Relativistic k.p Hamiltonians for centrosymmetric topological insulators from ab initio wave functions

We present a method to microscopically derive a small-size k$\cdot$p Hamiltonian in a Hilbert space spanned by physically chosen ab initio spinor wave functions. Without imposing any complementary symmetry constraints, our formalism equally treats three- and two-dimensional systems and simultaneously yields the Hamiltonian parameters and the true ${\mathbb{Z}}_2$ topological invariant. We consider bulk crystals and thin films of Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$. It turns out that the effective continuous k$\cdot$p models with open boundary conditions often incorrectly predict the topological character of thin films.