Studies of Topological Phases of Matter : Presence of Boundary Modes and their Role in Electrical Transport

Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second…