The first part of the project concerns the transport properties of interacting or disordered topological insulators and semimetals, while the second focuses on the derivation of effective theories for interacting Fermi gases in the mean-field regime or in the dilute regime.
About the first direction, the project combined rigorous renormalization group methods with other tools, to achieve new results in mathematical condensed matter physics. In particular, the combination with supersymmetry allowed to construct the scaling limit of a hierarchical model for weakly disordered three-dimensional semimetals, and to prove the irrelevance of disorder and the algebraic decay of correlations; the combination with Ward identities allowed to prove the universality of the lattice analogue of the chiral anomaly for interacting Weyl semimetals, and to compute the edge conductance for a generic class of interacting quantum Hall systems. This result provides a rigorous justification for the predictions about edge transport based on heuristic bosonization methods; moreover, it allows to extend the bulk-edge correspondence for quantum Hall systems from non-interacting models to the realm of weakly interacting systems. The approach has also been extended to the case of disordered edge modes, with quasi-periodic potentials. Furthermore, the project succeeded in justifying the expression of many transport coefficients used in physics, by proving the rigorous validity of Kubo formula for interacting, gapless quantum systems, at the basis of the widely used linear response approximation.
Concerning the second research direction, the project succeeded in characterizing the ground state properties of interacting fermionic systems in the mean-field regime beyond the Hartree-Fock approximation, by providing a rigorous, nonperturbative computation of the leading contribution to the correlation energy. The same techniques have then been used to derive a norm approximation for the many-body evolution of a class of fermionic states, in terms of the evolution of a suitable quasi-free Bose gas. The progress obtained in this second research direction is based on a rigorous bosonization approach to the study of excitations around the Fermi surface. Similar techniques have also been used to describe correlations in the somewhat opposite regime of dilute Fermi gases. Also, the project succeeded in deriving new analytic methods for the derivation of effective evolution equations for interacting fermions in the thermodynamic limit and at high density. Specifically, it allowed to obtain the first rigorous proof of the Hartree dynamics for extended Fermi gases.
The project opened new research directions beyond the foreseen ones, among which the rigorous analysis of the Z2 gauge theory coupled to fermionic matter in two dimensions, via the combination of reflection positivity, cluster expansion methods, quasi-adiabatic flow and renormalization group tools.
