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

TL;DR

This paper demonstrates the theoretical existence of three-dimensional higher-order topological insulators and semimetals in quasicrystals, revealing phases unattainable in crystalline materials through model Hamiltonians and topological invariants.

Contribution

It constructs and analyzes 3D quasicrystal models showing higher-order topological phases and Weyl-like semimetals without crystalline counterparts, introducing new topological invariants.

Findings

Existence of 3D second-order topological insulator with eight hinge modes

Quantized conductance of 4 e^2/h in the topological phase

Proposal of a higher-order Weyl-like semimetal with Fermi arcs

Abstract

Quasicrystals allow for symmetries that are impossible in crystalline materials, such as eight-fold rotational symmetry, enabling the existence of novel higher-order topological insulators in two dimensions without crystalline counterparts. However, it remains an open question whether three-dimensional higher-order topological insulators and Weyl-like semimetals 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 quadrupole moment. We further establish the existence of a second-order topological insulator with time-reversal symmetry, characterized by a $\mathbb{Z}_2$ 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.