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 ${\mathrm{Eu}}_{2n+1}{\mathrm{In}}_{2}$(As,${\mathrm{Sb})}_{2n+2}$ ($n=0,1,2$) 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 ${\mathrm{Eu}}_{3}{\mathrm{In}}_{2}{\mathrm{As}}_{4}$ 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.