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The noncommutative geometry of aperiodic solids. (English) Zbl 1055.81034

Cardona, Alexander (ed.) et al., Proceedings of the summer school on geometric and topological methods for quantum field theory, Villa de Leyva, Colombia, July 9–27, 2001. River Edge, NJ: World Scientific (ISBN 981-238-131-7/hbk). 86-156 (2003).
From the introduction: The main construction is the notion of a hull of an aperiodic solid. This is the content of Section 1 and Section 2. Several examples are proposed, impurities in a semiconductor, quasicrystals, tilings. It will be shown that the hull is actually determined by the Gibbs thermodynamical ground state of the set of atoms. This Gibbs state also determines in a unique way various thermodynamical properties such as the diffraction pattern, the electronic density of states or the vibrational density of states (phonon modes). Once the hull is constructed, it leads to the construction of the noncommutative Brillouin zone and its geometry. Then the description of electrons in the one-particle approximation, or of the phonons in the harmonic approximation follows easily. No attempt to account for the large number of results obtained in the eighties and later concerning the spectral properties for both electrons and phonons will be made here.
For the entire collection see [Zbl 1047.81003].

MSC:

81R60 Noncommutative geometry in quantum theory
82D20 Statistical mechanics of solids
81T75 Noncommutative geometry methods in quantum field theory
58B34 Noncommutative geometry (à la Connes)
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82D25 Statistical mechanics of crystals
82D37 Statistical mechanics of semiconductors
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