Topological Hopf-Chern insulators and the Hopf superconductor

We introduce new three-dimensional (3D) topological phases of two-band models using the Pontryagin-Thom construction. In symmetry class $A$ these are the infinitely many Hopf-Chern topological insulators, which are hybrids of layered Chern insulators and Hopf insulators. Being constructed by a modification of the experimentally observed Chern insulators, these provide promising candidates for the observation of a genuinely 3D topological phase in class $A$. In symmetry class $C$, there is a ${\mathbb{Z}}_2$ classification with the nontrivial topological phase, the Hopf superconductor, being realized by a construction that doubles a Hopf insulator in momentum space. For these new topological phases we introduce concrete tight-binding Hamiltonians and investigate their energy spectra in the presence of a boundary, revealing gapless surface modes in accordance with the bulk-boundary correspondence.

Figure 1

Ground-state Pontryagin manifolds of two phases in the sector $\mathbf{n}=(2,0,2)$, where $n_0\in {\mathbb{Z}}_4$. (a) Layered Chern insulator with $n_0=0$ realized by choosing hopping directions ${\mathbf{v}}_1=(1,0,-1)$ and ${\mathbf{v}}_2=(2,2,-2)$. (b) Corresponding Hopf-Chern insulator with Hamiltonian $\tilde{H}$ of Eq. (11). It has frame winding $n_0=2$, since its $gcd\left(\mathbf{n}\right)=2$ connected components have winding number 1 each.

Figure 2

Trivialization of a frame winding $n_0=2$ for $\mathbf{n}=(0,0,1)$, demonstrating that $n_0$ is only defined modulo $2gcd\left(\mathbf{n}\right)=2$. (a) Initial Pontryagin manifold. (b) Cobordism in Fig. 7 used. (c) Circle stretched using periodic boundary conditions. (d) Cobordism in Fig. 8 used. (e) Upper line moved using periodic boundary conditions. (f) Cobordism in Fig. 9 used. (g) Line “straightened”. (h) Framing “straightened”.

Figure 3

Comparison of phases $(0;0,0,1)$ (left column) and $(1;0,0,1)$ (right column) realized by the Hamiltonians in Eqs. (4) and (10), respectively. (a) Pontryagin manifolds of regular value $\mathbb{C}·\left(\begin{array}{c}0\\ 1\end{array}\right)$. (b) Energy spectra with boundary orthogonal to $(1,0,0)$ for a slab with 16 sites. Shown are the two energy eigenvalues closest to the Fermi energy $E=0$. For a boundary orthogonal to $(0,1,0)$ the spectra look identical with $k_2$ replaced by $k_1$. (c) Energy spectra with boundary orthogonal to $(0,0,1)$: gapped for the layered Chern insulator (left) and gapless with a ring of zero modes for the Hopf-Chern insulator (right).

Figure 4

Pontryagin manifold of the ground-state map ${\psi }_{\text{SC}}$ associated to $H_{\text{SC}}$ in Eq. (17). The choice of regular value is ${\psi }_{\text{SC}}(3\pi /4,0,0)$.

Figure 5

Triviality of superconductor with frame winding $\pm 2$. (a) Initial Pontryagin manifold with $m=1$ pair of connected components and frame winding ${\nu }_1=2$. (b) Deformation leading to the pinching off in (c) using the cobordism of Fig. 8. The resulting two pairs can be annihilated as indicated by the dashed arrows, avoiding the point $\mathbf{k}=0$ in the center.

Figure 6

Energy spectrum of the Hopf superconductor Hamiltonian in Eq. (17) in a 16-site thick slab geometry with boundary orthogonal to (a) the $(1,0,0)$ direction, (b) the $(0,1,0)$ direction, and (c) the $(0,0,1)$ direction. Shown are the two energy levels closest to the Fermi energy $E=0$. The gap closes in all three cases, creating the following Fermi surfaces of zero modes: (a) figure-eight pattern, (b) two Dirac points, and (c) two rings.

Figure 7

Contractible submanifolds with no frame windings are cobordant to the empty set.

Figure 8

Saddle cobordism.

Figure 9

Crossings can be undone via a cobordism at the expense of introducing a frame winding.