Surface states of gapped electron systems and semi-metals

With a general model for electrons occupying a semi-infinite crystal with a hard surface, we analyze the eigenstates of the system with a bulk band gap. We derive the criterion for the existence of the surface states and give physically transparent description on their formation. For electrons with spin-orbit interactions in a topological insulator with time reversal symmetry, we rigorously prove the correspondence between the change of Kramers degeneracy of the surface states and the bulk time-reversal $Z_2$ invariant. The analysis is also applicable to the semi-metals without gap existing as well. As examples, we solve the edge states of electrons under the spin-orbit interactions in graphene with a hard zigzag edge and that in $d$-wave superconductor with a (1,1,0) surface.