Topological defects and defect phases of rigid and flexibly bent-shaped liquid crystals are reviewed with emphasis on how they are affected by the departure of molecular shapes from a simple rod. The review discusses defects in bent-core uniaxial and hypothetical biaxial nematics, twist-bend nematic, and various frustrated layered bent-core liquid crystals, such as twist-grain boundary phase, nanoscale helical nanofilament phase, and the so-called B7 textures with helical ribbons.
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Statistics and topology of fluctuating ribbons
Ribbons are a class of slender structures whose length, width, and thickness are widely separated from each other. This scale separation gives a ribbon unusual mechanical properties in athermal macroscopic settings, for example, it can bend without twisting, but cannot twist without bending. Given the ubiquity of ribbon-like biopolymers in biology and chemistry, here we study the statistical mechanics of microscopic inextensible, fluctuating ribbons loaded by forces and torques. We show that these ribbons exhibit a range of topologically and geometrically complex morphologies exemplified by three phases—a twist-dominated helical phase (HT), a writhe-dominated helical phase (HW), and an entangled phase—that arise as the applied torque and force are varied. Furthermore, the transition from HW to HT phases is characterized by the spontaneous breaking of parity symmetry and the disappearance of perversions (that correspond to chirality-reversing localized defects). This leads to a universal response curve of a topological quantity, the link, as a function of the applied torque that is similar to magnetization curves in second-order phase transitions.
more » « less- Award ID(s):
- 2011754
- PAR ID:
- 10500037
- Publisher / Repository:
- PNAS
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 119
- Issue:
- 32
- ISSN:
- 0027-8424
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1073/pnas.2122907119
