FeSe, the simplest iron-based superconductor, reveals a variety of puzzling properties and features that could hold a key for the pairing mechanism in this family of superconductors. In particular, it’s complex electronic band structure differs essentially from the DFT calculated one in a specially way called the “red-blue shift” and, in addition, drifts anomalously with temperature. Here we study this band structure evolution in Fe(Se,Te) crystals and reveal essential downdrift of all the hole-like bands with increasing temperature, which is opposite to the “red-blue shift” expectation. We show that this drift cannot be described by temperature-dependent contribution to quasiparticle self-energy within the Fermi-liquid concept but could result in charge redistribution between the bulk and topological surface states. If such a scenario is confirmed, one can tune the topologically non-trivial bands near the Fermi level with temperature, potentially allowing temperature-induced crossover between different Fermi surface topologies.

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