AA 2024-25, Alessandro Olgiati, Massimo Moscolari

The course consists of two parts. The first part will be devoted to the analysis of one-particle Schrödinger operators modelling condensed matter systems with a particular emphasis on the Landau Hamiltonian as the main mathematical model for the Integer Quantum Hall Effect. The spectral and topological structure of the Landau levels will be described in detail. After a brief overview on linear response theory, the quantization of the Hall conductivity will be explained by using the concepts of Chern number and index of pair of projections. Finally, we will explain the phenomenon of bulk-edge correspondence.
In the second part we will switch to the Fractional Quantum Hall Effect. The Laughlin wave function, an ansatz for the ground state of the interacting many-body Landau Hamiltonian, will be introduced and discussed in detail. We will then address the mathematical derivation of the Laughlin wave function and the incompressibility of states describing its perturbations. If time allows we will discuss the ground state energy problem within such class of states and/or the connection with the emergence of anyonic quasi-particle excitations.