The geometrical structure of 2D bond-orientational order

Mark Bowick, Alex Travesset

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We study the formulation of bond-orientational order in an arbitrary two-dimensional geometry. We find that bond-orientational order is properly formulated within the framework of differential geometry with torsion. The torsion reflects the intrinsic frustration for two-dimensional crystals with arbitrary geometry. Within a Debye-Huckel approximation, torsion may be identified as the density of dislocations. Changes in the geometry of the system cause a reorganization of the torsion density that preserves bond-orientational order. As a byproduct, we are able to derive several identities involving the topology, defect density and geometric invariants such as Gaussian curvature. The formalism is used to derive the general free energy for a 2D sample of arbitrary geometry, in both the crystalline and hexatic phases. Applications to conical and spherical geometries are briefly addressed.

Original languageEnglish (US)
Pages (from-to)1535-1548
Number of pages14
JournalJournal of Physics A: Mathematical and General
Issue number8
StatePublished - Mar 2 2001

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy


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