Real Hopf insulator

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.