Unconventional topological phase transition of the Hopf insulator

The topological phase transition between two band insulators is mediated by a gapless state whose low-energy band structure normally contains sufficient information for describing the topology change. In this work, we show that there is a class of topological insulators whose topological phase transition cannot be explained by this conventional paradigm. Taking the Hopf insulator as a representative example, we show that the change of the Hopf invariant requires the information of wave functions as well as the gapless band structure simultaneously. More explicitly, the description of the Hopf invariant change requires us to trace not only the trajectory of Weyl points but also the evolution of the preimages for two distinct eigenstates. We show that such an unconventional topological phase transition originates from the fact that the Hopf invariant is well-defined when all the lower dimensional topological invariants are trivial, which in turn allows us to lift the classifying space of occupied state projectors to the corresponding universal 2-covering space of wave functions. Generalizing our theory to inversion-symmetric 10-fold Altland-Zirnbauer symmetry classes, we provide a complete list of symmetry classes, all of which turn out to have delicate band topology related to the Hopf invariant, where similar unconventional topological phase transitions can appear.