Three-band model for quantum Hall and spin Hall effects

The topological properties of a certain class of spinless three-band Hamiltonians are shown to be summed up by the skyrmion number in momentum space, analogous to the case of a two-band Hamiltonian. A topological tight-binding Hamiltonian on a kagome lattice is analyzed from this viewpoint. When such a Hamiltonian is "folded," the two bands with opposite Chern numbers merge into a degenerate band exhibiting a non-Abelian gauge connection. A conserved pseudospin current operator can be constructed in this case and used to compute the pseudospin Hall conductance.