Higher-order topological insulators in three dimensions without crystalline counterparts

Quasicrystals allow for symmetries that are impossible in crystalline materials, such as eightfold rotational symmetry, enabling the existence of novel higher-order topological insulators in two dimensions without crystalline counterparts. However, the specific structure of the ${\mathbb{Z}}_{2}$ topological invariant in two dimensions makes it impossible to be generalized to the three-dimensional case. Consequently, it remains unclear whether three-dimensional higher-order topological insulators without crystalline counterparts can exist. Here, we demonstrate the existence of a second-order topological insulator by constructing and exploring a three-dimensional model Hamiltonian in a stack of Ammann-Beenker tiling quasicrystalline lattices. The topological phase has eight chiral hinge modes that lead to quantized longitudinal conductances of $4{e}^{2}/h$. We show that the topological phase is characterized by the winding number of the generalized quadrupole moment. We further establish the existence of a second-order topological insulator with time-reversal symmetry, characterized by a generalized ${\mathbb{Z}}_{2}$ topological invariant. Finally, we propose a model that exhibits a higher-order Weyl-like semimetal phase, demonstrating both hinge and surface Fermi arcs. Our findings highlight that quasicrystals in three dimensions can give rise to higher-order topological insulators and semimetal phases that are unattainable in crystals.