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 Z classification of such an N-band Hopf insulator is related to the quantized isotropic magnetoelectric coefficient of its bulk. The boundary of an N-band Hopf insulator can be fully gapped, and we find that there is no unique way of dividing a finite system into bulk and boundary. Despite this nonuniqueness, we find that the magnetoelectric coefficient of the bulk and the anomalous Hall conductivity of the boundary are quantized to the same integer value. We propose an experiment where the quantized boundary effect can be measured in a nonequilibrium state.

Automated summary

The study generalizes the three-dimensional two-band Hopf insulator to many bands, finding the Z classification of an N-band Hopf insulator related to the quantized isotropic magnetoelectric coefficient of its bulk. Despite the nonuniqueness in dividing a finite system into bulk and boundary, it is discovered that the magnetoelectric coefficient of the bulk and the anomalous Hall conductivity of the boundary are quantized to the same integer value. An experiment is proposed to measure the quantized boundary effect in a nonequilibrium state.