Hierarchy and disorder in non-crystalline structures

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Abstract
The structure of glasses and amorphous materials is constructed explicitly and described geometrically and algebraically in a very simple model of a succession of decurving operations on a crystal (polytope) in curved space. Two kinds of disorder can be introduced: (1) in the succession of decurvings—this yields an ensemble of hierarchical structures; (2) locally, by commuting the order of two decurving operations at different points in space. The structure can be patched up geometrically, and a connection denned, leaving behind defects, which are disclination lines (2π-disclinations in the Euclidean limit). The defect sizes and the barriers between configurations are themselves hierarchic, and depend at which stage in the succession of decurvings local commutation has been introduced. Two-level systems (tunnelling modes) are associated with the defects, and o⋅cur at all scales. Algebraically, local disorder is a gauge transformation, and a gauge-invariant energy of glass is given. The gauge transformation is elementary, if non-Abelian. The influence of local disorder in destroying orientational order is also analysed.