While physics is grappling with new problems dominated by strong interactions and (or) correlations, two main strategies are available to the theorists. The first consists in developing sophisticated approximations and numerical techniques for studying ab-initio models; the second in elaborating analytical methods to solve exactly the main models that capture the physics of interest. This latter approach is less general, but essential for our fundamental understanding, for developing controlled approximations, and to provide benchmarks for the numerical simulations. The purpose of this proposal is to obtain exact results for two types of problems. The first concerns phase transitions in 2+1 dimensional electrons gases in presence of disorder and mostly without interactions. A classical example is the transition between plateaux in the integer quantum Hall effect, where, despite of a wealth of numerical and experimental data, the values of the critical exponents are not known exactly. The study of the corresponding conformal field theories is made difficult by the non-unitarity and non-compactedness of the associated target space. In the last few years, steady progress has occurred on these questions. The understanding of sigma models on supergroups has improved thanks to developments around the AdS/CFT conjecture. Deep relations have been uncovered between the non-simplicity of lattice algebras and logarithmic properties of the associated conformal field theories. The formalism of Schramm-Loewner evolutions has transformed our understanding of fractal properties and geometrical phase transitions. We plan to build on this progress to further our understanding of critical properties in disordered electronic systems. In particular, we plan to understand better the role of continuous symmetries and indecomposability in logarithmic theories, the topology of renormalization group flows, and the probabilistic nature of wave functions and electronic trajectories. The second problem concerns out of equilibrium transport in nanosystems such as quantum dots. This topic, of crucial importance for applications and experiments, is even more challenging that the foregoing one. Indeed, the physical phenomena involved are not suited to the traditional methods of solid state physics; they often are non-perturbative (implying spin-charge separation, charge fractionalization and non Fermi liquid strong coupling fixed points), and are difficult to study numerically. Following works by some of us, and independently by the group of N. Andrei at Rutgers, it seems possible to find realistic models where transport can be studied exactly. We plan to obtain a complete solution for these models - from the I-V characteristic to the full counting statistics and the entanglement entropy. We plan moreover to use these solutions to obtain benchmarks for new numerical techniques using time-dependent density matrix renormalization group. Finally, we plan to explore more fundamental properties, such as renormalization and fixed points out of equilibrium, the integrability of the Keldysh formalism, and fluctuation-dissipation relations à la Gallavotti-Cohen. Finally, the two problems share common features, which we plan to study as well - from the role of non-compact target spaces in the description of transport to the reformulation of the replica method using contours `a la Keldysh'. This project brings together two groups (IPhT and LPTENS) whose expertises are complementary, and who have in the past accomplished important progress on related problems.

The gravitational wave radiation is one of the most important consequences of the Einstein’s theory of General Relativity. The direct detection of gravitationnal waves (GW) will not only give a strong confirmation of the theory but it will open a new window to observe the universe, complementary to the electromagnetic one. The first generation of gravitational wave detectors, such GEO, LIGO and Virgo are under operation and they have collected several years of data. Due to the small rate of the astrophysical events accessible with the current sensitivities, the detection is possible but not probable. A 2nd generation GW detectors (Advanced Virgo and Advanced LIGO), with an order of magnitude better sensitivity, is under preparation and it will probably be operated around 2015. Furthermore, the design study of a third generation detector, The Einstein Telescope, has been funded by the European Union and the project is part of the ASPERA roadmap. The thermal noise is one of the main noise sources of the detector. It is due to the Brownian vibrations of the materials of the mirrors and of the suspension wires and its reduction is the subject of this research project. The main technique to reduce the thermal noise is the use of materials with very high quality factors. The use of a cryogenic detector has also been proposed, but this option is considered not enough mature for advanced Virgo and maybe considered for a third generation detectors. Another way to decrease the effect of thermal noise is to reduce the coupling between the mirror thermal noise itself and the beam sensing the position of the mirror. This is naturally done by an increase the beam size with respect to the mirror dimensions, thus averaging the thermal noise on a bigger surface. The limitation of this technique relies, of course, in the mirror dimensions. The Gaussian beams, currently used in the interferometric detectors and in almost all the experiments which uses coherent source of electromagnetic radiation, concentrate almost all the power in a small region around the center (the beam radius), but with long tails which make necessary the use of mirrors much larger than the beam radius. To overcome to this problem the use of non gaussian beams has been proposed. The Laguerre-Gauss (LG) modes are a solution of the propagation equation of the electromagnetic waves, in the paraxial approximation. High-order LG modes have a ring structure allowing a more uniform power distribution on the mirror’s surface and then a reduction of the thermal noise. Furthermore, having spherical wavefronts, these beams can be used with spherical mirrors. Furthermore, due to their wider power distribution, the LG modes will also allow to decrease the mirrors deformation due to the power transferred from the beam to the mirror materials. This thermal lensing effect is one of the main problems in the current gravitational wave interferometers. In conclusion, the great interest of the LG modes are on their possibility to decrease the thermal noise and the thermal deformations of the mirrors with a minimum impact on the detector configuration, but only changing the shape of the beam before it enters in the interferometer. The goal of the LAGUERRE project is to demonstrate the feasibility of this technique. First, we plan to produce high-order Laguerre-Gaussian modes with high efficiency and purity. Then, we plan to build a table-top LG interferometer with a similar optical scheme of the GW detectors.