Topology and Geometry of Smectic Order on Compact Curved Substrates |
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Authors: | Xiangjun Xing |
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Institution: | (1) Department of Physics, Syracuse University, Syracuse, NY 13244, USA |
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Abstract: | Smectic order on arbitrary curved substrate can be described by a differential form of rank one (1-form), whose geometric
meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is the local
dislocation density. Elastic deformations are described by superposition of exact differential forms. We use the formalism
of differential forms to systematically classify and characterize all low energy smectic states on torus as well as on sphere.
A two dimensional smectic order confined on either manifold exhibits many topologically distinct low energy states. Different
states are not accessible from each other by local fluctuations. The total number of low energy states scales as the square
root of the system area. We also address the energetics of 2D smectic on a curved substrate and calculate the mean field phase
diagram of smectic on a thin torus. Finally, we discuss the motion of disclinations for spherical smectics as low energy excitations,
and illustrate the interesting connection between spherical smectic and the theory of elliptic functions. |
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Keywords: | Smectic Translational order Degeneracy Nematic Geometry Topology Topological defects Differential forms Curvature Cohomology Topological order Topological quantum number Topological invariant Phase diagram Chirality Phase transition Braiding Anyon Spiral Quasi-baseball |
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