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Topological Defects 18.354 L24 Order Parameters, Broken Symmetry, and Topology James P. Sethna Laboratory of Applied Physics, Technical University of Denmark, DK-2800 Lyngby, DENMARK, and NORDITA, DK-2100 Copenhagen , DENMARK and Laboratory

  1. Topological Defects 18.354 L24 Order Parameters, Broken Symmetry, and Topology James P. Sethna Laboratory of Applied Physics, Technical University of Denmark, DK-2800 Lyngby, DENMARK, and NORDITA, DK-2100 Copenhagen Ø, DENMARK and Laboratory of Atomic and Solid State Physics (LASSP), Clark Hall, Cornell University, Ithaca, NY 14853-2501, USA (Dated: May 27, 2003, 10:27 pm) dunkel@mit.edu

  2. • work hardening, etc • optical effects order-parameter fields are discontinuities in Topological defects "umbilic defects" in a nematic liquid crystal

  3. order = symmetry = invariance � (under certain group actions ) symmetry groups can be discrete, continous, Lie-groups, ….

  4. More or less symmetric ?

  5. More or less symmetric ? Mg 2 Al 4 Si 5 O 18 http://www.doitpoms.ac.uk/tlplib/crystallography3/printall.php

  6. More or less symmetric ? broken continuous translation/rotation symmetry (invariance)

  7. Order parameters: 2D crystal 4 � dx ( κ / 2)( du/dx ) 2 . E = u + a ˆ x = ⃗ u + ma ˆ x + na ˆ ⃗ u ≡ ⃗ y.

  8. Order parameters: magnets tant.

  9. Order parameters: nematic liquid crystals “projective plane” = half-sphere with opposite points on equator identified

  10. Topological defects

  11. Work hardening

  12. Disclinations edge screw

  13. Disclineations

  14. Bacterial vortices PIV +1 -1 -1 +1 Dunkel et al PRL 2013

  15. Active nematics Dogic lab (Brandeis) Nature 2012

  16. Active nematics Giomi et al PRL 2012

  17. Defects in nematics winding number

  18. Defects in nematics winding number

  19. Two-Dimensional Nematic Colloidal Crystals Self-Assembled by Topological Defects Igor Musevic et al. Science 313 , 954 (2006); DOI: 10.1126/science.1129660 s bilized ). taneously geometric l- col- p fabrication loids re- l- ecision micro- a- ce d s. ic id

  20. Two-Dimensional Nematic Colloidal Crystals Self-Assembled by Topological Defects Igor Musevic et al. Science 313 , 954 (2006); DOI: 10.1126/science.1129660

  21. Reconfigurable Knots and Links in Chiral Nematic Colloids Uros Tkalec et al. Science 333 , 62 (2011); DOI: 10.1126/science.1205705 efect ts s. g nt rs a he on e ori- o- Defect dimer, re nknot. e topologi- o loops. d- ns numerical- u-de el l- nd r- the fu- are by using a program for representing knots ( 33 ) to show the relaxation

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