Hamiltonian decomposition for bulk and surface states

We demonstrate that a tight-binding Hamiltonian with nearest- and next-nearest-neighbor hopping integrals can be decomposed into bulk and boundary parts in a general lattice system. The Hamiltonian decomposition reveals that next nearest-neighbor hopping causes sizable changes in the energy spectrum of surface states even if the correction to the energy spectrum of bulk states is negligible. By applying the Hamiltonian decomposition to edge states in graphene systems, we show that the next nearest-neighbor hopping stabilizes the edge states.