Topological Defects 18.354 L24 Order Parameters, Broken Symmetry, - - PowerPoint PPT Presentation

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Topological Defects 18.354 L24 Order Parameters, Broken Symmetry, - - PowerPoint PPT Presentation

Mar 24, 2023 39 likes •250 views

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

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SLIDE 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

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Topological defects are discontinuities in

  • rder-parameter fields
  • optical effects
  • work hardening, etc

"umbilic defects" in a nematic liquid crystal

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  • rder = symmetry = invariance
  • (under certain group actions )

symmetry groups can be discrete, continous, Lie-groups, ….

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More or less symmetric ?

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SLIDE 5

http://www.doitpoms.ac.uk/tlplib/crystallography3/printall.php

Mg2Al4Si5O18

More or less symmetric ?

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SLIDE 6

More or less symmetric ?

broken continuous translation/rotation symmetry (invariance)

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Order parameters: 2D crystal

4

⃗ u ≡ ⃗ u + aˆ x = ⃗ u + maˆ x + naˆ y.

E =

  • dx (κ/2)(du/dx)2.
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SLIDE 8

Order parameters: magnets

tant.

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Order parameters: nematic liquid crystals

“projective plane” = half-sphere with opposite points on equator identified

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Topological defects

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Work hardening

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Disclinations

edge screw

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Disclineations

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Dunkel et al PRL 2013

Bacterial vortices

PIV

+1

  • 1
  • 1

+1

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Active nematics

Dogic lab (Brandeis) Nature 2012

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Active nematics

Giomi et al PRL 2012

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Defects in nematics

winding number

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Defects in nematics

winding number

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SLIDE 19

s bilized ). taneously geometric l- col- p fabrication loids re- l- ecision micro- a- ce d s. ic id

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

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SLIDE 20

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

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SLIDE 21

DOI: 10.1126/science.1205705 , 62 (2011); 333 Science et al. Uros Tkalec Reconfigurable Knots and Links in Chiral Nematic Colloids

efect ts s. g nt rs a he

  • n

e

  • ri-
  • Defect

dimer, re nknot. e topologi-

  • 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