Hidden wallpaper fermion and third-order topological insulator

Abstract

Nonsymmorphic symmetry can induce exotic wallpaper fermions, e.g., hourglass fermion, fourfold-degenerate Dirac fermion, and Möbius 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 Möbius 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.

References in corpus (10)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Topological Crystalline Insulators
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Higher-order Topology of Axion Insulator EuIn${}_2$As${}_2$
• Realization of an acoustic third-order topological insulator
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Maximally Localized Wannier Functions within the FLAPW formalism