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 twoand 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 Z2 index. These Hamiltonians can be viewed from the dimensional reduction of the high-dimensional quantum Hall states in threeand four-dimensional flat spaces. These states can be realized with ultracold fermions inside harmonic traps with synthetic gauge fields.