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Summer Research Fellowship Programme of India's Science Academies

Topological Phases of Matter

Raghav Chaturvedi

Indian Institute of Science Education and Research, Maruthamala PO, Vithura, Thiruvananthapuram, Kerala 695551.

Professor Sumathi Rao

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, 211019. India.

Abstract

Ever since the prediction of topologically ordered phases of matter in the late 1980's, research in the field of topological insulators and superconductors has advanced substantially. With the vast amount of literature available and the interdisciplinary background needed to study these exciting phases, a need for a compiled text aimed at introducing the necessary ideas and motivation to study problems of mesoscopic physics, specifically the sub-field of topological insulators, is in order and this article seeks to provide the same. For theoretical and experimental physicists alike, any pedagogical description of topological band theory and its consequences in mesoscopic physics and quantum computation is incomplete without a series of mathematical and physical constructs. In this report, we begin our discussion with briefly introducing the formalism and ideas borrowed from several branches of physics and mathematics including Topology, Differential Geometry, Group Theory, Quantum Mechanics, Many-body Physics, Quantum Field Theory, Solid State Physics and Statistical Physics. Readers are expected to have a general background in quantum mechanics and are advised to look at references provided in each subsection to get an extended picture of the ideas presented here. Next, we lay down the framework to explain transport in mesoscopic systems, bridging the gap between classical and quantum pictures, wherein we introduce the Landaur-Buttiker, NEGF, and Kubo formalisms. Further, we provide the necessary motivation to establish the principles of topological band theory. Here, we also present the Haldane and Kane Models for graphene. Following this we look at trivial and non-trivial Topological Insulators and provide a clearer picture of the integer and fractional quantum Hall Effects. We then go on to study the fractional statistics associated with Abelian and Non- Abelian Anyons, which have been proposed as possible candidates for topological Quantum computation.

Keywords: Mesoscale and Nanoscale Physics, Electron transport, Topological Band theory, Quantum-Hall effect, Topological Insulators, Anyon Physics

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