Rotation Anomaly and Topological Crystalline Insulators

TL;DR

This paper demonstrates that certain rotation and time-reversal symmetries impose a multiple-of-$2n$ fermion flavor constraint in 2D lattices, revealing anomalies linked to novel topological crystalline insulators with characteristic surface states.

Contribution

It establishes a stronger fermion doubling theorem under rotation and time-reversal symmetries and connects violations to anomalies on topological crystalline insulator surfaces.

Findings

Number of fermion flavors must be a multiple of 2n for n-fold rotation symmetry.

Violations indicate anomalies and surface states with n Dirac cones.

Surface states feature n helical edge modes connecting top and bottom surfaces.

Abstract

We show that in the presence of $n$-fold rotation symmetries and time-reversal symmetry, the number of fermion flavors must be a multiple of $2n$ ($n=2,3,4,6$) on two-dimensional lattices, a stronger version of the well-known fermion doubling theorem in the presence of only time-reversal symmetry. The violation of the multiplication theorems indicates anomalies, and may only occur on the surface of new classes of topological crystalline insulators. Put on a cylinder, these states have $n$ Dirac cones on the top and on the bottom surfaces, connected by $n$ helical edge modes on the side surface.