Dual topological insulator with mirror symmetry protected helical edge states

Dual topological insulators (DTIs) are simultaneously protected by time-reversal and crystal symmetries, representing advantageous alternatives to conventional topological insulators. By combining ab initio calculations and the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ approach, here, we investigate the electronic band structure of a ${\mathrm{Na}}_{2}\mathrm{CdSn}$ triatomic layer and derive a low-energy $4\ifmmode\times\else\texttimes\fi{}4$ effective model consistent with all the symmetries of this material class. We obtain the effective Hamiltonian using the L"owdin perturbation theory, the folding-down technique, and the theory of invariants and determine its parameters by fitting our analytical dispersion relations to those of ab initio calculations. We then calculate the bulk topological invariants of the system and show that the ${\mathrm{Na}}_{2}\mathrm{CdSn}$ triatomic layer is a giant-gap (hundreds of millielectronvolts) quasi-two-dimensional DTI characterized by both spin and mirror Chern numbers $\ensuremath{-}2$. In agreement with the bulk-boundary correspondence theorem, we find that a finite-width strip of ${\mathrm{Na}}_{2}\mathrm{CdSn}$ possesses two pairs of counterpropagating helical edge states per interface. We obtain analytical expressions for the edge state energy dispersions and wave functions, which are shown to agree with our numerical calculations. Our work opens an avenue for further studies of ${\mathrm{Na}}_{2}\mathrm{CdSn}$ as a potential DTI candidate with room-temperature applications in areas of technological interest, such as nanoelectronics and spintronics.