Inversion-Asymmetric Chiral Hinge States Due to the Inversion-Z₄Topology

We study positions of chiral hinge states in higher-order topological insulators with inversion symmetry, and we find inversion-asymmetric hinge states, which are different from the conventional inversion-symmetric hinge states in the higher-order topological insulators. First, we find all possible configurations of the hinge states in the higher-order topological insulators in all type-I magnetic space groups with inversion symmetry. In particular, in a layered antiferromagnet, we find a difference in the hinge states between the cases with an even and odd number of layers. In the case of an even number of layers, which does not preserve inversion symmetry, positions of hinge states are not inversion symmetric. Nonetheless, these inversion-asymmetric hinge states result from the bulk topology. We show that their inversion-asymmetric configurations are uniquely determined from the symmetries and the topological invariant. In addition, we discuss the emergence of inversion-asymmetric hinge states in the axion insulator EuIn₂As₂.