Two- and three-dimensional topological insulators with isotropic and parity-breaking Landau levels

We investigate topological insulating states in both two and three dimensions with the harmonic potential and strong spin-orbit couplings breaking the inversion symmetry. Landau-level-like quantization appear with the full two- and three-dimensional rotational symmetry and time-reversal symmetry. Inside each band, states are labeled by their angular momenta over which energy dispersions are strongly suppressed by spin-orbit coupling, to being nearly flat. The radial quantization generates energy gaps between neighboring bands at the order of the harmonic frequency. Helical edge or surface states appear on open boundaries characterized by the $Z_2$ index. These Hamiltonians can be viewed from the dimensional reduction of the high-dimensional quantum Hall states in three- and four-dimensional flat spaces. These states can be realized with ultracold fermions inside harmonic traps with synthetic gauge fields.

Figure 1

Energy dispersions of the solutions for the first four LLs to the 2D reduced Hamiltonian, Eq. (7) (solid lines), and those for Eq. (1) (dashed lines). The value of $\alpha =l_T/l_{\mathrm{so}}=35$. The lowest LLs of Eq. (7) are dispersionless with respect to $j_z$. Please note that the overall shift of the zero-point energy difference $\frac{1}{2}\hslash \omega$ is performed for the spectra of Eq. (1) for a better illustration.

Figure 2

Level diagram for atom-laser coupling. Four lower energy levels are coupled to two excited levels to compose a hybrid tripod and tetrapod configuration.

Figure 3

Energy levels of the atom-laser coupling Hamiltonian, Eq. (40).

Figure 4

Energy level scheme for alkali atoms ${}^{40}$K. The Zeeman sublevels of two hyperfine states $F=\frac{9}{2}$ and $F=\frac{7}{2}$ can be used to fulfill our requirements. Lines or curves with an arrow indicate effective transitions between different magnetic levels which can be implemented using resonant Raman processes. Other levels, which are not involved in the scheme, are not shown.