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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.