Hidden wallpaper fermion and third-order topological insulator

TL;DR

This paper reveals that symmorphic wallpaper groups can host exotic wallpaper fermions and uncovers the first real material example of a third-order topological insulator with unique corner states, expanding the understanding of topological phases.

Contribution

It extends the concept of wallpaper fermions to symmorphic groups and identifies a real material hosting third-order topological insulator states, previously thought exclusive to nonsymmorphic groups.

Findings

Discovery of wallpaper fermions in symmorphic groups

Identification of Tl4XTe3 as a third-order topological insulator

Presence of 16 helical corner states in the material

Abstract

Nonsymmorphic symmetry can induce exotic wallpaper fermions, e.g., hourglass fermion, fourfold-degenerate Dirac fermion, and M\"obius fermion, as commonly believed only in nonsymmorphic wallpaper groups. Here, we extend the notion of wallpaper fermions to symmorphic wallpaper groups, and remarkably uncover the emergence of long-awaited third-order topological insulators. The symmetry analysis and k $\cdot$ p models reveal that nonessential symmetries play an essential role for obtaining the previously overlooked hidden surface spectrum. Based on this, we present the hourglass fermion, fourfold-degenerate Dirac fermion, and M\"obius fermion in the (001) surface of Tl$_4$XTe$_3$ (X = Pb/Sn) with a symmorphic wallpaper group $p4m$. Remarkably, 16 helical corner states reside on eight corners in Kramers pair, rendering the first real electronic material of third-order topological insulator. A time-reversal polarized octupole polarization is defined to uncover the nontrivial third-order topology, as is implemented by the 2$^{nd}$ and 3$^{rd}$ order Wilson loop calculations. Our results could considerably broaden the range of wallpaper fermions and lay the foundation for future experimental investigations of third-order topological insulators.