The classification of topological insulators and superconductors

We use the process of band crossings during quantum phase transitions to explain the periodic table of topological insulators and superconductors. This is achieved by showing how irreducible representations of the real and complex Clifford algebras are related to the 10 Altland-Zirnbauer symmetry classes of Hamiltonian matrices which are associated with time reversal, particle-hole, and chiral symmetries. The representation theory not only reveals why a unique topological invariant ($0, Z_2, Z$) exists for each specific symmetry class and dimension, but also shows the interplay between quantum phase transitions, topologically protected boundary modes, and topological invariants.