Evidence of Hofstadter's Fractal Energy Spectrum in the Quantized Hall Conductance

Laterally modulated 2D electron system in the extreme quantum limit

We report on magnetotransport of a two-dimensional electron system (2DES), located 32 nm below the surface, with a surface superlattice gate structure of periodicity 39 nm imposing a periodic modulation of its potential. For low Landau level fillings nu, the diagonal resistivity displays a rich pattern of fluctuations, even though the disorder dominates over the periodic modulation. Theoretical arguments based on the combined effects of the long-wavelength, strong disorder and the short-wavelength, weak periodic modulation present in the 2DES qualitatively explain the data.

Thermodynamics of critical strange nonchaotic attractors

The thermodynamic formalism is applied to dynamical attractors which have fractal geometry and on which all Lyapunov exponents are exactly zero. Such critical strange nonchaotic attractors (SNAs) which arise, for example, in the Harper system exhibit a static phase transition in the free energy. The Tsallis nonextensive entropy which is known to characterize the thermodynamics of systems with leading Lyapunov exponent zero is found to be subadditive for the critical states. These properties are shared by other quasiperiodic systems with critical SNAs.

Nonperturbative solution for BLOCH electrons in constant magnetic fields

A general theoretical approach for the nonperturbative Bloch solution of Schrödinger's equation in the presence of a constant magnetic field is presented. Using a singular gauge transformation based on a lattice of magnetic flux lines, an equivalent quantum system with a periodic vector potential is obtained. For rational magnetic fields this system forms a magnetic superlattice for which Bloch's theorem then applies. Extensions of the approach to particles with spin and many-body systems and connections to the theory of magnetic translation groups are discussed.

Deriving exact energy solutions to the symmetrized q-difference Harper equation

This paper deals with the application of the q calculus to second order q-difference equations, like the q symmetrized form of the Harper equation. One obtains three-term recurrence relations, for which a symmetrized version is written down. This opens the way to establish explicit energy results that are dependent on the commensurability parameter. The continuous realization of such energy results exhibits interesting patterns characterized by hierarchical structures.

Interplays between Harper and Mathieu equations

This paper deals with the application of relationships between Harper and Mathieu equations to the derivation of energy formulas. Establishing suitable matching conditions, one proceeds by inserting a concrete solution to the Mathieu equation into the Harper equation. For this purpose, one resorts to the nonlinear oscillations characterizing the Mathieu equation. This leads to the derivation of two kinds of energy formulas working in terms of cubic and quadratic algebraic equations, respectively. Combining such results yields quadratic equations to the energy description of the Harper equation, incorporating all parameters needed.

Critical states and fractal attractors in fractal tongues: localization in the Harper map

Localized states of Harper's equation correspond to strange nonchaotic attractors in the related Harper mapping. In parameter space, these fractal attractors with nonpositive Lyapunov exponents occur in fractally organized tongue-like regions which emanate from the Cantor set of eigenvalues on the critical line epsilon=1. A topological invariant characterizes wave functions corresponding to energies in the gaps in the spectrum. This permits a unique integer labeling of the gaps and also determines their scaling properties as a function of potential strength.