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Disorders with creating the localized states, which are the states with limited spatial distribution, cause carriers of these states become limited to small part of space and cannot participate in the transport. If energy of these states is near the Fermi energy, the Anderson metal-insulator phase transition occurs. In this thesis,we studied type of disorders and determining their Hamiltonian and by studying definition of localization from different aspects, we attained a better explanation of disorders effects on materials. With studying the localization scaling theory, we ex-amined conditions of localization in different dimensions. Then for determining the green function and therefor the... 

After the discovery of integer and fractional quantum Hall effects, a comprehensive effort started to understand the physics behind these findings thoroughly. At the end, these efforts resulted in the introduction of topological phases of matter and the notion of topological phase transition. With the recent developments in this field, as the introduction and synthesis of 3D topological insulators, many researchers began inspecting the effects of disorders and impurities in the lattice structure on the topological phases, As these imperfections are inevitable in the lab-created instances. Since, in the previous works only on-site disorders was studied, the concentration of this thesis is on... 

In this thesis we would like to investigate structural disorder of Graphene and its effect on electronic properties. Graphene is a one-layer crystal of Carbon atoms which was the first two dimensional crystal observed experimentally. Low energy current carriers on Graphene behave like mass-less Dirac fermions and this interesting electronic property attracts much attention on studying Graphene. Structural disorder of Graphene highly affects its electronic properties. We study thermal fluctuations of Graphene sheet, disorder super lattice, and static ripples, as structural disorders of Graphene. Disorders due to thermal fluctuations of graphene induce a disordered gauge field on the graphene... 

The zero-energy excitations of topological superconductor promising the design of quantum computers have received a great deal of attentions in recent years. But the materialization has remained to be elusive. Our focus in this thesis is on the one- and two-dimensional systems where the necessary ingredients may conspire to realize topological superconductors. In particular, we will focus on the role of reduction of spatial symmetries near the interfaces. We will start with a one-dimensional system at the boundary of a two-dimensional system with particular orientation so that the full translational symmetry is replaced with a quasi-periodic symmetry. We show that the Fibonacci sequence...