Hopf insulators and their topologically protected surface states

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed $U\left(1\right)$ charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the $U\left(1\right)$ charge conservation that is implicit in all kinds of topological insulators.

Figure 1

Plot of the Hopf index and the Chern number in the $z$ direction for different $(p,q)$. The Hopf index and the Chern number converge rapidly as the number of grids increases in discretization. The parameters $t$ and $h$ are chosen as $(t,h)=(1,1.5).$

Figure 2

Phase diagrams of the Hamiltonian for different $(p,q)$. The values of $(p,q)$ in (a), (b), (c), and (d) are chosen to be $(1,1)$, $(1,2)$, $(3,1)$, and $(2,3)$, respectively.

Figure 3

Surface states and zero-energy modes in the $\left(001\right)$ direction for a 200-site-thick slab. The parameters $t$ and $h$ are chosen as $(t,h)=(1,1.5)$ for all the figures. We have $(p,q)=(1,2)$ for (a) and (b), and $(p,q)=(1,3)$ for (c) and (d). In panels (b) and (d), we add random perturbations to the Hamiltonian, but otherwise keep the same parameters as (a) and (c). The left diagrams in (a)–(d) plot the energy spectrum of all 400 states at a fixed $(k_x,k_y)=(0.72,0.72)$ for easy visualization. The points inside the gap represent the energies of the surface states. There are four [six] surface states in (a) and (b) [(c) and (d)]. The right diagrams in (a)–(d) show the wave functions of a surface state (upper one) and a bulk state (lower one).