Symmetry breaking in smectics and surface models of their singularities
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Edited by Noel A. Clark, University of Colorado, Boulder, CO, and approved July 20, 2009 (received for review May 13, 2009)
Abstract
The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the interaction between translations and rotations. Dislocations are associated, as usual, with branch points in a phase field, whereas disclinations arise as critical points and singularities in the phase field. We introduce a three-dimensional model for two-dimensional smectics that clarifies the topology of disclinations and geometrically captures known results without the need to add compatibility conditions. Our work suggests natural generalizations of the two-dimensional smectic theory to higher dimensions and to crystals.
Footnotes
- 1To whom correspondence should be addressed. E-mail: kamien{at}physics.upenn.edu
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Author contributions: B.G.C., G.P.A., and R.D.K. designed research, performed research, and wrote the paper.
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The authors declare no conflict of interest.
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This article is a PNAS Direct Submission.
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↵* The astute reader might wonder about the restriction of paths for a spontaneously broken gauge theory where the Goldstone mode is also absent. There are no restrictions: In the gauged case, the winding is not a nontrivial loop in a ground state manifold connecting equivalent but distinct ground states, but rather a winding on a gauge orbit connecting different representations of the same physical state.
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↵† Here and throughout ∇ acts on the two-dimensional xy-plane.
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↵‡ More precisely, we are considering the n-jet of the configuration at P. Strictly speaking, for 1-jets, we should keep track of the magnitude |∇φ|x=P as well. But because |∇φ| is an element of ℝ, which is contractible, the topological behavior of this component would be trivial. An argument of Kléman, Michel, and Toulouse (8) (suitably reinterpreted) shows that jets higher than first order are not needed for a similar reason.
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↵§ Note that, although the branch point of a helicoid is quite apparent when viewed in three dimensions, in any given slice of it, the branch point is not apparent because the layers all remain smooth; see, particularly, images in ref. 19.
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¶ There may be other types of deformations that we choose to exclude due to energetic or other physical concerns. For instance, it might be convenient to consider deformations up to those that preserve the number of layers, which would prevent us from pushing the surfaces in the vertical direction.
