Lavrentovich, O.D. (eds) Soft Matter Physics: An Introduction. Partially Ordered Systems. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21759-8_12
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DOI: https://doi.org/10.1007/978-0-387-21759-8_12
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