Topology in Quantum Matter

Basic concepts necessary to understand the electronic structure of certain new types of materials called Topological Insulators are presented. These materials were predicted, first in 2D and then in 3D, and were later found experimentally. Their salient feature is that, although insulating in the bulk, they exhibit gapless (metallic) surface states with a remarkable robustness to a wide class of "deformations" of the Hamiltonian. The root cause of this robustness can be traced back to properties of the eigenstates of the systems that can be understood in relatively simple geometric terms.