Time-reversal invariant $\text{SU}\left(2\right)$ Hofstadter problem in three-dimensional lattices

We formulate the three-dimensional $\text{SU}\left(2\right)$ Landau level problem in cubic lattices with time-reversal invariance. By taking a Landau-type $\text{SU}\left(2\right)$ gauge, the system can be reduced into one dimension, as characterized by the $\text{SU}\left(2\right)$ generalization of the usual Harper equations with a periodic spin-dependent gauge potential. The surface spectra indicate the spatial separation of helical states with opposite eigenvalues of a lattice helicity operator. The band topology is investigated from both the analysis of the boundary helical Fermi surfaces and the calculation of the ${\mathbb{Z}}_2$ index based on the bulk wave functions. The transition between a three-dimensional weak topological insulator to a strong one is studied as varying the anisotropy of hopping parameters.

Figure 1

The configuration of Landau-type $\text{SU}\left(2\right)$ gauge fields $\vec{A}(n_x,n_y,n_z)$ defined on the bonds of the cubic lattice, which has coordinate dependence only on $n_z$.

Figure 2

The 2D surface Brillouin zone and the time-reversal invariant points $\mathrm{\Gamma }=(0,0), X_1=(\pi ,0), X_2=(0,\pi )$, and $M=(\pi ,\pi )$.

Figure 3

The bulk and surface spectra along the high-symmetry lines connecting time-reversal invariant points in the Brillouin zone. (a) $X_1-\mathrm{\Gamma }-X_1$, (b) $M-\mathrm{\Gamma }-M$, and (c) $M-X_2-M$, respectively. The red and blue lines represent the surface states with positive and negative helicities, respectively. Here, only surface modes localized at the lower boundary are plotted, and those of the upper surface are of the same values in the energy spectrum but with opposite lattice helicities. The following parameters are used: $t_x=t_y=0.5, t_z=1, p/q=2/7$, and $L_z=56$.

Figure 4

The boundary Fermi surface for states localized at the lower surface. Here, blue and red represent the negative and positive helicity states, respectively. (a) $E_F=-1.5t_{\parallel }$ lying in the second gap. Only one boundary helical Fermi surface appears. (b) $E_F=-0.47t_{\parallel }$ lying in the third gap. There are four boundary helical Fermi surfaces. Parameter values are $t_x=t_y=0.5, t_z=1, p/q=2/7$, and $L_z=56$.

Figure 5

The bulk and surface spectra along the time-reversal invariant paths of (a) $X_1-\mathrm{\Gamma }-X_1$, (b) $M-\mathrm{\Gamma }-M$, and (c)$X_2-\mathrm{\Gamma }-X_2$, respectively. The red and blue lines represent helical modes on the lower boundary with positive and negative helicities, respectively. The following parameters are used: $t_x=t_z=1, t_y=0.2, p/q=2/7$, and $L_z=56$.

Figure 6

The bulk and surface spectra along the time-reversal invariant paths. (a) $X_1-\mathrm{\Gamma }-X_1$, (b) $M-\mathrm{\Gamma }-M$, and (c)$X_2-\mathrm{\Gamma }-X_2$ for $t_x=t_z=1$ and $t_y=0.8$; (d) $X_2-\mathrm{\Gamma }-X_2$ for $t_x=t_z=1$ and $t_y=0.5$. The red and blue lines represent helical modes on the lower boundary with positive and negative helicities, respectively. The following parameters are used: $p/q=2/7$ and $L_z=28$.