Duality and integer quantum Hall effect in isotropic 3D crystals

We show here a series of energy gaps as in Hofstadter's butterfly, which have been shown to exist by Koshino et al [Phys. Rev. Lett. 86, 1062 (2001)] for anisotropic three-dimensional (3D) periodic systems in magnetic fields \Vec{B}, also arise in the isotropic case unless \Vec{B} points in high-symmetry directions. Accompanying integer quantum Hall conductivities $(\sigma_{xy}, \sigma_{yz}, \sigma_{zx})$ can, surprisingly, take values $\propto (1,0,0), (0,1,0), (0,0,1)$ even for a fixed direction of \Vec{B} unlike in the anisotropic case. We can intuitively explain the high-magnetic field spectra and the 3D QHE in terms of quantum mechanical hopping by introducing a ``duality'', which connects the 3D system in a strong \Vec{B} with another problem in a weak magnetic field $(\propto 1/B)$.Comment: 7 pages, 6 figure

Topological delocalization of two-dimensional massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which e.g., underlies the theoretical description of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing $g$. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of states of the massless Dirac Hamiltonian can be localized.Comment: 4 pages, 2 figure

Chiral orbital current and anomalous magnetic moment in gapped graphene

We present a low-energy effective-mass theory to describe chiral orbital current and anomalous magnetic moment in graphenes with band gap and related materials. We show that a Bloch electron generally contains an anomalous current density due to inter-band matrix elements, which describes a chiral current circulation associated with magnetic moment. In gapped graphenes, the chiral current is opposite between different valleys, and corresponding magnetic moment accounts for valley splitting of Landau levels. In gapped bilayer graphene, in particular, the valley-dependent magnetic moment is responsible for divergence of paramagnetic susceptibility at the band bottom, and full valley polarization is achieved in relatively small magnetic field range. The formulation also applies to the gapped surface states of three-dimensional topological insulator, where the anomalous current is related to the magneto-electric response in spatially-modulated potential.Comment: 10 pages, 3 figure

Quantum Hall effect and Landau level crossing of Dirac fermions in trilayer graphene

We investigate electronic transport in high mobility (\textgreater 100,000 cm$^2$/V$\cdot$s) trilayer graphene devices on hexagonal boron nitride, which enables the observation of Shubnikov-de Haas oscillations and an unconventional quantum Hall effect. The massless and massive characters of the TLG subbands lead to a set of Landau level crossings, whose magnetic field and filling factor coordinates enable the direct determination of the Slonczewski-Weiss-McClure (SWMcC) parameters used to describe the peculiar electronic structure of trilayer graphene. Moreover, at high magnetic fields, the degenerate crossing points split into manifolds indicating the existence of broken-symmetry quantum Hall states.Comment: Supplementary Information at http://jarilloherrero.mit.edu/wp-content/uploads/2011/04/Supplementary_Taychatanapat.pd

Phase Diagram for the Hofstadter butterfly and integer quantum Hall effect in three dimensions

We give a perspective on the Hofstadter butterfly (fractal energy spectrum in magnetic fields), which we have shown to arise specifically in three-dimensional(3D) systems in our previous work. (i) We first obtain the `phase diagram' on a parameter space of the transfer energies and the magnetic field for the appearance of Hofstadter's butterfly spectrum in anisotropic crystals in 3D. (ii) We show that the orientation of the external magnetic field can be arbitrary to have the 3D butterfly. (iii) We show that the butterfly is beyond the semiclassical description. (iv) The required magnetic field for a representative organic metal is estimated to be modest ($\sim 40$ T) if we adopt higher Landau levels for the butterfly. (v) We give a simpler way of deriving the topological invariants that represent the quantum Hall numbers (i.e., two Hall conductivity in 3D, $\sigma_{xy}, \sigma_{zx}$, in units of $e^2/h$).Comment: 8 pages, 8 figures, eps versions of the figures will be sent on request to [email protected]

Topological Phase Transition and Electrically Tunable Diamagnetism in Silicene

Silicene is a monolayer of silicon atoms forming a honeycomb lattice. The lattice is actually made of two sublattices with a tiny separation. Silicene is a topological insulator, which is characterized by a full insulating gap in the bulk and helical gapless edges. It undergoes a phase transition from a topological insulator to a band insulator by applying external electric field. Analyzing the spin Chern number based on the effective Dirac theory, we find their origin to be a pseudospin meron in the momentum space. The peudospin degree of freedom arises from the two-sublattice structure. Our analysis makes clear the mechanism how a phase transition occurs from a topological insulator to a band insulator under increasing electric field. We propose a method to determine the critical electric field with the aid of diamagnetism of silicene. Diamagnetism is tunable by the external electric field, and exhibits a singular behaviour at the critical electric field. Our result is important also from the viewpoint of cross correlation between electric field and magnetism. Our finding will be important for future electro-magnetic correlated devices.Comment: 4 pages,5 figure