# Higher-order Topology of Axion Insulator EuIn$_2$As$_2$

## TL;DR

This paper predicts EuIn$_2$As$_2$ as a high-order axion insulator with quantized topological magneto-electric effect, gapless surface states, and hinge states, based on first-principles calculations and symmetry analysis.

## Contribution

It introduces EuIn$_2$As$_2$ as a novel axion insulator with high-order topology and detailed surface and hinge state characterizations.

## Key findings

- EuIn$_2$As$_2$ has a $ heta=\pi$ topological magneto-electric effect.
- Gapless surface states are protected by mirror symmetries.
- Hinge states emerge as a hallmark of high-order topology.

## Abstract

Based on first-principles calculations and symmetry analysis, we propose that EuIn$_2$As$_2$ is a long awaited axion insulator with antiferromagnetic (AFM) long range order. Characterized by the parity-based invariant $\mathbb Z_4=2$, the topological magneto-electric effect is quantized with $\theta=\pi$ in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moment of the AFM phase is along $a/b$ axis, it's also a TCI phase. Gapless surface states emerge on (100), (010) and (001) surfaces, protected by mirror symmetries (nonzero mirror Chern numbers). When the magnetic moment is along $c$ axis, the (100) and (001) surfaces are gapped. As a consequence of a high-order topological insulator with $\mathbb Z_4=2$, the one-dimensional (1D) chiral state can exist on the hinge between those gapped surfaces. We have calculated both the topological surface states and hinge state in different phases of the system, respectively, which can be detected by ARPES or STM experiments.