$N$-band Hopf insulator

We study the generalization of the three-dimensional two-band Hopf insulator to the case of many bands, where all the bands are separated from each other by band gaps. The obtained $\mathbb{Z}$ classification of such a $N$-band Hopf insulator and the formulation of its bulk-boundary correspondence are our main results. We find that the quantized boundary effect can only be measured in a non-equilibrium state.