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.

Figure 1

(a) A vortex and an antivortex. The direction of the arrow represents the phase of the Higgs field $⟨\tilde{b}⟩$. The dotted line is the mirror axis. (b) Illustration of the fermion spectrum flow from the left to the right as we create a vortex-antivortex pair from the vacuum and move them far apart. In this process, four vortex core states are separated from the bulk spectrum, two from the conducting band and two from the valence band, and become degenerate zero modes. As illustrated in the inset, the core states are twofold degenerate with spin $s=\pm 1$.

Figure 2

Two mirror-symmetric edges of a ${\mathbb{Z}}_2$ surface topological order. The mirror symmetry maps $x$ to $-x$ with respect to the mirror plane marked by the red disk at $x=0$. The surface topological order marked by the shade terminates at edges against two ordered regions with $⟨b⟩=\pm 1$, respectively.