$N$-band Hopf insulator

TL;DR

This paper generalizes the three-dimensional two-band Hopf insulator to many bands, revealing a $ ext{Z}$ classification linked to quantized magnetoelectric effects, and proposes an experiment to measure boundary phenomena.

Contribution

It introduces a $ ext{Z}$ classification for multi-band Hopf insulators and explores their boundary properties and experimental detection methods.

Findings

$ ext{Z}$ classification extends to multi-band Hopf insulators.

Bulk magnetoelectric coefficient is quantized and related to boundary Hall conductivity.

Proposes an experimental setup to measure boundary effects in non-equilibrium states.

Abstract

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 is related to the quantized isotropic magnetoelectric coefficient of its bulk. The boundary of a $N$-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 non-uniqueness, 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 non-equilibrium state.