Abstract:Année de publication : 2015
The critical points of statistical mechanical systems in 2 dimensions or quantum mechanical systems in 1+1 dimensions, as well as some aspects of non interacting systems in 2+1 dimensions, are eciently tackled by the methods of conformal eld theory (CFT) and integrability, which have witnessed spectacular progress during the past 40 years. Diculties are met in the non unitary case, frequent in statistical mechanics or condensed matter (as for instance in the Integer Quantum Hall Eect plateau transitions), where the CFTs are logarithmic, therefore very dicult to classify algebraically, and most often non compact, which has long been considered as an obstacle to their understanding through lattice discretizations. The connection between discrete models and non compact CFTs has indeed never been clear, in particular it was long believed that the latter could not arise as the continuum limit of `usual' spin chains or lattice models built out of a compact set of degrees of freedom. In this thesis, we show that the world of compact discrete models with a non compact continuum limit extends much further than the few examples known to this date and encompasses, in contrast with earlier expectations, cases of great physical relevance. We propose an exact solution of an innite family of models, the so-called a(2) n and b(1) n models, all of which are argued to allow for a non compact regime. Some of these models are related to problems of polymer physics, coupled Potts models and a tentative description of the quantum Hall plateau transition by some compact geometrical truncation. We show that the existence of an unsuspected non compact continuum limit for such models can have dramatic practical eects, for instance in the numerical determination of the critical exponents or in Monte-Carlo simulations. We use our results for a better understanding of the controversial theta transition describing the collapse of polymers in two dimensions, and draw perspectives on a possible understanding of the quantum Hall plateau transition from exactly solvable lattice models.