Nonunique connection between bulk topological invariants and surface physics

At the heart of the study of topological insulators lies a fundamental dichotomy: Topological invariants are defined in infinite systems but surface states as their main footprint only exist in finite systems. In the slab geometry, namely, infinite in two planar directions and finite in the perpendicular direction, the 2D topological invariant was shown to display three different types of behavior. The perpendicular Dirac velocity turns out to be a critical control parameter discerning between different qualitative situations. When it is zero, the three types of behavior extrapolate to the three 3D topologically distinct phases: trivial, weak, and strong topological insulators. We show analytically that the boundaries between types of behavior are topological phase transitions of particular significance since they allow us to predict the 3D topological invariants from finite-thickness transitions. When the perpendicular Dirac velocity is not zero, we identify a new phase with surface states but no band inversion at any finite thickness, disentangling these two concepts which are closely linked in 3D. We also show that at zero perpendicular Dirac velocity, the system is gapless in the 3D bulk and therefore not a topological insulating state, even though the slab geometry extrapolates to the 3D topological phases. Finally, in a parameter regime with strong dispersion perpendicular to the surface of the slab, we encounter the unusual case that the slab physics displays nontrivial phases with surface states but nevertheless extrapolates to a 3D trivial state.