The recent discovery of a topological phase transition in IV-VI narrow-gap semiconductors has revitalized the decades-old interest in the bulk band inversion occurring in these materials. Here we systematically study the (001) surface states of Pb{1-x}Sn{x}Se mixed crystals by means of angle-resolved photoelectron spectroscopy in the parameter space 0 <= x <= 0.37 and 300 K >= T >= 9 K. Using the surface-state observations, we monitor directly the topological phase transition in this solid solution and gain valuable information on the evolution of the underlying fundamental band gap of the system. In contrast to common model expectations, the band-gap evolution appears to be nonlinear as a function of the studied parameters, resulting in the measuring of a discontinuous band inversion process. This finding signifies that the anticipated gapless bulk state is in fact not a stable configuration and that the topological phase transition therefore exhibits features akin to a first-order transition.
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