Exact Random Walk Distributions using Noncommutative Geometry

Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $N$ on a two-dimensional square lattice for large $N$, taking into account finite size contributions.Comment: Latex, 3 pages, 1 figure, to be published in J. Phys. A : Math. Ge

Two interacting Hofstadter butterflies

The problem of two interacting particles in a quasiperiodic potential is addressed. Using analytical and numerical methods, we explore the spectral properties and eigenstates structure from the weak to the strong interaction case. More precisely, a semiclassical approach based on non commutative geometry techniques permits to understand the intricate structure of such a spectrum. An interaction induced localization effect is furthermore emphasized. We discuss the application of our results on a two-dimensional model of two particles in a uniform magnetic field with on-site interaction.Comment: revtex, 12 pages, 11 figure

Double butterfly spectrum for two interacting particles in the Harper model

We study the effect of interparticle interaction $U$ on the spectrum of the Harper model and show that it leads to a pure-point component arising from the multifractal spectrum of non interacting problem. Our numerical studies allow to understand the global structure of the spectrum. Analytical approach developed permits to understand the origin of localized states in the limit of strong interaction $U$ and fine spectral structure for small $U$.Comment: revtex, 4 pages, 5 figure

Semiclassical methods in solid state physics : two examples

We present here a review of two problems motivated by 2D models for high T, superconductivity. The first part concerns the energy spectrum of 2D Bloch electrons in a uniform magnetic field. A semiclassical analysis provides a qualitative as well as a quantitative understanding of this spectrum. In the second part we make the case for the application of “Quantum Chaos" to strongly correlated fermion systems. It is illustrated by the level spacing distribution for the $t - J$ model in two dimensions.Ce travail est une revue de deux problèmes motivés par l'étude des modèles bidimensionnels pour la supraconductivité à haute température critique. La première partie concerne l'étude du spectre d'énergie pour des électrons de Bloch bidimensionnels soumis à un champ magnétique uniforme. Une analyse semi-classique permet d'en comprendre les propriétés qualitatives et quantitatives. La deuxième partie est un plaidoyer pour l'utilisation des méthodes du “Chaos Quantique" dans l'étude des systèmes de fermions fortement corrélés. La distribution des écarts de niveaux d'un modèle $t - J$ en deux dimensions, en fournit une illustration

Braid-structure in a harper model as an example of phase space tunneling

We consider an electron on a square lattice in a uniform magnetic field. For small fields a sufficient large second neighbour coupling splits the lowest Landau levels into a braid like structure. Phase space tunneling in the spirit of Wilkinson's extension of the WKB approximation is verified numerically to explain the phenomenon. Furthermore the high fields effects are also discussed.Nous considérons un modèle d'électron sur réseau carré en champ magnétique uniforme. A la limite des champs faibles, une interaction aux seconds voisins suffisamment grande entraîne une dégénérescence des niveaux de Landau les plus bas en quatre sous-niveaux tressés. Un effet tunnel dans l'espace des phases, vu sous le jour de l'extension de la méthode BKW initiée par M. Wilkinson, explique numériquement ce phénomène. Les effets dus à des champs magnétiques plus élevés sont examinés