  Topological states of matter are characterized by global topological
invariants which change their value across a topological quantum phase
transition. It is commonly assumed that the transition between topologically
distinct noninteracting gapped phases of fermions is necessarily accompanied by
the closing of the band gap as long as the symmetries of the system are
maintained. We show that such a quantum phase transition is possible without
closing the gap in the case of a three-dimensional topological band insulator.
We demonstrate this by calculating the free energy of the minimal model for a
topological insulator, the Bernevig-Hughes-Zhang model, and show that as the
band curvature continuously varies, a jump between the band gap minima
corresponding to the topologically trivial and nontrivial insulators occurs.
Therefore, this first order phase transition is a generic feature of
three-dimensional topological band insulators. For a certain parameter range we
predict a re-entrant topological phase transition. We discuss our findings in
connection with the recent experimental observation of a discontinuous
topological phase transition in a family of topological crystalline insulators.
