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Quasicrystals allow for symmetries that are impossible in crystallinematerials, such as eight-fold rotational symmetry, enabling the existence ofnovel higher-order topological insulators in two dimensions without crystallinecounterparts. However, it remains an open question whether three-dimensionalhigher-order topological insulators and Weyl-like semimetals withoutcrystalline counterparts can exist. Here, we demonstrate the existence of asecond-order topological insulator by constructing and exploring athree-dimensional model Hamiltonian in a stack of Ammann-Beenker tilingquasicrystalline lattices. The topological phase has eight chiral hinge modesthat lead to quantized longitudinal conductances of $4 e^2/h$. We show that thetopological phase is characterized by the winding number of the quadrupolemoment. We further establish the existence of a second-order topologicalinsulator with time-reversal symmetry, characterized by a $\mathbb{Z}_2$topological invariant. Finally, we propose a model that exhibits a higher-orderWeyl-like semimetal phase, demonstrating both hinge and surface Fermi arcs. Ourfindings highlight that quasicrystals in three dimensions can give rise tohigher-order topological insulators and semimetal phases that are unattainablein crystals.

Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts