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 Z₂ topological order, where all anyons carry mirror-symmetry fractionalization M²=-1. The identification of such anomalous crystalline symmetry fractionalization implies that in a two-dimensional Z₂ spin liquid, the vison excitation cannot carry M²=-1 if the spinon carries M²=-1 or a half-integer spin.