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.
Highlights
Topological materials exhibit robust boundary effects that promise many applications
More energyefficient microelectronics can be designed by making use of backscattering-free edge modes, i.e., chiral modes appearing in the quantum Hall systems, and the surface states of three-dimensional Z2 topological insulators can serve as a good catalyst [1]
These robust boundary effects are guaranteed by the bulk-boundary correspondence [4,8,18] that holds for the tenfold-way topological classification
Summary
Topological materials exhibit robust boundary effects that promise many applications. Majorana zero-energy states can appear as corner states of either twoor three-dimensional systems These robust boundary effects are guaranteed by the bulk-boundary correspondence [4,8,18] that holds for the tenfold-way topological classification. The Hopf insulator can be turned into a stable topological phase through additional symmetry constraints [29], here we take a different route and relax the requirements of the delicate topological classification to allow for a trivial band to be added if separated by the gaps from all the other bands; see Fig. 1(d). We consider three-dimensional systems with no additional symmetry constraints, and find that a delicate multigap topological classification is the same as the classification of the Hopf insulator.
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