A criterion of the non-existence of surface states in a semi-infinite crystal

An infinite crystal can be constructed by an infinite number of parallel two-dimensional (hkl) crystal planes coupled to each other. For crystals with negligible spin-orbit coupling, we report a rigorous proof of a criterion on the non-existence of surface states in a semi-infinite crystal with the crystal symmetry. The forward transfer to be the same as the backward one, called as F-B dynamical symmetry, is key to realize the criterion. Based on lattice model Hamiltonian with coupling between the nearest neighbor crystal planes only, we prove that a cut crystal will not be able to accommodate any surface states if the original infinite crystal has reflection symmetry about every crystal plane which results in F-B symmetry. The criterion provide a platform to simply conclude whether surface states exist or not in a cut crystal. For any such crystals, the non-existence or existence of surface states depends on the cut direction of the crystal plane. Since the spin-orbit coupling breaks the chiral symmetry, resulting in the F-B asymmetry, surface states can emerge in the (hkl) cut crystal with spin-orbit coupling.