Symmetry-protected topological Hopf insulator and its generalizations

We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry ${\mathcal{C}}^{\prime }$, a generalized particle-hole symmetry that gives the Hopf insulator a ${\mathbb{Z}}_2$ classification. The $4d$ analog of the Hopf insulator with symmetry ${\mathcal{C}}^{\prime }$ has the same ${\mathbb{Z}}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as “Chern $\mathrm{insulator}⋊S^1$” and “Chern $\mathrm{insulator}⋊T^2$” respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points toward a direction for its possible experimental realization.

Figure 1

An illustration of the Hopf map in the momentum space, which is equivalent to $T^3$. The domain wall $n_3=0$ forms a two-torus $T^2$, and $(n_1,n_2)$ winds around both directions of the torus. Thus intuitively the Hopf insulator can be viewed as Chern $\mathrm{insulator}⋊S^1$. The symbol $⋊$ represents the ‘winding of the Chern insulator along $S^1$.

Figure 2

The Fermi ring at the XY boundary of the minimal two-band model of the Hopf insulator for $m=2$ in Eq. (5).

Figure 3

The boundary zero-energy states of the $4d$ minimal two-band model for Hopf insulator with $m=2$ in Eq. (15), plotted in the $3d$ boundary Brillouin zone. Because this model can be heuristically viewed as Chern $\mathrm{insulator}⋊T^2$, its boundary has a torus of zero-energy states.