In the Bloch Hamiltonian formalism for periodic crystals, where the Hamiltonian H ( k ) acts on modes of crystal momentum k , the chiral symmetry, TRS, and PHS conditions become U S H ( k ) U S − − 1 = − − H ( k ) , U T H ( k ) ∗ ∗ U T − − 1 = H ( − − k ) and U C H ( k ) ∗ ∗ U C − − 1 = − − H ( − − k ) .