Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice
We consider periodic Hamiltonians on a three dimensional (3 D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level. By using K theory applied for the quarter plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non triviality of the other means that the corner Hamiltonian is gapless. We prove a correspondence between these two invariants. Such gapped Hamiltonians can be constructed from Hamiltonians of 2 D type A and 1 D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases.