The Topological Character of Smectics
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Though the systematic use of topology to understand defects in ordered matter is now nearly 50 years old, the original work failed to completely characterize systems with broken translational order, i.e., crystals. Smectics are the simplest example of crystals and we have employed new mathematical tools to study and classify the allowed point and line defects in them. The theory reduces to the time-honored homotopy theory if we ignore the periodic order of the smectic but offers a refinement -- though the smectic can support all the defect structure and algebra of the nematic phase that sits above it, the defects have further structure that we have uncovered. This has allowed us to understand previously open puzzles, including the nature of composite dislocations in smectics.
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