This proposal concerns the development of a mathematical tool liable to describe the structure and the transport properties of solids that are not periodic. It will consider the following aspects: (i) In such solids there is a new mode of atomic diffusion called phason transport that is dominant at low temperature. Using a new recent diffusive random process, the PI expect to give a precise description of such transport. (ii) To extend this mathematical description to solids more complex, namely those which are neither random nor rigid, and for which the number of local patches of a given size can be infinite. An example of such solids could be amorphous silicon. (iii) To describe electronic transport at very low temperature, such as the so-called Mott's variable range hopping transport, which is responsible, in particular, for the unusual accuracy of the measurement of the resistance standard in the Quantum Hall Effect.
This proposal is the continuation of a 30 years program intending to develop the mathematical tools allowing the study of such solids. Because of the lack of translation symmetry, the usual tools of calculation in Solid State Physics cannot be used anymore and require the development of more sophisticated methods. Examples of solids covered by this study are semi-conductors at very low temperatures, or quasi-crystalline. The PI has used successfully Non-commutative Geometry, a theory developed by the Field Medalist Alain Connes, in several aspects of this program since 1980. At the interface between Mathematics and Physics, this program requires the use of a broad list of mathematical tools including Number Theory, Algebraic Topology, Analysis, Probability Theory, and should be applicable to a broad list of materials that cannot be studied by usual theoretical methods.
This project is a contribution to a long term program with aim to complete a mathematical theory for describing the physical properties of non periodic solids such as (i) crystal in magnetic field (ii) semiconductors at very low temperature (including the Quantum Hall Effect, the Gap labeling Theorem, Topological Insulators) (iii) aperiodic solids with long range order (quasicrystals) (iv) solids without long range ordre such as amorphous materials, bulk metallic glasses (v) liquids can be included in some cases. The present project contributions include: (i) a better description of the topology and the geometry of the Hull and the Tiling space of the atomic array (ii) a contribution on the atomic motion in aperiodic solids (phason modes or flip-flops), leading to what is called nowadays the Pearson Lapacian (John Pearson has been a PhD Student advised by the PI on this result). (iii) a contribution to electronic transport in semiconductors at very low temperature, and to the impact of impurities on electron scattering, (iv) the computation of topological invariants for topological insulators. (v) the begining of a mathematical description of bulk metallic glasses and liquids at the atomic scale The present state of knowleadge on aperiodic Solids includes contributions from the PI and from a now large community of experts worldwide, that were produced since 1979. The time is probably ripe for these results to be collected in a book that should lay the basis of the Mathematical Theory of Aperiodic Solids.
