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The Geometrical Structure of 2d Bond-Orientational Order
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.Comment: 22 pages, LaTeX, 4 eps figures Published versio