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Recent acoustic and electrical-circuit experiments have reported the third-order (or octupole) topological insulating phase, while its counterpart in classical magnetic systems is yet to be realized. Here we explore the collective dynamics of magnetic vortices in three-dimensional breathing cuboids, and find that the vortex lattice can support zero-dimensional corner states, one-dimensional hinge states, two-dimensional surface states, and three-dimensional bulk states, when the ratio of alternating intralayer and interlayer bond lengths goes beyond a critical value. We show that only the corner states are stable against external frustrations because of the topological protection. Full micromagnetic simulations verify our theoretical predictions with good agreement.
We study the topological phase in dipolar-coupled two-dimensional breathing square lattice of magnetic vortices. By evaluating the quantized Chern number and $mathbb{Z}_{4}$ Berry phase, we obtain the phase diagram and identify that the second-order
A second-order topological insulator (SOTI) in $d$ spatial dimensions features topologically protected gapless states at its $(d-2)$-dimensional boundary at the intersection of two crystal faces, but is gapped otherwise. As a novel topological state,
The three dimensional (3D) topological insulators are predicted to exhibit a 3D Dirac semimetal state in critical regime of topological to trivial phase transition. Here we demonstrate the first experimental evidence of 3D Dirac semimetal state in to
From the analysis of the cyclotron resonance, we experimentally obtain the band structure of the three-dimensional topological insulator based on a HgTe thin film. Top gating was used to shift the Fermi level in the film, allowing us to detect separa
The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topologi