Spontaneous Breaking of Euclidean Invariance and Classification of Topologically Stable Defects and Configurations of Crystals and Liquid Crystals
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 26 (1978), Exposé no. 4, 4 p.
@article{RCP25_1978__26__45_0,
author = {Kl\'eman, Maurice and Michel, Louise},
title = {Spontaneous {Breaking} of {Euclidean} {Invariance} and {Classification} of {Topologically} {Stable} {Defects} and {Configurations} of {Crystals} and {Liquid} {Crystals}},
journal = {Les rencontres physiciens-math\'ematiciens de Strasbourg -RCP25},
note = {talk:4},
publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
volume = {26},
year = {1978},
language = {en},
url = {http://www.numdam.org/item/RCP25_1978__26__45_0/}
}
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Kléman, Maurice; Michel, Louise. Spontaneous Breaking of Euclidean Invariance and Classification of Topologically Stable Defects and Configurations of Crystals and Liquid Crystals. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 26 (1978), Exposé no. 4, 4 p. http://www.numdam.org/item/RCP25_1978__26__45_0/

1 D. Finkelstein, J. Math. Phys. (N.Y.) 7, 1218 (1966); | Zbl 0151.44203

D. Finkelstein and J. Rubinstein, J. Math. Phys. (N.Y.) 9, 1762 (1968). | MR 234695 | Zbl 0167.56205

2 Yu. S. Tyupkin, V. A. Fateev, and A. S. Shvarts, Pis'ma Zh. Eksp. Teor. Fiz. 21, 91 (1975) [JETP Lett, 21, 42 (1975)].

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6 G. Toulouse and M. Kleman, J. Phys. (Paris), Lett. 37, L149 (1976).

7 We disagrée with D. Rogula, in Trends in Application of Pure Mathématics to Méchanics, edited by G. Fichera (Pitman, New York, 1976), conceraing his point of view for the homotopic classification of crystal defects. | Zbl 0327.00011

8 L. Michel, in Proceedings of the Sixth International Colloquium Group Theoretical Methods in Physics, Tübingen, West Germany, 1977 (to be published).

9 M. Kleman, L. Michel, and G. Toulouse, J. Phys, (Paris), Lett. 38, L195 (1977),

10 G. E. Volovik and V. P. Mineev, "Study of singularities in ordered Systems by homotopic topology methods" (to be published).

11 G. Toulouse, J. Phys, (Paris), Lett. 38, L67 (1977).

12 M. Kleman and L. Michel, J. Phys. (Paris), Lett. 39, L29 (1977).

13 J. Kukula, senior the sis, Princeton University (unpublished).

14 V. Poenaru and G. Toulouse, J. Phys. (Paris) 38, 887 (1977).

15 A. T. Garel, "Boundary conditions for textures and defects" (to be published),

16 D. Finkelstein has coined the name kinks for non-singular topologically stable configurations. The ter m of texture has been introduced by P. W. Andersen and G. Toulouse, Phys. Rev. Lett. 38, 508 (1977), to connote nonsingular non-topologically-stable configurations (eg., S = 1 lines in the nematics, 4π rotation iines in superfluid He3-A phase). There is at the présent présent time some laxity in terminology, in that physicists tend to call textures what Finkelstein called kinks. We propose to use "configuration," because the words kinks and textures have both been used extensively for a long time with a very précise meaning in the physics of dislocations [a kink being a spécial type of accident on a line of dislocation and a texture being an exten-sive word to connote an assembly of defects in, e.g., J. Friedel, Dislocations (Pergamon, Oxford, 1964)].

17 D. Kastler, G. Loupias, M. Mebkhout, and L. Michel, Commun. Math. Phys. 27, 195 (1972). | MR 432108 | Zbl 0239.46069

18 R and Z are, respectively, the additive group of the real numbers and the integers. For the other groups we use the notation of L. D. Landau and E. M. Lifshitz, Quantum Méchantes (Pergamon, New York, 1977), 3rd éd., Chap. 12.

19 U. Essmann and H. Träuble, Phys. Lett. 24A, 256 (1967).

This pattern was predicted by A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) [JETP Lett. 5, 1174 (1957)],

but with a square lattice while the observed hexagonal one was proposed by W. M. Kleiner L. M. Roth, and S. H. Autler, Phys. Rev. 133A, 1226 (1964).

20 V. Luzzati and A. Tardieu, Annu, Rev. Phys. Chem. 25, 79 (1974).

21 N. Steenrod, The Topology of Fibre Bundles (Princeton Univ. Press, Princeton, N. J., 1957). | MR 39258 | Zbl 0054.07103

22 The topoiogical space of a semidirect product of groups is a topoiogical product of the group spaces. In such a product we can omit contractible factors Rk since their homotopy is trivial and the homotopy groups of a topoiogical product are direct products of the homotopy group of the factors.

23 E. Cartan, La Topologie des Espaces Représentatifs des Groupes de Lie (Hermann, Paris, 1936). | JFM 62.0441.02 | Zbl 0015.20401

24 R. Bott, Proc. Nat. Acad, Sci. U.S.A. 40, 586 (1954). | MR 62750 | Zbl 0057.02201

25 G. Friedel, Leçons de Cristallographie (Berger-Levrault, Paris, 1929).

26 S. Chandrasekhar, B. K. Sadashiva, and K. A. Surech, Pramana 9, 471 (1977). They claim that this phase formed by a benzene-hexa-n -alkanoate belongs to case III of Table I: $\left(R*{Z}^{2}\right)\square {D}_{6k}$.

27 J. Billard, J. C. Dubois, Nguyen Huu Tinh, and A. Zann, in Proceedings of the Europe an Congress on Smectics, Madonna di Campiglio, January, 1978 (unpublished). They have observed the phase of some hexa-n-oxytriphenylene and they do not exclude the possibility that it could be a nematic with symmetry group $R\square {D}_{3k}$.

28 R. Thom, to be published.