${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump

Abstract

We introduce a time-reversal-symmetric analog of the Hopf insulator that we call a spin Hopf insulator. The spin Hopf insulator harbors nontrivial Kane-Mele ${\mathbb{Z}}_2$ invariants on its surfaces, and is the first example of a nonmagnetic delicate topological insulator with spin-orbit coupling. We show that the Kane-Mele ${\mathbb{Z}}_2$ topology on the surface is generically unstable, but can be stabilized by the addition of a composition of the particle hole and spatial inversion symmetry. Such a symmetry not only protects the surface ${\mathbb{Z}}_2$ invariant, but also protects gapless helical hinge states on the spin Hopf insulator. Furthermore, we show that in the presence of four-fold rotational symmetry, the spin Hopf insulator exhibits a returning Thouless pump, as well as surface states on sharp boundary terminations.

9+4 pages, 6+1 figures

References in corpus (15)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Field Theory of Time-Reversal Invariant Insulators
• Dissipationless Quantum Spin Current at Room Temperature
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Topological Acoustics
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Topological acoustics
• Wannier representation of Z_2 topological insulators
• On the $Z_2$ classification of Quantum Spin Hall Models
• Topological surface states in three-dimensional magnetic insulators
• Tuning phase transition between quantum spin Hall and ordinary insulating phases
• Surface theorem for the Chern-Simons axion coupling
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Bulk-and-edge to corner correspondence

Cited by in corpus (11)

• Non-Abelian Hopf-Euler insulators
• Real Hopf Insulator
• Dipolar Weyl semimetals
• Berry-dipole Semimetals
• Scattering theory of delicate topological insulators
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Quantized crystalline-electromagnetic responses in insulators
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• PT-symmetric Non-Hermitian Hopf Metal
• Non-Hermitian Hopf insulators
• Correlated Hopf insulators