We formulate the tight-binding model for cubic $\alpha$-Sn based on the density functional theory calculations. In the model, we incorporate a variable bond angle, which allows us to simulate the effect of the in-plane strain. In the bulk, we demonstrate the presence of the ${\mathbb{Z}}_2$ topological invariant and a nonzero mirror Chern number, making $\alpha$-Sn one of the rare cases where dual topology can be observed. We calculate the topological phase diagram of multilayer $\alpha$-Sn as a function of strain and number of layers. We find that a nontrivial quantum spin Hall state appears only for compressive strain above five layers of thickness. Quite surprisingly, both in the trivial and nontrivial phases, we find a plethora of edge states with energies inside the bulk gap of the system. Some of these states are localized at the side surfaces of the slab and some of them prefer top and bottom surfaces. We find that they can be adiabatically connected to topological states in the chiral limit, where the model supports multiple one-dimensional winding numbers that take different values in different directions in the Brillouin zone. Finally, around high-symmetry points their presence can also be explained by the mirror Chern number and Wilson-loop invariants.

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• Band gap
• Edge states
• Local density of states
• Surface states
• Topological materials
• First-principles calculations
• Tight-binding model
• Wannier function methods