## Abstract

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 language | English (US) |
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Pages (from-to) | 1535-1548 |

Number of pages | 14 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 34 |

Issue number | 8 |

DOIs | |

State | Published - Mar 2 2001 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)