Hybrid-order topology with tunable chiral hinge modes and unpinned Dirac surface states in the altermagnetic insulator Eu$_{3}$In$_{2}$As$_{4}$

The exploration of magnetic topological states is instrumental in exploring axion electrodynamics and intriguing transport phenomena, such as the quantum anomalous Hall effect. Here, we predict that the recently-synthesized material Eu$_{3}$In$_{2}$As$_{4}$ exhibits as both an axion insulator and a 3D Stiefel-Whitney insulator with an altermagnetic order. When spins align in the $ab$ plane, we find an unpinned surface Dirac cone on the $ab$ plane and chiral hinge states along the $c$ direction, where hinge states can generate a half-quantized surface anomalous Hall effect on the $ac$ and $bc$ facets. When spins align along $c$, we observe a mirror-protected topological crystalline insulator. Furthermore, the ferromagnetic phase, in which spins are aligned in the same direction by an external in-plane magnetic field, presents an ideal Weyl semimetal with a single pair of type-I Weyl points and no extra Fermi pocket. Our work predicts rich topological states tuned by magnetic structures in Eu$_{3}$In$_{2}$As$_{4}$, supporting the further study of the topological transport and Majorana fermions in proximity to a superconductor.