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, both in the crystalline and hexatic phases. Applications to conical and spherical geometries are briefly addressed.
Bowick, Mark and Travesset, Alex, "The Geometrical Structure of 2d Bond-Orientational Order" (2001). Physics. 160.
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