Recent years have witnessed an increasing and crucial role of mathematics in condensed matter and many body theory.
While most mathematical results in the field apply only to non-interacting systems, this project focuses on the role of the interaction between particles, unavoidable in real systems and providing a higher level of complexity, which results in several of the most interesting macroscopic features.
We take advantage of recent developments in the theory of renormalization both in quantum field theory and in dynamical systems and probability. We will focus on transport, and in particular on two of the most remarkable macroscopic consequences of the quantum nature of the constituents of matter: first, the issue of universality of transport in topological insulators, such as graphene or semimetals; second, the problem of persistence of localization in interacting disordered systems (many-body localization).
The phenomenon of universality consists in the independence of transport coefficients from microscopic details, and appears in a class of materials, ranging from graphene to topological insulators, of crucial relevance for nanotechnologies. Topology has provided an elegant explanation of the phenomenon in the single-particle case, but a full understanding in the presence of interactions is still lacking. Recently, new approaches have been proposed, obtained by combining apparently unrelated tools and ideas, ranging from renormalization group methods, to quantum and statistical field theory techniques, to generalized central limits in probability; using these methods, we will investigate the basic mechanism of universality in the presence of many-body interactions.
The problem of localization of interacting quantum particles in the presence of quasi-periodic or random disorder (many-body localization) is a question that attracted a great interest for quantum computing as well as for foundational problems.
While the theory of single-particle quantum localization (Anderson localization) is mathematically well developed and advanced, the interplay between disorder and many-body interaction is largely unknown. Particularly interesting is the case of quasi-periodic disorder, characterized by the analogue of small divisors for dynamical systems and the necessity of number-theoretical properties.
Recent results suggest that methods coming from dynamical systems, such as quasi-integrability and Kolmogorov-Arnold-Moser theorem, combined with renormalization group techniques, offer the key for advance in the problem of many-body localization — we aim to pursue such ideas in this project.