We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth Sobolev norms solutions. also exhibit explicit unbounded trajectories show that these are optimal.

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• Introduction • Strong Magnetic Fields (a) The Landau problem (b) The lowest Landau level (c) Field theory • String Theory and D-Branes (a) Noncommutative geometry in string theory (b) D-branes (c) String theory in magnetic fields • Noncommutative Quantum Field Theory (a) Fundamental aspects (b) UV/IR mixing (c) Gauge interactions (d) Violations of special relativity • The Fractional Quantum H...

The longitudinal resistivity at transitions between integer quantum Hall states in two-dimensional electrons confined to AlAs quantum wells is found to depend on the spin orientation of the partially filled Landau level in which the Fermi energy resides. The resistivity can be enhanced by an order of magnitude as the spin orientation of this energy level is aligned with the majority spin. We di...

We consider a clean two-dimensional electron liquid in a weak magnetic field where N@1 lower Landau levels are completely filled, while the upper level is only partially filled. Due to a screening by the lower Landau levels, the repulsive interaction between any two electrons at the upper level as a function of the separation between the guiding centers of their cyclotron orbits abruptly drops ...

We report the observation of an acute sensitivity of the anisotropic longitudinal resistivity of twodimensional electron systems in half-filled high Landau levels to the magnitude and orientation of an in-plane magnetic field. In the third and higher Landau levels, at filling fractions n 9 2, 11 2, etc., the in-plane field can lead to a striking interchange of the “hard” and “easy” transport di...

We have measured the quantum-Hall activation gaps in graphene at filling factors nu=2 and nu=6 for magnetic fields up to 32 T and temperatures from 4 to 300 K. The nu=6 gap can be described by thermal excitation to broadened Landau levels with a width of 400 K. In contrast, the gap measured at nu=2 is strongly temperature and field dependent and approaches the expected value for sharp Landau le...

We construct continuum models of 3D and 4D topological insulators by coupling spin-1/2 fermions to an SU(2) background gauge field, which is equivalent to a spatially dependent spin-orbit coupling. Higher dimensional generalizations of flat Landau levels are obtained in the Landau-like gauge. The 2D helical Dirac modes with opposite helicities and 3D Weyl modes with opposite chiralities are spa...

We show that the Landau levels cease to be eigenvalues if we perturb the 2D Schrödinger operator with constant magnetic field, by bounded electric potentials of fixed sign. We also show that, if the perturbation is not of fixed sign, then any Landau level may be an eigenvalue of the perturbed problem. AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20