New classes of topological crystalline insulators having surface rotation anomaly

We discover new types of quantum anomalies in two-dimensional systems with time-reversal symmetry (T) and discrete rotation symmetry with order of n = 2, 4, and 6 (C ). The new anomalous states have n flavors of massless Dirac fermions protected by T and C , whereas any two-dimensional lattices having the two symmetries must have a multiple of 4, 8, and 12 Dirac cones for n = 2, 4, and 6, respectively. We show that these anomalous states are physically realized on the surface of new classes of topological crystalline insulators, normal to the rotation axis. Moreover, these topological crystalline insulators support n gapless one-dimensional helical mode on the otherwise fully gapped side surface, connecting the anomalous two-dimensional states on the top and bottom surfaces. The presence of these helical modes enables a new quantum device made from a topological crystalline insulator nanorod, a "helical nanorod," which has a quantized longitudinal conductance of ne ²/h.