Spectral Properties of Aperiodic Solids

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This proposal deals with developing mathematical tools that can be used to model the transport of energy through a solid. Such modeling activities are of great importance, e.g. one wishes to model the transport of electric current through various materials on the surface of a computer chip. The problem is highly dependent on the structure of the solid: for material that "has a pattern" a very successful theory- Bloch theory- has been developed that serves as a very good physical and mathematical model. The goal of this research is to develop an analog to Bloch theory for those solids that are without any repeating pattern. Besides its own intrinsic scientific interest, such a theory could have technological implications in electronics, instrumentation, and the like. During the 20th century, a lot of information about the electronic behavior of periodic crystals has been obtained through the so-called "Bloch Theory". It has led to a substantial understanding of properties of metals, insulators, and semiconductors as long as they are periodic crystals and the electron interactions can be neglected. In particular, numerical methods are routinely used to compute the band spectrum of such materials. Bloch Theory is more efficiently justified through the "Wannier transform". In particular, it allowed to a theoretical description of impurities in semiconductors, leading to substantial progress in their practical use. Nothing like the Bloch theory exists for aperiodic materials, namely for most of materials available in material science. Physicists have relied upon tricks, approximations, effective models and numerical methods. This led sometimes to seminal results like the theory of Anderson localization, Renormalization Group Analysis and spectacular effects like the Quantum Hall effect that led to a change in the definition of the resistance standard. The discovery of quasicrystals in the mid eighties changed the trend dramatically: the lack of Bloch theory made it an outstanding challenge to predict the electronic properties of such materials. Various approximation schemes have been used, but the accuracy of the results is questionable and the heaviness of the software makes it essentially impossible to go beyond two dimensions. There is an urgent need of a robust mathematical theory to overcome these difficulties. The present project is one step towards such a goal. Using the full strength of progress made during the last thirty years, partly by the PI, the present proposal proposes a construction of the Wannier transform for aperiodic solids having "finite local complexity". The construction relies heavily upon the developments made in the eighties by the PI, using the tools of Noncommutative Geometry. It also uses the developments made in the nineties by a small community of younger mathematicians, including the PI, to describe precisely tilings with finite local complexity. This proposal intends to go beyond the Wannier transform to give a tool to predict the nature of the electronic spectrum, hopefully leading to numerical codes that will enable to compute the properties of specific materials.