Topological Insulators with SU(2) Landau Levels

We construct continuum models of 3D and 4D topological insulators by coupling spin-$\frac{1}{2}$ fermions to an SU(2) background gauge field, which is equivalent to a spatially dependent spin-orbit coupling. Higher dimensional generalizations of flat Landau levels are obtained in the Landau-like gauge. The 2D helical Dirac modes with opposite helicities and 3D Weyl modes with opposite chiralities are spatially separated along the third and fourth dimensions, respectively. Stable 2D helical Fermi surfaces and 3D chiral Fermi surfaces appear on open boundaries, respectively. The charge pumping in 4D Landau level systems shows quantized 4D quantum Hall effect.

Figure 1

3D and 4D LLs for $H_{LL}^{3\mathrm{D}}$ and $H_{LL}^{4\mathrm{D}}$ as spatially separated 2D helical SO plane wave modes localized along the $z$ axis (a), and 3D Weyl modes localized along the $u$ axis (b), respectively. Their central locations are $z_0({\overrightarrow{k}}_{2\mathrm{D}},\Sigma )=\Sigma l_{so}^2{k}_{2\mathrm{D}}$ and $u_0({\overrightarrow{k}}_{3\mathrm{D}},\Sigma )=\Sigma l_{so}^2{k}_{3\mathrm{D}}$, respectively. Note that 2D Dirac modes with opposite helicities and the 3D ones with opposite chiralities are located at opposite sides of $z=0$ and $u=0$ planes, respectively.

Figure 2

The central positions $u_0(m,k_z,\nu )$ of the 4D LLs in the presence of the magnetic field $\overrightarrow{B}=B\widehat{z}$. Only the branch of $m=0$ [shown by the red (solid) line] runs across the entire $u$ axis, which gives rise to quantized charge transport along $u$ axis in the presence of $\overrightarrow{E}\parallel \overrightarrow{B}$ as indicated in Eq. (12). This plot takes parameters $l_{so}=l_B$.