There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian H ^ ^ = ∑ ∑ i , j H i j c i † † c j , (where c i , and c i † † , are the annihilation and creation operators of mode i , in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.