Chapter 5 Topological insulators and superconductors Learning goals

Kane Mele model. • We can derive the topological index based on time reversal polarization. • We understand the entries of the periodic table for topological insulators. • We know what a px + ipy superconductor is. • We are acquainted with the Kitaev wire. • We know the Su-Schrie↵er-Heeger model. • M. König et al., Science 318, 766 (2007) In this chapter we try to understand what topological properties can arise for free fermion sys-tems subject to some symmetry constraints. The exposition starts from a historical perspective with the first time-reversal symmetric topological insulator in two dimensions by Kane and Mele [1]. We then motivate on physical grounds how one can construct a topological index charac-terizing this new type of band insulator. Our derivation follows the historically motivated path covered in the book by Bernevig and Hughes [2]. The so derived topological index for two-dimensional systems readily generalizes to three dimensions. Once we established the presence of two- and three-dimensional topological insulators protected by time-reversal symmetry we take a somewhat more formal perspective and discuss the periodic table for topological insula-tors which catalogues all symmetry protected topological free fermion systems. We learn how to read and use the table and relate its entries to experimentally relevant response functions. Finally, we conclude this chapter by covering three archetypal models in di↵erent symmetry classes. 5.1 The Kane-Mele model In the last chapter we have seen that we can construct lattice models where the Bloch bands have a non-vanishing Chern number despite the absence of a net magnetic field. Here we try to build a time-reversal invariant version based on Haldane’s honeycomb model for a Chern insulator. We start from the low energy version of graphene H0 = i~v