# Topological Crystalline Insulators with $C_2$ Rotation Anomaly

## TL;DR

This paper identifies a new class of topological crystalline insulators with $C_2$ rotation anomaly in Zintl compounds, featuring unique surface Dirac cones and hinge states, and demonstrates how strain can tune their topological phases.

## Contribution

It introduces a novel $C_2$ rotation anomaly in TCIs, develops a Wilson loop method to locate surface states, and shows strain-induced topological phase transitions.

## Key findings

- Discovery of $C_2$ rotation anomaly in specific Zintl compounds.
- Existence of surface Dirac cones protected by $C_2$ and time-reversal symmetry.
- Presence of helical hinge states along specific hinges.

## Abstract

Based on first-principles calculations and symmetry-based indicator analysis, we find a class of topological crystalline insulators (TCIs) with $C_2$ rotation anomaly in a family of Zintl compounds, including $\mathrm{Ba}_{3}\mathrm{Cd}_{2}\mathrm{As}_{4}$, $\mathrm{Ba}_{3}\mathrm{Zn}_{2}\mathrm{As}_{4}$ and $\mathrm{Ba}_{3}\mathrm{Cd}_{2}\mathrm{Sb}_{4}$. The nontrivial band topology protected by coexistence of $C_2$ rotation symmetry and time-reversal symmetry $T$ leads to two surface Dirac cones at generic momenta on both top and bottom surfaces perpendicular to the rotation axis. In addition, ($d-2$)-dimensional helical hinge states are also protected along the hinge formed by two side surfaces parallel with the rotation axis. We develop a method based on Wilson loop technique to prove the existence of these surface Dirac cones due to $C_2$ anomaly and precisely locate them as demonstrated in studying these TCIs. The helical hinge states are also calculated. Finally, we show that external strain can be used to tune topological phase transitions among TCIs, strong Z$_2$ topological insulators and trivial insulators.