Surface states in Dirac metals and topological crystalline insulators

We reconsider the problem of surface states spectrum in type Ι Dirac metal. We find that its surface states, despite being gapped, always form branches terminating at Dirac points. Furthermore, we consider evolution of the surface states in the case, when rotational symmetry is broken, and as a result, Dirac points are gapped. We find, that in this case, special role is played by mirror symmetry relative to the plane connecting the Dirac points. When it is present, the resulting gapped state is a topological crystalline insulator, which surface spectrum can contain either one or three Dirac points, two of which are protected solely by the mirror symmetry. The Dirac metal can be viewed as a topological phase transition between two phases with different mirror Chern numbers.