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While stable and fragile topological insulators (TIs) are blind to unoccupied states, defining a delicate TI depends critically on the conduction band. Here, the authors report a new delicate TI, dubbed a ``real Hopf insulator'' (RHI), in a spinless P0{0ex}T-symmetric fermion system in three dimensions. Having two topological invariants characterizing both conduction and valence bands, the RHI features a bulk-boundary correspondence and reverting Thouless pump that must be described by using both bands simultaneously. This sheds light on the generic properties of delicate TIs.
Lim et al. (Fri,) studied this question.