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 Eu2⁢𝑛+1⁢In2(As,Sb)2⁢𝑛+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 𝑎⁢𝑐/𝑏⁢𝑐 facets due to Berry curvature, while the unpinned Dirac surface states on the gapless 𝑎⁢𝑏 facet generate an intrinsic nonlinear anomalous Hall effect due to the quantum metric. When Eu3⁢In2⁢As4 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.