Quantum spin Hall effect in monolayer and bilayer TaIrTe$_{4}$

Generally, stacking two quantum spin Hall insulators gives rise to a trivial insulator. Here, based on first-principles electronic structure calculations, we confirm that monolayer ${\mathrm{TaIrTe}}_{4}$ is a quantum spin Hall insulator and remarkably find that bilayer ${\mathrm{TaIrTe}}_{4}$ is still a quantum spin Hall insulator. Theoretical analysis indicates that the covalentlike interlayer interaction in combination with the small band gap at the time-reversal invariant $\mathrm{\ensuremath{\Gamma}}$ point results in new band inversion in bilayer ${\mathrm{TaIrTe}}_{4}$, namely, the emergence of quantum spin Hall phase. Meanwhile, a topological phase transition can be observed by increasing the interlayer distance in bilayer ${\mathrm{TaIrTe}}_{4}$. Considering that bulk ${\mathrm{TaIrTe}}_{4}$ is a type-II Weyl semimetal, layered ${\mathrm{TaIrTe}}_{4}$ thus provides an ideal platform to realize different topological phases at different dimensions.