DESCRIPTION OF THE PROJECT:

The proposed research aims at investigating several problems arising in the study of quantum interacting particles and (classical and quantum) spin systems, with the goal of enhancing the comprehension of phenomena and techniques that are still far from being fully understood, even from a heuristic point of view. E.g., non-Fermi liquid phases in interacting Fermi systems, graphene physics, quantum Hall effect, stripes and periodic patterns formation in dipolar systems, Bose-Einstein condensation in homogeneous Bose gases, the bosonization procedure and the use of conformal field theories in the study of 2D non-integrable critical spin systems.

Moreover, the project aims at developing new methods based on the combination of the techniques that are currently applied in the rigorous mathematical analysis of quantum and classical phase transitions, such as constructive renormalization group, functional inequalities, localization bounds, reflection positivity. At present these approaches are somewhat disjoint and their specific applicability, which is still the subject of intense research, is often restricted to distinct problems and different ranges of parameters. However, these methods share common underlying features, such as the identification of relevant length scales, the averaging over unimportant degrees of freedom and the reduction to simpler effective theories, either by means of symmetry and positivity, or by means of functional estimates, or by means of rigorous perturbative arguments. Presumably, future progress on the mathematical theory of quantum and classical many body systems will result from a deeper understanding of the relations and common features shared by these methods.

The research will develop along four parallel and interconnected lines:

low temperature properties of interacting Fermi systems;

spontaneous formation of periodic patterns in systems with competing interactions;

study of spin-spin correlations and related critical indices in 2D critical spin models;

low temperature properties of homogeneous Bose gases.