Topological Quantum Matter

This advanced course covers the fundamentals of the thriving field of Topological States of Matter. We discuss both theoretical and experimental aspects and emphasize the physical picture over the technical details. The course is divided to nine units: The Integer and Fractional Quantm Hall Effects are covered in the first unit, including the concepts of edge states, localization, fractional charges, composite fermions, and non-abelian states; Topological Superconductivity is covered in the 2^nd unit, including the concepts of the Thouless pump, Majorana zero modes, and realizations in one and two dimensions; Topological Universe on a Graphene Sheet is offered by the 3^rd unit, including the concepts of Dirac cones, Klein tunneling and Chern bands, as well as the rich world of twisted bi-layer graphene; Topological Insulators are covered by the 4^th unit, including two- and three- dimensional systems, as well as topological crystalline insulators; The 5^th unit, on Topological Classification, puts all examples of the previous units into a unified framework, introducing the periodic table of gapped topological systems with no topological order; The 6^th unit expands the course into the realm of Gapless Topological Phases, covering Dirac and Weyl semi-metals, both in their bulk and surface; The 7^th unit covers the theoretical tools for Predicting Topological Materials, with an emphasis on Density Functional Theory, and the quantities that need to be calculated to probe the topological characteristics of a material; The 8^th unit dives into the abstract world of Topological Order, from the Toric Code all the way to a brief discussion of Topological Quantum Computation; And finally, the 9^th unit describes some Experimental Tools that are of wide use in the study of topological states of matter, and makes connection between measurements and their interpretation.