学术活动发布与订制系统

The usual 2D Landau levels arise from the cyclotron motion of electrons in magnetic fields which crucially rely on the planar geometry. The complex analyticity of the lowest Landau level wavefunctions is essential for the study of fractional quantum Hall states. On the other hand, the current study of 3D topological insulators is largely confined to lattice systems. The complicated Bloch-wave wavefunctions and dispersive energy spectra are an obstacle for the study of fractional high dimensional topological states.

We would like to go back to Landau levels for high dimensional topological states because they are explicit and elegant. We identify their connections to quaternions which are the first non-commutative division algebra discovered by Hamilton in 1843 and whose analytic properties were developed by Feuter. Simple Hamiltonians are constructed in the continuum by coupling spin-1/2 fermions with the SU(2) Aharanov-Casher potential. They exhibit flat SU(2) Landau levels in which orbital angular momentum and spin are coupled with a fixed helicity. The lowest Landau level wavefunctions satisfy the Cauchy-Riemann-Fueter condition of quaternionic analyticity. Each Landau level contributes one branch of gapless helical Dirac modes to the surface spectra. These results are also generalized to Dirac electrons, which can be viewed as a quaternionic generalization of the 2D Dirac Landau level problem. The zeroth Landau levels of Dirac fermions are a branch of half-fermion Jackiw-Rebbi modes which are degenerate over all the high dimensional angular momentum quantum numbers. We have also studied the 4D quantum Hall effects of the SU(2) Landau levels in the Landau-type gauge, which exhibit quantized non-linear electromagnetic response as a spatially separated chiral anomaly. We expect that the quaternionic analytical properties of Landau levels and the spectra flatness will further facilitate the study of high dimensional fractional topological states.