Visibility diagrams and experimental stripe structure in the quantum Hall effect

We analyze various properties of the visibility diagrams that can be used in the context of modular symmetries and confront them to some recent experimental developments in the Quantum Hall Effect. We show that a suitable physical interpretation of the visibility diagrams which permits one to describe successfully the observed architecture of the Quantum Hall states gives rise naturally to a stripe structure reproducing some of the experimental features that have been observed in the study of the quantum fluctuations of the Hall conductance. Furthermore, we exhibit new properties of the visibility diagrams stemming from the structure of subgroups of the full modular group.Comment: 8 pages in plain TeX, 7 figures in a single postscript fil

Modular Groups, Visibility Diagram and Quantum Hall Effect

We consider the action of the modular group $\Gamma (2)$ on the set of positive rational fractions. From this, we derive a model for a classification of fractional (as well as integer) Hall states which can be visualized on two ``visibility" diagrams, the first one being associated with even denominator fractions whereas the second one is linked to odd denominator fractions. We use this model to predict, among some interesting physical quantities, the relative ratios of the width of the different transversal resistivity plateaus. A numerical simulation of the tranversal resistivity plot based on this last prediction fits well with the present experimental data.Comment: 17 pages, plain TeX, 4 eps figures included (macro epsf.tex), 1 figure available from reques

Γ(2)\Gamma(2) modular symmetry, renormalization, group flow and the quantum Hall effect

We construct a family of holomorphic $\beta$-functions whose RG flow preserves the $\Gamma(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.Comment: 14 pages plain TeX, 7 figures in a postscript fil

The group Γ(2)\Gamma (2) and the fractional quantum hall effect

We analyze the action of the inhomogeneous modular group $\Gamma (2)$ on the three cusps of its principal fundamental domain in the Poincare half plane. From this, we obtain an exhaustive classification of the fractional quantum Hall numbers. This classification is somehow similar to the one given by Jain. We also present some resulting remarks concerning direct phase transitions between the different quantum Hall states.Comment: 11 pages Tex, 2 figures from reques

A history of astronomical optics in France

International audienceIn this paper, we explore the advances made in France in astronomical optics. They were quite important in the seventeenth century thanks to Mersenne, Auzout and Picard. Progress in the eighteenth century was less remarkable and many instruments had to be imported from England, but in the following century the developments were again significant thanks in particular to Arago, Foucault, Fizeau, Loewy and the Henry brothers, working with very competent instrument builders. The French school of optics flourished at the end of the nineteenth century and during the twentieth century, thanks to Fabry, his collaborators and successors, as illustrated by Perot, Chrétien, Lyot, Danjon, Paul, Couder, Courtès, Connes, Labeyrie, Baranne and Lemaitre, among others. However, from time to time there was a trend to build instruments that were too original and with little or no future, at the expense of more conventional but better-adapted instruments

A history of astronomical optics in France

International audienceIn this paper, we explore the advances made in France in astronomical optics. They were quite important in the seventeenth century thanks to Mersenne, Auzout and Picard. Progress in the eighteenth century was less remarkable and many instruments had to be imported from England, but in the following century the developments were again significant thanks in particular to Arago, Foucault, Fizeau, Loewy and the Henry brothers, working with very competent instrument builders. The French school of optics flourished at the end of the nineteenth century and during the twentieth century, thanks to Fabry, his collaborators and successors, as illustrated by Perot, Chrétien, Lyot, Danjon, Paul, Couder, Courtès, Connes, Labeyrie, Baranne and Lemaitre, among others. However, from time to time there was a trend to build instruments that were too original and with little or no future, at the expense of more conventional but better-adapted instruments

Interferometry and monochromatic imaging at Marseille Observatory

International audienceWe first give a brief history of the astronomical observatory in Marseilles, which was founded in 1702. Then, we describe the first attempt to measure in this Observatory the angular diameter of stars by interferometry, in 1873–1874. Because the size of the remarkable Foucault telescope that was used by édouard Stéphan for this program was only 80 cm, none of the bright stars was resolved, and the upper limit to their diameters was given as 1/6 of an arc second. This result was however a very significant advance, as only fancy figures had been given previously for stellar diameters. The next incursion of the Marseilles Observatory in interferometry took place in 1911–1914, when Charles Fabry and Henri Buisson measured with the same telescope the radial velocity and the temperature of the Orion Nebula, using the Pérot-Fabry interferometer developed at the Marseilles University. After WW2, the Observatory experienced a complete renewal. Then Georges Courtès used interference filters to obtain deep photographs of HII regions, and Pérot-Fabry interferometers for measuring their radial velocities. We describe the very important instrumental advances realized for this program, in particular the focal reducers that allowed a considerable increase in sensitivity. The final result obtained by Courtès and his collaborators was a complete H survey of the Milky Way, which was the basis for a new description of the structure of our Galaxy, with four spiral arms, and a detailed H survey of the Magellanic Clouds. The distribution of HII regions in the closest galaxies was also observed and their velocity field determined. In 1963, Courtès built the first integral field spectrograph, based on an array of micro-lenses; it had a great success, so that similar instruments are mounted at the focus of the largest present and future telescopes

A history of astronomical optics in France

International audienceIn this paper, we explore the advances made in France in astronomical optics. They were quite important in the seventeenth century thanks to Mersenne, Auzout and Picard. Progress in the eighteenth century was less remarkable and many instruments had to be imported from England, but in the following century the developments were again significant thanks in particular to Arago, Foucault, Fizeau, Loewy and the Henry brothers, working with very competent instrument builders. The French school of optics flourished at the end of the nineteenth century and during the twentieth century, thanks to Fabry, his collaborators and successors, as illustrated by Perot, Chrétien, Lyot, Danjon, Paul, Couder, Courtès, Connes, Labeyrie, Baranne and Lemaitre, among others. However, from time to time there was a trend to build instruments that were too original and with little or no future, at the expense of more conventional but better-adapted instruments