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 for honeycomb lattice systems. 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. The application of Hamiltonian decomposition to a general lattice system is discussed.