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In this lecture, we discuss the general classification of free fermionic symmetry-protected topological (SPT) phases of matter using the two antiunitary symmetries, viz, time-reversal and particle-hole conjugation. We outline the derivation of the periodic table of topological insulators and discuss various examples. Since the particle-hole symmetry naturally occurs in the BdG (Bogoliubov-de Gennes) description of superconductivity, we also discuss some examples of topological superconductors such as the Kitaev chain in one dimension and the px ± ipy superconductors in two dimensions.
The video lecture is divided into three parts. You can also check out the video lectures at our Vimeo showcase.
In the first section, we introduce the ten families of Hamiltonians that originate from the time-reversal and particle-hole symmetries ― the Altland-Zirnbauer (AZ) symmetry classes ― and relates these families to the ten families of symmetric spaces defined as cosets of Lie groups.
In this section, we introduce the periodic table of topological insulators and superconductors which classifies topological phases for the 10 symmetry classes in a given number of dimensions, and see how the lattice models we have studied so far fit into this periodic table.
In this section, we derive the BdG Hamiltonian for superconductors starting from a mean-field description and look at some of its features. We also introduce two lattice models that exhibit topological superconductors: the Kitaev chain in 1d and the p±ip superconductors in 2d.
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