Disorder and interactions in one dimensional Bose gas

One-dimensional quantum fluids are studied using the bosonization formalism combined with a non perturbative and functional implementation of the Wilson renormalization group (NPFRG). These tools are applied to the sine-Gordon model which describes the Mott-insulator phase, stabilized by a periodic potential. Our results are in good agreement with the exact solution for the mass of the first excited state and confirm the Lukyanov-Zamolodchikov conjecture on the amplitude of quantum fluctuations. Then, the effect of disorder is considered by coupling the NPFRG formalism to the replica trick. We are able to describe the localised and disordered phase, called Bose glass in bosonic systems. Our approach allows us to get the fixed point associated to the Bose-glass phase and shows the importance of metastable states which are responsible for the glassy properties. The chaotic behaviour of the Bose-glass phase, i.e. its extreme sensitivity to microscopic parameters, is revealed : two copies of the system with slightly different disorder configurations become uncorrelated at large distances. The universal features of the chaotic behaviour is also described. Finally, the effect of long-range interactions is studied. In the absence of disorder, these interactions favor crystalline order, and the system is a Wigner crystal. We show that the presence of disorder can stabilize a Mott glass, the existence of which was a subject of debate in one dimension. (author)