Type-II quantum spin Hall insulator

Quantum spin Hall effect is usually realized in two-dimensional materials with time-reversal symmetry, but whether it can be realized without symmetry protection remains unexplored. Here, we propose type-II quantum spin Hall insulator with quantized spin Hall conductivity, whose edge states with opposite chirality and polarization, distributed in different Brillouin zone regions, connect the conduction and valence bands at the boundary. Thus, the type-II quantum spin Hall insulator does not require any symmetry protection other than translational symmetry. Then, based on symmetry analysis and the first-principles electronic structure calculations, we demonstrate that type-II quantum spin Hall insulator can be realized in both altermagnetic materials and Luttinger compensated magnetic materials. Furthermore, based on lattice model, we find that as long as $U(1)$ symmetry exists, type-II quantum spin Hall insulator phase can always exist stably. However, if $U(1)$ symmetry is broken, type-II quantum spin Hall insulator phase transforms into an obstructed atomic insulator phase as spin-orbit coupling effect is enhanced. Therefore, our work not only proposes a new mechanism for realizing the quantum spin Hall effect, but also enriches the types of unconventional magnetic topological phases.