Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

TL;DR

This paper demonstrates that the surface of certain topological crystalline insulators can host anomalous symmetry fractionalization, revealing constraints on excitations in related two-dimensional quantum spin liquids.

Contribution

It identifies a specific anomalous mirror symmetry fractionalization on the surface of a topological crystalline insulator with mirror Chern number 4, linking surface phenomena to bulk topological properties.

Findings

Surface can host a $ ext{Z}_2$ topological order with all anyons carrying $M^2=-1$

Implication that visons in 2D spin liquids cannot carry $M^2=-1$ if spinons do

Shows a connection between surface anomalies and bulk topological invariants

Abstract

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.