Symmetry tuning topological states of the axion insulator candidate EuIn$_2$As$_2$ by its noncollinear magnetic order
Topological properties of quantum materials are intimately related to symmetry. Here, we tune the magnetic order of the axion insulator candidate EuIn$_2$As$_2$ from its broken-helix ground state to the field-polarized phase by applying an in-plane magnetic field. Using results from neutron diffraction and magnetization measurements with ab inito theory and symmetry analysis, we assume a commensurate magnetic ground state and predict how the field tunes the magnetic symmetry within individual magnetic domains and examine the resulting changes to the topological surface states and hinge states existing on edges shared by certain surfaces hosting gapped Dirac states. We predict complex field-tunable domain-specific hinge-state patterns, with some crystal surfaces undergoing a field induced topological phase transition. We further find that domain walls can have pinned hinge states when intersecting certain crystal surfaces, providing another channel for tuning the chiral-charge-transport pathways.