3D and 4D Topological Insulators - Coming Back to Landau Levels by Yi Li from UC San Diego

Wednesday, November 21, 2012 - 4:00pm to 5:00pm
Condensed Matter Physics Journal Club

The study of topological electron states started from the 2D quantum Hall effect under external magnetic fields, where the Landau level (LL) quantization arises from the cyclotron motion of electrons. However, the usual LLs crucially relies on the planar structure, and it is not obvious to identify its 3D counterparts. On the other hand, the current study of 3D topological insulators is largely defined on lattice systems with Bloch-wave bands. In this talk, we present simple continuous Hamiltonians generalizing LLs to the 3D flat space for both Schroedinger and Dirac fermions, which are not layered construction based on 2D LLs but are genuinely 3D with fully rotational symmetry in the "symmetric-like" gauge version or translational invariant along xy in the "Landau-like" gauge version. For the 3D LLs of spin-1/2 Schroedinger fermions, we employ the SU(2) Aharanov-Casher potential, which is equivalent to a 3D harmonic oscillator with spin-orbit coupling. Each filled LL gives rise to a 2D helical Dirac Fermi surface at open surface, which demonstrates its Z2 topological nature. For the 4D LLs, the integer quantized topological properties can be manifested explicitly from quantized non-linear response. The 3D Dirac LL Hamiltonian is a square-root problem of the above Schroedinger one, which can be viewed as the Dirac equation defined in the phase space. These 3D LLs can be systematically generalized to an arbitrary high dimension. Due to the elegant analytical properties of LLs and the flatness of LL spectra, we expect these high dimensional LLs will further facilitate the future study of high dimensional fractional topological states.