Homotopy Theory of Topological Insulators

We transfer the classification results for disordered free fermions to the setting of translation-invariant ground states and complete the framework developed by Kitaev in which true symmetries are encoded as pseudo-symmetries satisfying Clifford algebra relations. In this mathematical setting, we give a homotopy theoretic proof of the Periodic Table for topological insulators and superconductors in its generalized form allowing for the presence of a defect. Permitting arbitrary numbers of valence and conduction bands, we extend the homotopy classification to include the Hopf insulator and a newly identified topological phase we call the Hopf superconductor. In this general setting, we show that the distinction between strong and weak topological phases needs to be altered in order to prevent strong phases from being realized by stacking lower dimensional phases