Spectral approach to transport problems in two-dimensional disordered lattices : physical interpretation and applications.
This paper introduces the spectral approach to transport problems in infinite disordered systems characterized by an Anderson-type Hamiltonian. The spectral approach determines (with probability 1) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. Here we focus on the critical 2D case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems where interaction with various types of disorder is considered. The first chapter provides a general overview of Anderson-type transport problems and the most relevant theoretical, numerical, and experimental developments in this field of research. Chapter 2 introduces the mathematical formulation of the spectral approach together with a physical interpretation and discussion on its applicability in physical systems. The third chapter presents a numerical study of delocalization in the 2D disordered honeycomb, triangular, and square lattices. In chapter 4, we investigate transport in the quantum regime by a spectral analysis of the quantum percolation and binary alloy problems. Next, the applicability of the method is extended to the classical regime, where we examine diffusion of lattice waves in the 2D disordered complex plasma crystal. In the final chapter, we summarize the results and propose future developments in the study of complex transport problems using spectral theory.