Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous ${\mathbb{Z}}_2$ topological order, where all anyons carry mirror-symmetry fractionalization $M^2=-1$. The identification of such anomalous crystalline symmetry fractionalization implies that in a two-dimensional ${\mathbb{Z}}_2$ spin liquid the vison excitation cannot carry $M^2=-1$ if the spinon carries $M^2=-1$ or a half-integer spin.