Higher-Order Topology of the Axion Insulator EuIn_{2}As_{2}.

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 Z_{4}=2, the topological magnetoelectric effect is quantized with θ=π in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moments of the AFM phase are along the a or b axis, it is also a topological crystalline insulator phase with gapless surface states emerging on (100), (010), and (001) surfaces. When the magnetic moments are along the c axis, both the (100) and (001) surfaces are gapped, and the material can also be viewed as a high-order topological insulator with one-dimensional chiral states existing on the hinges between those gapped surfaces. We have calculated both the topological surface states and the hinge state in different phases of the system, respectively, which can be detected by angle-resolved photoemission spectroscopy or STM experiments.