Hybrid-order topology in unconventional magnets of Eu-based Zintl compounds with surface-dependent quantum geometry

The exploration of magnetic topological insulators is instrumental in exploring axion electrodynamics and intriguing transport phenomena, such as the quantum anomalous Hall effect. Here, we report that a family of magnetic compounds Eu2n+1In2(As,Sb)2n+2 (n=0,1,2) exhibit both gapless Dirac surface states and chiral hinge modes. Such a hybrid-order topology hatches surface-dependent quantum geometry. By mapping the responses into real space, we demonstrate the existence of chiral hinge modes along the c direction, which originate from the half-quantized anomalous Hall effect on two gapped ac/bc facets due to Berry curvature, while the unpinned Dirac surface states on the gapless ab facet generate an intrinsic nonlinear anomalous Hall effect due to the quantum metric. When Eu3In2As4 is polarized to the ferromagnetic phase by an external magnetic field, it becomes an ideal Weyl semimetal with a single pair of type-I Weyl points and no extra Fermi pocket. Our work predicts rich topological states sensitive to magnetic structures, quantum geometry-induced transport, and topological superconductivity if proximitized with a superconductor.